Thinking Better: The Art of the Shortcut
Date: 2021-08-18 23:00:00+00:00
Keywords: kindle; nonfiction
Publisher: HarperCollins Publishers
Rights: All rights reserved

THINKING BETTER

The Art of the Shortcut

Marcus du Sautoy

Copyright

4th Estate

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This eBook first published in Great Britain by 4th Estate in 2021

Copyright © Marcus du Sautoy

Cover design by Steve Leard

Marcus du Sautoy asserts the moral right to be identified as the author of this work

A catalogue record for this book is available from the British Library

Figures in the book redrawn by Martin Brown, from files generated by the author or those in the public domain. Figure 5.1 reproduced courtesy of Andrew Viner. Figure 5.5 created by Joaquim Alves Gaspar. Figure 5.14 taken from https://commons.wikimedia.org/wiki/File:Doughnut_(economic_model).jpg, uploaded by DoughnutEconomics, published under Creative Commons Licence. Figure 9.5 created by Arpad Horvath.

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Source ISBN: 9780008393915

Ebook Edition © August 2021 ISBN: 9780008393939

Version: 2021-07-05

Dedication

To all mathematics teachers
but especially Mr Bailson who showed me
my first mathematical shortcut

Contents

Cover

Title Page

Copyright

Dedication

Departure

1 The Pattern Shortcut

Pitstop: Music

2 The Calculated Shortcut

Pitstop: Start-up

3 The Language Shortcut

Pitstop: Memory

4 The Geometric Shortcut

Pitstop: Journeys

5 The Diagram Shortcut

Pitstop: Economics

6 The Differential Shortcut

Pitstop: Art

7 The Data Shortcut

Pitstop: Therapy

8 The Probable Shortcut

Pitstop: Finance

9 The Network Shortcut

Pitstop: Neuroscience

10 The Impossible Shortcut

Arrival

Acknowledgements

Index

About the Author

Also by Marcus du Sautoy

About the Publisher

Departure

You have a choice. The obvious path is a long slog, with no beautiful vistas on the way. It is going to take you for ever and sap all your energy but it will at least eventually get you to your destination. But there is a second path. You’ve got to be sharp to spot it veering off the main track, seemingly taking you away from your goal. Then you spot the signpost that says ‘shortcut’. This is promising a quicker, off-road route that will get you to your destination faster and with minimal energy. There might even be the chance of a stunning view on the way. It’s just that you are going to have to keep your wits about you to navigate this path. It’s your choice. This book is pointing you towards that second path. It’s your shortcut to the better thinking you’ll need to negotiate this unorthodox route and to get you to where you want to go.

It was the lure of the shortcut that made me want to become a mathematician. As a rather lazy teenager, I was always after the most efficient path to my destination. It’s not that I was after cutting corners. I just wanted to achieve my goal with as little effort as possible. So when my mathematics teacher revealed to me at the age of twelve that the subject we were learning in school was really a celebration of the shortcut, my ears pricked up. It started with a simple story, featuring a nine-year-old boy called Carl Friedrich Gauss. The year was 1786 and our teacher transported us back to the boy’s classroom in the town of Brunswick near Hanover where the young Gauss grew up. It was a small place and the local school had only one teacher, Herr Büttner, who had to somehow teach the town’s one hundred children in just one classroom.

My own teacher, Mr Bailson, was a rather dour Scot who kept strict discipline, but it sounded like he was a softy compared to Herr Büttner. Gauss’s teacher would stride up and down the benches brandishing a cane to maintain discipline among the rowdy class. The classroom itself, which I’ve subsequently visited on a mathematical pilgrimage, was a drab room with a low ceiling, little light and uneven floors. It felt like a medieval prison and Büttner’s regime seems to have matched the setting.

The story goes that during one arithmetic lesson Büttner decided to set the class a rather tedious task to occupy them so that he could grab forty winks. ‘Class … I want you to add up the numbers from 1 to 100 on your slates,’ instructed Büttner. ‘As soon as you are done, bring your slates to the front of the class and place them on my desk.’

Before the teacher had even finished the sentence, Gauss was on his feet and had placed his slate on the desk declaring in low German: ‘Ligget se.’ There it is. Büttner looked at the boy, shocked at his impertinence. The cane in Büttner’s hand quivered but he decided to wait until all the students had submitted their slates for inspection before upbraiding the young Gauss. Eventually the rest of the class finished and Büttner’s desk was a tower of slates covered in chalk and calculations. The teacher began to work his way through the pile, starting with the last slate that had been placed on top. Most of the calculations were wrong, the students having invariably made some arithmetic slip on the way.

Eventually Büttner arrived at Gauss’s slate. He was bracing himself for a rant at the young upstart but as he turned over the slate he saw the correct answer: 5050. With no extraneous calculation. Büttner was shocked. How had this young boy found the answer so quickly?

The story goes that the precocious young student had spotted a shortcut that helped him avoid the hard work of actually doing much arithmetic. What he had realised was that if you add up the numbers in pairs:

1 + 100

2 + 99

3 + 98

…

the answer was always 101. But there were 50 pairs. Hence the solution was

50 × 101 = 5050

I remember being electrified by this story. To see Gauss’s insight into how to shortcut all this horribly tedious and labour-intensive work was a revelation.

Although the story of Gauss’s schoolroom shortcut is probably more legend than fact, it nonetheless captures beautifully an important point: mathematics is not a subject of tedious calculation as so many think, but rather one of strategic thinking.

‘That, my dear students, is mathematics,’ announced my teacher. ‘The art of the shortcut.’

Hello, my twelve-year-old self thought … tell me more!

Getting further faster

Humans use shortcuts all the time. We have to. We have a short amount of time to make decisions. We have limited mental capacity to navigate complex problems. One of the first strategies that we developed as a pathway to solving complicated challenges was the idea of heuristics: the process by which we make problems less complex by ignoring some of the information that’s coming into the brain, either consciously or unconsciously.

The trouble is that most of the heuristics that humans use lead to bad judgements and biased decisions, and generally aren’t fit for purpose. We might know one thing from experience and then try to extrapolate all of our other problems by comparing them to this one thing we know. We judge the global by knowledge of the local. This was fine when our environment didn’t extend too much beyond the small region of savannah we inhabited. But as our neighbourhood expanded, these heuristics didn’t give us good ways of understanding how things worked beyond our local knowledge. This was the moment when we started to develop better shortcuts. Those tools are what today we call mathematics.

To find good shortcuts requires an ability to lift yourself out of the geography you’re trying to traverse. If you are in the landscape then often you can only rely on what you see around you. Although each step feels like it is taking you in the right direction, the path that results might take you the long way round to your destination or lead you astray completely. That’s why humans developed a better way of thinking: the ability to take ourselves out of the minutiae of the task at hand and understand that there might be an unexpected path that could get us to our target more efficiently and faster.

This is what Gauss did with the challenge his teacher set the class. While the other students started plodding from one number to the next, adding each new one they encountered to the tally, Gauss surveyed the problem in its totality, understanding how to use the beginning and the end of the journey to his advantage.

Mathematics is all about this ability to substitute higher-level thinking that sees structure where before we just saw random meandering pathways. To lift us up and out of the landscape so that we can look down from a great height to see the true lie of the land. Shortcuts emerge when problems are mapped in this way. And once we started to exploit the ability to see structure in our mind’s eye without physically encountering it, this capacity for abstract thinking unleashed human civilisation’s extraordinary advances over the centuries.

The journey to better thinking began 5000 years ago around the Nile and the Euphrates. Humans wanted to find cleverer ways of building the city states that were blossoming alongside these rivers. How many blocks of stone would be needed to build a pyramid? What area of land had to be cultivated with crops to feed a city? What changes in river height were indicators of a forthcoming flood? Those who had the tools to find shortcuts to solving these challenges were those who rose to prominence in these emerging societies. The success of mathematics as a shortcut to the rapid development of these civilisations launched the subject as a powerful tool for those wishing to get further faster.

Time and again a gear change in civilisation was effected by the discovery of new mathematics. The explosion of mathematics during the Renaissance and beyond, which gave us tools like calculus, gave scientists extraordinary shortcuts to efficient engineering solutions. And today mathematics is behind all the algorithms that are being implemented on our computers to assist us through the digital jungle, literally shortcutting our way to the best routes to our destinations, the best websites for our internet searches and even the best partners for a journey through life.

It is interesting to note, however, that humans weren’t the first to exploit the power of mathematics to access the best way to tackle a challenge. Nature was using mathematical shortcuts to solve problems long before we arrived. Many of the laws of physics are based on nature always finding the shortcut. Light travels along the path that gets it to its destination fastest, even if that involves bending round a large object like the Sun. Soap films create the shapes that cost the lowest amount of energy: the bubble makes a sphere because this symmetrical shape is the one with the smallest surface area and therefore the minimum energy. Bees make hexagonal hives because the hexagon uses the least amount of wax to contain a fixed area. Our bodies have found the most energy-efficient way of walking to transport us from A to B.

Nature is lazy, like humans, and wants to find the low-energy solution. As the eighteenth-century mathematician Pierre Louis Maupertuis wrote: ‘Nature is thrifty in all its actions.’ It is extremely good at sniffing out the shortcut. Invariably that has a mathematical rationale to it. And often the discoveries of shortcuts by humans materialise out of our observations of how nature has solved a problem.

The journey ahead

What I want to do in this book is to share with you the arsenal of shortcuts that mathematicians like Carl Friedrich Gauss have developed over the centuries. Each chapter will introduce a different sort of shortcut with its own particular flavour. But all of them have the aim of transforming you from being someone who has to slog through the hard work of solving a problem to someone who can place their slate down with the answer before everyone else.

I have chosen to take Gauss as a companion on our journey. His classroom success launched him on a career that for me marks him out as the prince of the shortcut. Indeed the plethora of breakthroughs that he made during his lifetime span many of the different shortcuts that I will introduce in the book.

By telling the stories of the shortcuts that mathematicians have amassed over the years, I hope this book will act as a toolkit for all those who want to save time doing one thing so that they can spend more time doing something more exciting. Very often these shortcuts are transferable to problems that at first sight don’t seem mathematical in nature. Yet mathematics is a mindset for navigating a complex world and finding the pathway to the other side.

This is why mathematics really deserves its place as a core subject in the educational curriculum. Not because it is absolutely essential that the population know how to solve a quadratic equation. Frankly, when has anyone ever needed to know that! The essential skill in solving such a problem is understanding the power that algebra and algorithms hold.

I will begin the journey to better thinking with one of the most powerful shortcuts mathematicians have developed: patterns. A pattern is often the best sort of shortcut. See the pattern and you’ve found the shortcut to continuing the data into the future. This ability to identify an underlying rule is the basis of mathematical modelling.

Quite often the role of the shortcut is to understand the founding principle that unites a whole slew of seemingly unrelated problems. The beauty of Gauss’s shortcut is that even if the teacher tries to make it harder by asking you to add the numbers up to a thousand or a million, the shortcut still works. While adding the numbers up one by one would get increasingly more time consuming, Gauss’s trick is unaffected: to add the numbers up to a million, just pair them up again to get 500,000 pairs that each add up to 1,000,001. Multiply these two together and bingo: you’ve got your answer. Like the tunnel that provides a shortcut through a mountain: the road is unaffected even if the mountain was somehow to get taller.

The power of creating and changing language turns out to be a very effective shortcut too. Algebra helps us to recognise the underlying principles behind a whole range of different-looking problems. The language of coordinates turns geometry into numbers and frequently reveals shortcuts that were not visible in the geometric setting. Creating language can be an amazing tool for understanding. I remember wrestling with an extraordinarily complex set-up that needed many conditions to pin it down. The liberating effect of my doctoral supervisor saying ‘give it a name’ was a revelation. It truly allowed me to shortcut thought.

Whenever I mention the idea of the shortcut, people invariably think that I am trying to cheat somehow. The word ‘cut’ sounds like you could be cutting corners, so it’s important right from the outset to distinguish between shortcuts and cutting corners. I’m after the clever path to get to the correct solution. I’m not interested in finding some shoddy approximation to the answer. I want complete understanding but without unnecessary hard work.

That said, some shortcuts are about approximations that are good enough to solve the problem at hand. In some sense, language itself is a shortcut. The word ‘chair’ is a shortcut to a whole host of different sorts of things that we can sit on, for example. But it is not efficient to come up with a different word for every distinct instance of an example of a chair. Language is a very clever low-dimensional representation of the world around us that allows us to communicate with others efficiently and facilitates our path through the multifaceted world we live in. Without the shortcut of single words for multiple instances, we would be overwhelmed by noise.

In mathematics too I will reveal how throwing away information is often essential to finding a shortcut. The idea of topology is geometry without measurement. If you are on the London Underground, a map that shows how stations are connected is much more useful for finding your way around the city than a geographically accurate map. Diagrams can also be a powerful shortcut. Again the best of them discard anything that is extraneous to navigating the problem at hand. But as I shall illustrate, there is often a fine line between a good shortcut and the danger of cutting a corner.

Calculus is one of humankind’s greatest inventions to find shortcuts. Many engineers depend on this bit of mathematical magic to find the optimal solution to an engineering challenge. Probability and statistics have been a shortcut to knowing a lot about a huge data set. And mathematics can often help you to find the most efficient path through a complex geometry or tangled network. One of the staggering revelations I had as I fell in love with mathematics was its ability to even find shortcuts to navigate the infinite. A shortcut to get from one end of an endless path to the other.

Each chapter begins not with an epigram but a puzzle. Often these puzzles involve a choice: the long slog or, if you can find it, the shortcut. Each has a solution that takes advantage of the shortcut that is at the heart of that puzzle’s chapter. They are worth tackling before the shortcut is revealed as often the more time you spend battling to get to your destination the more you appreciate the shortcut when it is finally revealed.

What I have also discovered on my own journey is that there are different sorts of shortcut. It is therefore about highlighting the multiple approaches that you can take to the journey you are about to embark on and that you will get to your destination faster by using the shortcut. There are shortcuts waiting there in the terrain to take advantage of. It’s just that you might need a signpost to point you in the right direction or a map to show you the way. There are shortcuts that don’t exist without you doing a lot of hard work to carve them out: the tunnel that takes years to dig but once there allows everyone else to follow you through to the other side. There is the shortcut that requires totally escaping the space you are in: the worm hole from one side of the universe to the other. There is the extra dimension that shows how two things are much closer than you imagined, provided you can step out of the confines of the current world. There are shortcuts that speed things up, shortcuts that cut down the distance you need to travel, or the amount of energy expended. There is a saving somewhere that is worth the time to find the shortcut.

But I’ve also recognised that there are times when the shortcut misses the point. Maybe you want to be taking your time. Maybe the journey is the thing. Maybe you want to expend energy in an attempt to lose weight. Why go on a walk in nature for the day if you will curtail the pleasure by taking a shortcut home? Why read a novel rather than a synopsis on Wikipedia? But even then, it’s still good to know you’ve got the option of a shortcut even if you decide to ignore it.

To some extent the shortcut is about our relationship to time. What do you want to spend your time doing? Sometimes it is important to experience something in time and there is little value in finding a shortcut that cheats you of the feeling. Listening to a piece of music can’t be short-circuited. But on other occasions life is too short to spend time getting to where you want to be. A film can condense a life into ninety minutes. You don’t want to witness every action of the character you are following. Taking a flight to the other side of the world is a shortcut to walking there and means that you can begin your holiday sooner. If you could shorten the flight even more, most would. But there are times when people want to experience the slow version of getting to their destination. Pilgrimage abhors the shortcut. I never watch film trailers as they shortcut the film too much. Yet it is still worth having the choice.

Shortcuts in literature are invariably paths that lead to disaster. Little Red Riding Hood would never have met the wolf if she hadn’t strayed from the path in search of a shortcut through the wood. In John Bunyan’s The Pilgrim’s Progress, those who take a shortcut round Difficulty Hill get lost and perish. In The Lord of the Rings Pippin warns that ‘Short cuts make long delays’, although Frodo counters that inns make even longer ones. Homer Simpson swears after his disastrous detour on the way to Itchy and Scratchy Land, ‘Let us never speak of the shortcut again.’ The dangers inherent in taking shortcuts are well summed up in the film Road Trip: ‘Of course it’s difficult, it’s a shortcut. If it was easy it would just be “the way”.’ This book looks to rescue the idea of a shortcut from these literary tropes. Rather than the road to disaster, the shortcut is the road to freedom.

Human versus machine

One of the things that sparked my desire to write this book celebrating the art of the shortcut is the increasing perception that the human race is about to be superseded by a new species that doesn’t need to bother with shortcuts.

We are now living in a world where computers are able to do more computations in an afternoon than I can do in a lifetime. Computers can analyse the entirety of the world’s literature in the time that it takes me to read one novel. They are able to analyse so many variations of a game of chess compared to the few moves that I can hold in my head. Computers can explore the contours and pathways covering Planet Earth faster than it takes me to walk to the corner shop.

Would a computer, today, come up with Gauss’s shortcut? Why would it bother, when it can add up the numbers from 1 to 100 in the nth of an nth of a blink of an eye?

What hope is there for the human race to keep apace of the extraordinary speed and near infinite memory of our silicon neighbours? The computer in the 2013 film Her declares to its human owner that the pace of human interaction is so slow that it prefers to spend time with other operating systems that can match its speed of thought. The computer considers humans the way we might a mountain as it slowly emerges or erodes.

But maybe there is something that gives the human race the edge. The limitations of our brain to perform millions of computations simultaneously, the physical shortcomings of our body compared with the strength of a mechanical robot, these will force humans to stop and think whether there is a way to avoid all the steps that a computer or robot finds trivial.

Faced with a seemingly unassailable mountain, humans will instead seek out the shortcut. Rather than trying to go over the top, is there perhaps a sneaky way around? And often it is the shortcut that leads to a truly innovative way to solve a problem. While the computer ploughs on, flexing its digital muscles, the human sneaks to the finishing line thanks to revealing the cunning shortcut that avoided all the hard grafting.

Slackers take note. I think laziness is our saving grace against the onslaught of the machine. Human laziness is a really important part of finding good, new ways to do things. I often look at something and think: This is just getting too complicated – let me try to step back and figure out a shortcut. We know what the computer will say: ‘Well, I’ve got these tools and I can just bash on, deep into the problem.’ But because it doesn’t get tired and it’s not going to be lazy, maybe it will miss things that our laziness will take us to. Because we don’t have the ability to storm deep into things, we’re forced to find clever ways to do them.

There are many stories of innovation and progress coming from laziness and a desire to avoid hard work. Scientific discovery has often emerged from a mind left idling. The German chemist August Kekulé is said to have come up with the ring structure of benzene after falling asleep and dreaming of a snake swallowing its own tail. The great Indian mathematician Srinivasa Ramanujan talked often of his family goddess Namagiri writing equations in his dreams. As he wrote: ‘I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.’ A new invention has often been born of someone who couldn’t be bothered to do things the hard way. Jack Welch, as chairman and CEO of General Electric, spent an hour each day in what he called ‘looking out of the window time’.

Laziness doesn’t mean that you do nothing. And this is a really important point. Finding shortcuts often requires hard slog. It’s something of a paradox. Strangely, although the motivation to find the shortcut might come from the desire to avoid work, it can frequently lead to intense, explosive periods of deep thought not only to avoid that boring work but to cope with the boredom that idleness instils. There is a fine line between idleness and boredom and this is often the catalyst for the hunt for the shortcut that can then involve a great amount of labour. As Oscar Wilde wrote, ‘to do nothing at all is the most difficult thing in the world, the most difficult and the most intellectual’.

Doing nothing is often a precursor to great mental progress. A paper published in 2012 in the journal Perspectives on Psychological Science entitled ‘Rest Is Not Idleness’ revealed how important our default mode of neural processing is to cognitive ability. This mode is often suppressed when our attention is too focused on the outside world. The recent surge of mindfulness promotes stilling the mind from invasive thoughts as the pathway to enlightenment. Often it means you prefer to play rather than work. But play can be the place to foster creativity and new ideas, rather than the dull mechanistic world of work. It is one of the reasons that the offices of start-ups and maths departments are habitually filled with pool tables and board games as much as they are desks and computers.

Perhaps society’s disapproval of laziness is a way of controlling and curtailing those who prefer not to conform. The real reason the lazy person is regarded with suspicion is that it is the mark of someone who is not prepared to play by the rules of the game. Carl Friedrich Gauss’s teacher saw his pupil’s shortcut to doing hard work as a threat to his authority.

Idleness has not always been shunned. Samuel Johnson very eloquently argued in favour of laziness: ‘The Idler not only escapes labours which are often fruitless, but sometimes succeeds better than those who despise all that is within their reach.’ As Agatha Christie admitted in her autobiography, ‘Invention arises directly from idleness, possibly also from laziness. To save oneself trouble.’ Babe Ruth, one of the biggest hitters of home runs baseball has ever seen, was apparently motivated to hit the ball out of the stadium because he hated having to run between bases.

Choosing to work

I do not wish to imply that all work is bad. Indeed many people get great value out of the work they do. It defines their identity. It gives them purpose. But the quality of the work is important. Generally this is not tedious mindless work. Aristotle distinguished between two different sorts of work: praxis, which is action done for its own sake, and poiesis, or activity aimed at the production of something useful. We are happy to look for shortcuts in the case of the second sort of work but there seems little point chasing the shortcut if the pleasure is actually in doing the work for its own sake. Most work seems to fall into the second category. Yet surely the ideal is to aspire to work of the first kind. That is where the shortcut aims to take you. The shortcut is not after eliminating work, it wants to lead you on a path to meaningful work.

The goal of the new political movement fully automated luxury communism is that the advances of AI and robotics will mean that menial work is taken away from humans and done by machines, leaving time for us to indulge in work we find meaningful. Work becomes a luxury. The cultivation of good shortcuts should be added to the list of technologies steering us towards a future of work that is undertaken for the joy of it rather than as a means to an end. This was Karl Marx’s aim with communism: to remove the difference between leisure and work. ‘In a higher phase of communist society,’ he wrote. ‘labour has become not only a means of life but life’s prime want.’ The shortcuts we have created promise to take us away from what Marx called the ‘realm of necessity’ and lead us instead into the ‘realm of freedom’.

But are there some places where you can’t get away from hard work? How can a lazy person learn a musical instrument? Write a novel? Climb Everest? Even here I shall illustrate how the combination of putting in the hours at your desk or in training and a good shortcut can maximise the value of the time you put in. The book is punctuated by conversations I’ve had with other practitioners to see if shortcuts are possible in their professions or if you just can’t avoid the 10,000 hours that the writer Malcolm Gladwell talks about to get to the top.

I have been intrigued to see whether practitioners in other professions use any shortcuts that resonate with those I’ve learned in my own subject of mathematics. Or whether there might be new sorts of shortcuts of which I’ve not been aware that could prompt new modes of thinking in my own work. But I’m also fascinated by those challenges in which no shortcuts are possible. What is it about certain domains of human activity that preclude the power of the shortcut? Time and again it turns out that the human body is the limiting factor. To change or train or push the body to do new things quite frequently takes time and repetition, and there are no shortcuts to speed up those physical transformations. As I take you on the journey through the different shortcuts that mathematicians have discovered, each chapter is also interspersed with a pitstop to explore the shortcuts, or lack of them, in other fields of human activity.

Gauss’s schoolroom success at adding the numbers from 1 to 100 using his cunning shortcut fuelled his desire to pursue his mathematical talent. His teacher Herr Büttner wasn’t up to the task of cultivating the budding young mathematician but Büttner had an assistant, seventeen-year-old Johann Martin Bartels, who was equally passionate about mathematics. Although he’d been employed to cut quill pens for the students and assist them in their first attempts at writing, Bartels was more than happy to share his mathematical texts with the young Gauss. Together they explored the mathematical terrain, enjoying the shortcuts that algebra and analysis provided to reach their destinations.

Bartels soon realised that Gauss needed a more challenging environment to test his skills. He managed to get Gauss an audience with the Duke of Brunswick. The duke was so taken with the young Gauss that he agreed to become his patron, funding his education at the local college and then at the University of Göttingen. It was here that Gauss began to learn some of the great shortcuts that mathematicians had developed over the centuries which would soon become the springboard for his own exciting contributions to mathematics.

This book is my curated guide through 2000 years of better thinking. It has taken me decades to learn how to navigate these cunning tunnels or hidden passes through the landscape, and it took mathematicians through history thousands of years to piece them together. But in this book I’ve tried to distil some of these clever strategies for attacking the complex problems we encounter in everyday life. This is your shortcut to the art of the shortcut.

1

The Pattern Shortcut

You have a flight of stairs in your house with 10 steps. You can take one or two steps at a time. For example, you could do 10 one-steps to get to the top or 5 two-steps. Or combinations of one-steps or two-steps. How many different possible combinations are there to get to the top? You could do this the long way and try to find all the combinations, running up and down the stairs. But how would our young Gauss do it?

Want to know a shortcut to getting an extra 15 per cent salary for doing exactly the same work? Or perhaps a shortcut to growing a small investment into a large nest-egg? How about a shortcut to understanding where a stock price might be heading in the coming months? Do you feel like you are sometimes reinventing the wheel again and again yet sense there is something that connects all these different wheels you are making? What about a shortcut to help you with your terrible memory?

I’m going to dive in and share with you one of the most potent shortcuts that humans have discovered. It is the power of spotting a pattern. The ability of the human mind to glean a pattern in the chaos around us has provided our species with the most amazing shortcut: knowing the future before it becomes the present. If you can spot a pattern in data describing the past and the present then by extending that pattern further you have the chance to know the future. No need to wait. The power of the pattern is, for me, the heart of mathematics and its most effective shortcut.

Patterns allow us to see that even though the numbers might be different, the rule for how they grow can be the same. Spotting the rule underlying the pattern means that I don’t have to do the same work every time I encounter a new set of data. The pattern does the work for me.

Economics is full of data with patterns that if read properly can guide us to a prosperous future. Although as I shall explain, some patterns can be misleading, as the world witnessed in the financial crash of 2008. Patterns in the number of those falling ill with a virus mean we can understand the trajectory of a pandemic and intervene before it kills too many people. Patterns in the cosmos allow us to understand our past and our future. Looking at the numbers that describe the way stars are moving away from us has revealed a pattern that tells us our universe began in a big bang and will end with a cold future called the heat death of the universe.

It was this ability to sniff out the pattern in astronomical data that launched the aspiring young Gauss onto the world stage as the master of the shortcut.

Planetary patterns

On New Year’s Day, 1801, an eighth planet was detected orbiting around the Sun somewhere between Mars and Jupiter. Christened Ceres, its discovery was regarded by everyone as a great omen for the future of science at the beginning of the nineteenth century.

But excitement turned to despair a few weeks later, when the small planet (which was in fact just a tiny asteroid) disappeared from view near the Sun, lost among a plethora of stars. The astronomers had no idea where it had gone.

Then news arrived that a twenty-four-year-old from Brunswick had announced that he knew where to find this missing planet. He told the astronomers where to point their telescopes. And lo, as if by magic, there was Ceres. The young man was none other than my hero Carl Friedrich Gauss.

Since his classroom successes aged nine, Gauss had gone on to make numerous fascinating mathematical breakthroughs, including the discovery of a way to construct a 17-sided figure using only a straight edge and a compass. This challenge had been outstanding for 2000 years ever since the ancient Greeks had started finding clever ways to draw geometric shapes. He was so proud of this feat that he started a mathematical diary, filling it over the ensuing years with his amazing discoveries about numbers and geometry. But it was the data from this new planet that fascinated Gauss. Was there a way to find some rationale in the readings that were taken before Ceres disappeared behind the Sun that would reveal where to find it? Eventually he’d cracked the secret.

Of course, his great act of astronomical prediction was not magic. It was mathematics. The astronomers had discovered Ceres by chance. Gauss used mathematical analysis to work out the underlying pattern behind the numbers describing the asteroid’s location to know what it would do next. He was not the first to spot patterns in the dynamic cosmos, of course. Astronomers have been using this shortcut for navigating the changing night sky to make predictions and plan the future ever since our species understood that the future and the past were connected.

Patterns in the seasons meant that farmers could plan when to plant crops. Each season was matched to a particular configuration of stars. Patterns in the behaviour of animals migrating and mating allowed early humans to hunt at the most opportune moment, expending the least amount of energy for the most gain. Being able to predict eclipses elevated the status of the predictor to important member of the tribe. Indeed Christopher Columbus famously exploited his knowledge of an imminent lunar eclipse to save his crew when they were captured by local inhabitants after he had become stranded on Jamaica in 1503. The locals were so awestruck by his ability to predict the disappearance of the Moon that they acquiesced to his demands for freedom.

What’s the next number?

The challenge of looking for patterns is perfectly encapsulated in those problems you probably had at school where you were given a sequence of numbers and instructed to determine the next in the sequence. I used to love the challenges our teacher would chalk up on the blackboard. The longer it took me to spot the pattern the more rewarding was the experience of uncovering the shortcut. This is a lesson I learned early on. The best shortcuts often take a long time to uncover. They take work. But once revealed they become part of your repertoire of ways of seeing the world and can be tapped into again and again.

To get your pattern shortcut neurons firing, here are a few challenges. What is the next number in this sequence?

1, 3, 6, 10, 15, 21 …

Not too difficult. You probably spotted that you are just adding another number on each time. So 28 is the next number because it’s 21 + 7. These are called the triangular numbers because they represent the number of stones you need to build a triangle, adding on another row each time. But is there a shortcut to finding the 100th number on this list without having to work your way through all the preceding 99? This is in fact the challenge that Gauss was faced with when his teacher gave him the task of adding up the numbers from 1 to 100. Gauss found the clever shortcut of adding the numbers up in pairs to get the answer. More generally, if you want the nth triangular number, Gauss’s trick translates into the formula:

½ × n × (n + 1)

These triangular numbers continued to fascinate Gauss after he’d first encountered them in Herr Büttner’s class. Indeed one of the entries in his mathematical diary on 10 July 1796 declares excitedly in Greek ‘Eureka!’ followed by the formula:

num = Δ + Δ + Δ

Gauss had discovered the rather extraordinary fact that every number can be written as three triangular numbers added together. For example, 1796 = 10 + 561 + 1225. This kind of observation can lead to powerful shortcuts because rather than proving that something is true for all numbers, it might be enough to prove it for triangular numbers and then exploit Gauss’s discovery that every number is the sum of three triangular numbers.

Here’s another challenge. What’s the next number in this sequence:

1, 2, 4, 8, 16 …

Not too tricky. It’s 32 next. This sequence is doubling each time. Called exponential growth, this controls the way a lot of things can grow and it’s important to understand how this kind of pattern evolves. For example, the sequence looks quite innocent to start with. That’s certainly what the Indian king thought when he agreed to pay the creator of the game of chess the price he demanded for his game. The inventor had asked for a single grain of rice to be placed on the first square of the chessboard and then to double the number of grains of rice on each subsequent square on the board. The first row looked quite innocent. Only a total of 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255 grains of rice. Barely enough for a piece of sushi.

But as the king’s servants added more and more rice to the board, they very quickly ran out of supplies. To get to the halfway point needs about 280,000 kilograms of rice. And that’s the easy half of the board. How many grains of rice does the king need in total to pay the inventor? This is looking like one of those problems Herr Büttner might have given his poor students. There is the hard way to do it: add up the 64 different numbers. Who wants to do such difficult work? How might Gauss have gone about this sort of challenge?

There is a beautiful shortcut to making this calculation but at first sight it looks like I’m making life harder. Shortcuts often begin by seeming to head in the opposite direction from your destination. First I’m going to give the total grains of rice a name: X. It’s one of our favourite names in mathematics, and is in itself a powerful shortcut in the mathematician’s arsenal, as I shall explain in Chapter 3.

I am going to kick off by doubling the amount that I am trying to work out:

2 × (1 + 2 + 4 + 8 + 16 + … + 2^62 + 2^63)

This looks like it has made life more difficult. But stick with me. Let’s multiply this out:

= 2 + 4 + 8 + 16 + 32 + … + 2^63 + 2^64

Now comes the smart bit. I am going to take X away from this. At first it looks like I’ve just got us back to where we started: 2X − X = X. So how does that help? A bit of magic happens when I replace 2X and X with the sums I’ve got:

2X − X = (2 + 4 + 8 + 16 + 32 + … + 2^63 + 2^64) − (1 + 2 + 4 + 8 + 16 + … + 2^62 + 2^63)

Most of these terms cancel! There is just the 2^64 in the first part and 1 in the second part that don’t get cancelled. So all I am left with is

X = 2X − X = 2^64 −1

Instead of lots of calculating all I need to do is this one calculation to discover that the number of grains of rice that the king needed in total to pay the inventor of chess is:

18,446,744,073,709,551,615

That’s more rice than has been produced on our planet in the last millennium. The message here is that sometimes you can play hard work off against hard work and be left with something that is much simpler to analyse.

As the king learned to his cost, doubling starts off looking innocent and then ramps up very quickly. This is the power of exponential growth. The effect is felt by those who take out loans to cover debt. At first sight the offer from a company of a £1000 loan at 5 per cent interest each month might be a life saver. After one month you only owe £1050. But the trouble is that each month this gets multiplied by 1.05 again. After two years you already owe £3225. By the fifth year, the debt would be £18,679. Great for the person who’s lending money to you but not so great for the borrower.

The fact that people in general don’t understand this pattern of exponential growth means that it can be a shortcut to penury. Payday loan companies have successfully exploited this inability to read the pattern into the future to suck vulnerable punters into a contract that initially looks quite attractive. The dangers of doubling and the path that it takes us down are important to know before we find ourselves lost and helpless with no way back to safety.

We all learned the frightening rate of exponential growth to our cost too late with the coronavirus pandemic of 2020. The number of people infected doubled every three days on average. And this resulted in healthcare systems being overwhelmed.

On the other hand the power of the exponential can also help to explain why there are (probably) no vampires on Earth. Vampires need to feed on the blood of a human being at least once a month to survive. The trouble is that once you have feasted on the human, the victim becomes a vampire too. So next month there are twice as many vampires in the search for human blood to feast on.

The world’s population is estimated to be 6.7 billion. Each month the population of vampires doubles. Such is the devastating effect of doubling that within 33 months a single vampire would end up transforming the world’s population into vampires.

Just in case you ever meet a vampire, here is a useful trick from the mathematician’s arsenal to ward off the blood-sucking monster. In addition to the classic use of garlic, mirrors and crosses, one rather unusual way to fend off the Prince of Darkness is to scatter poppy seeds around his coffin. It turns out that vampires suffer from a condition called arithmomania: a compulsive desire to count things. Theoretically, before Dracula finishes trying to count how many poppy seeds are scattered around his resting place, the Sun will have driven him back to his coffin.

Arithmomania is a serious medical condition. The inventor Nikola Tesla, whose studies into electricity gave us alternating current, suffered from the syndrome. He was obsessed with numbers divisible by three: he insisted on 18 clean towels a day and counted his steps to make sure they were divisible by three. Perhaps the most famous fictional depiction of arithmomania is the Muppets’ Count von Count, a vampire who has helped generations of viewers in their first steps along the mathematical path.

Urban patterns

Here’s a slightly more challenging sequence of numbers. Can you sniff out the pattern here?

179, 430, 1033, 2478, 5949 …

The trick is to divide each number by the number before it. This reveals that the multiplying factor is 2.4. Still exponential growth but what is intriguing is what these numbers actually represent.

They are patents issued in cities of population size 250,000, 500,000, 1 million, 2 million, 4 million. It turns out that when you double the population of a city you don’t simply get a doubling of the number of patents, as you might expect. Larger cities seem to be more creative. The doubling of population appears to add an extra 40 per cent to creativity! And it isn’t just patents that show this pattern of growth.

Despite the huge cultural differences between Rio de Janeiro, London and Guangzhou, there is a mathematical pattern that connects all cities across the world from Brazil to China. We are used to describing them by their geography and history, traits that highlight the individuality of a place like New York or Tokyo. But those facts are mere details, interesting anecdotes that don’t explain very much. Look at the city through the eyes of a mathematician instead and a universal character begins to emerge that transcends political and geographic boundaries. This mathematical perspective unveils the appeal of the city … and proves that bigger is better.

The mathematics reveals that the growth of each resource in a city can be understood by a single magic number particular to that resource. Each time the population of a city doubles, the social and economic factors scale correspondingly not simply by doubling but by doubling and a bit more. Rather remarkably, for many resources that bit more is around 15 per cent. For example, if you compare a city with a population of 1 million people to a city of 2 million people, instead of the larger city having twice as many restaurants, concert halls, libraries, schools, you find an extra 15 per cent on top of what you would have expected from simply doubling the numbers.

Even salaries are affected by this scaling. Take two employees doing exactly the same job but in different-sized cities: the employee living in the city with a population of 2 million will on average have a salary 15 per cent higher than the salary of the employee in the city with 1 million inhabitants. Double the city size again to 4 million and the salary gets increased by another factor of 15 per cent. The bigger the city the more you’ll get paid for doing exactly the same job.

It’s spotting a pattern like this that can be the key to a business getting the most out of what it puts in. Cities come in lots of shapes and sizes. Understanding that the shape is irrelevant but the size matters means that a company could get much more for its buck by simply relocating to a city twice the size.

This strange universal scaling was discovered not by an economist or a social scientist but a theoretical physicist applying the same mathematical analysis as is typically used in the search for the fundamental laws that underpin the universe. Geoffrey West was born in the UK and after studying physics at Cambridge, went to do research at Stanford exploring the properties of fundamental particles. But it was his move to become president of the Santa Fe Institute that would be the catalyst for his discoveries about urban growth. The institute specialises in finding ways for people from different disciplines to mix and discuss ideas. Very often the shortcut to cracking the enigmas in your own area of research is to take a detour through someone else’s seemingly unrelated domain.

It was the mix of mathematics, physics and biology that was bubbling away at Santa Fe that led West to wonder whether there are universal characteristics to the cities that span the globe, just as an electron or a photon of light has universal properties regardless of where it is in the universe.

It is possible to believe that mathematics is at the heart of the fundamental laws of the universe, that maths can explain gravity or electricity. A city on the other hand seems like an incomprehensible mass of people with their own motivations and desires going about the business of their lives. But as we have tried to make sense of the world around us, we have discovered that mathematics is the code that controls not only our world and everything in it, but us as well. There is even a pattern to the forces that control the jumble of millions of individuals.

West and his team amassed data for thousands of cities across the world. They collected everything from the total amount of electrical wire in Frankfurt to the number of college graduates in Boise, Idaho. They recorded statistics on petrol stations, personal income, flu outbreaks, homicides, coffee shops, and even the walking speed of pedestrians. Not everything was on the web, though. West had to grapple with Mandarin in an attempt to decode a massive almanac of data for cities in provincial China. When they started to analyse the numbers, this hidden code began to emerge. If the population of a city was double that of a smaller city, regardless of where in the world the two cities were, social and economic factors scaled up by this magic number of an extra 15 per cent.

Over 50 per cent of the world’s population now lives in a city. This extra amount of exponential growth provided by West’s scaling factor could well be the key to why cities are so attractive. Once a population of people is brought together, you seem to get more out than you put in. This is probably why people move to the big cities. Because if a person relocates to a city that’s twice as big, all of a sudden they’ll get an extra 15 per cent more of everything.

Infrastructure too is affected by this scaling, but in the opposite direction. Instead of needing twice as much stuff when you double the size of a city, you find that you save on infrastructure. The cost per person of copper wire, tarmac, sewage pipes, goes down by 15 per cent. Contrary to popular belief, this implies that your personal carbon footprint is smaller the larger the city in which you live.

Unfortunately it isn’t always positive benefits that get scaled by the mathematics. Crime, disease and traffic go up by the same factor. For example, suppose you know the number of Aids cases in a city of 5 million, then to estimate the number of cases in a city of 10 million instead of simply doubling the first number you find you also need to add on an extra 15 per cent of cases. The magic number 15 per cent again.

Is there an explanation for this universal scaling across cities? Is there something like Newton’s law of gravity that applies to everything from apples to planets to black holes?

The key to understanding why a city depends on its population size rather than its physical dimensions is that a city consists not of the buildings and roads but the people who inhabit it. The city is the stage upon which the real actors are playing out the story of civilisation. Cities are valuable because they act as a network for facilitating human interactions.

This means we should model a city not by whether it is built on an island, sits in a valley or sprawls across a desert but by the network of interactions between its inhabitants. It seems to be the quality of the network arising from city interactions that has the property of universal scaling discovered by West. This is the power of mathematics. To see the simple structures that are at the heart of our complicated environment.

If I take the extreme case that as a city grows everyone has contact with everyone else, one can see why a large city would lead to super-linear growth. If it has a population of N people, how many different handshakes can these N people perform, a measure of the top-end connectivity of the inhabitants? Let’s line them up numbered from 1 to N. Citizen number 1 shakes all the hands of the people down the line, totalling N − 1 handshakes. Now citizen number 2 sets off down the line. They’ve already shaken citizen 1’s hand so they end up shaking N − 2 hands. As we go down the line, each citizen does one less handshake. The total number of handshakes is the sum from 1 to N − 1. Hello again! This is the calculation that Gauss was asked to perform. His shortcut produces a formula for this number:

½ × (N − 1) × N

What happens to this connectivity when I double N? The number of handshakes doesn’t double but goes up by a factor of 2 squared, i.e. 4. The number of handshakes is proportional to the square of the number of inhabitants.

This is a great example of why mathematics can spare us from having to reinvent the wheel continually. Although I was asking a completely different question about connections across a network, I found that I already had the tools from the analysis of the triangular numbers to know how this number grows. Time and again the characters might change but the script remains the same. Understand the script and you’ve got a shortcut to understanding the behaviour of any character inserted into the drama. In this case the number of connections between citizens grows quadratically with the number of inhabitants.

Of course there is no way that every inhabitant will know every other citizen in the city. A more conservative measure would be that they know the citizens in their local neighbourhood. But this would scale linearly; the overall size doesn’t really matter.

It looks like the connections that people have in cities are somewhere in between these two extremes. A citizen has all their local connections plus a number of more long-range ones across the city. It seems that the long-range connections are the ones that cause the growth of connectivity to reach this extra 15 per cent as the population doubles. As I will explain later in the book, this sort of network arises in many different scenarios and it turns out to be a very efficient set-up for creating shortcuts.

Misleading patterns

Although patterns are incredibly powerful, we should still be careful with how we use them. You can set off on a path and you think you might know where you are heading. But sometimes that path can veer off in a weird and unexpected direction. Take the sequence that I challenged you with earlier.

1, 2, 4, 8, 16 …

What if I were to tell you that 31, not 32, was the next number in the sequence?

If I take a circle and I add points on its edge and join up all the points with lines, what is the largest number of regions that the circle can be divided into? If I have just one point on the circle, there are no lines and I’ve got just one region. If I add a point I can join the two points to get two regions divided by the line I’ve drawn. Now add a third point. Draw in all the lines connecting points and I have a triangle figure with three sectors of the circle surrounding the triangle: four regions.

1.1 The first five circle division numbers.

If I keep doing this it seems like a pattern is emerging. Here is the data showing the number of regions as I add another point on the circle:

1, 2, 4, 8, 16 …

A good guess at this point would be that adding a point doubles the number of regions. The trouble is the pattern breaks down as soon as I add a sixth point on the circle. Hard as you try, the maximum number of regions cut out by the lines is 31. Not 32!

1.2 The sixth circle division number.

There is a formula that will give you the number of regions but it is a little more complicated than simple doubling. If there are N points on the circle, the maximum number of regions you’ll get when you join the points is

1/24(N^4 − 6N^3 + 23N^2 − 18N + 24)

The message here is that it is still important to know what your data is describing and not to rely simply on the numbers alone. Data science is dangerous if it is not combined with a deep understanding of where that data comes from.

Here is another warning about this shortcut. What is the next number in this sequence?

2, 8, 16, 24, 32 …

Lots of powers of 2 in there. But a slightly unusual 24? Well, if you could identify 47 as the next number in this sequence, I recommend you buy a lottery ticket next Saturday. These were the winning numbers for the UK’s National Lottery on 26 September 2007. We are so addicted to looking for patterns that we often see them in places where we can’t expect a pattern. Lottery tickets are random. No patterns. No secret formulas. No shortcuts to becoming a millionaire. But that said, I shall explain in Chapter 8 that even random things have patterns in them that we can exploit as potential shortcuts. The shortcut when it comes to randomness is to stand back and take the long view.

The concept of pattern can be used as a shortcut to understanding if something is truly random or not, and it relates to how memorable a sequence of numbers is.

A shortcut to a good memory

Given that there is so much data being spewed out every second on the internet, companies are on the lookout for clever ways to store it. Finding patterns in the data actually gives a way to compress the information such that you don’t need as much space to store it. This is the key behind technologies like JPEG or MP3.

Take an image that is made up of just black and white pixels. In any such picture there might be a large swathe of white pixels somewhere. Instead of recording each pixel as white, and using as much memory to store the picture as there is data in the image, you could take a potential shortcut. You could record instead the location of the boundary of the region of white pixels and just add the instruction to fill in the region with white. The bit of code that I can write to do this filling in will in general be much smaller than recording that each pixel in this area is white.

Any patterns like this that you can discern in the pixels can be exploited to write code that will record the picture using far less memory than saving the data pixel by pixel. For example, take a chessboard. The image has a very obvious pattern that allows us to write code saying simply repeat white-black 32 times across the board. Even if you had an enormous chessboard the code would not grow any bigger.

I believe patterns are also key to how humans store data. I must admit that I have a very bad memory. I think it was one of the reasons that I was drawn to mathematics. Maths has always been my weapon against my terrible memory for names and dates and random information that I can make no logical sense of. In history I haven’t a clue what date Queen Elizabeth I died and if you told me it was 1603 I’ll have forgotten it ten minutes later; in French I always had difficulty recalling all the different forms of the irregular verb aller; in chemistry, was it potassium or sodium that burns purple? But in mathematics I could reconstruct everything from the patterns and logic that I’d identified in the subject. Spotting patterns replaced the need for a good memory.

I suspect that this is one of the ways we store memories. Memory depends on our brains’ ability to identify pattern and structure to help retain a condensed program from which to regenerate the memory. Here’s a little challenge. Stare at the squiggles contained in the following 6 × 6 grid. Then close the book. Can you reproduce it from memory? The key is not to try to remember each of the 36 squares in the image individually but to find a pattern that helps you to generate the image.

1.3 Can you memorise where the squiggles are?

Although this image has roughly the same proportion of squiggles as a 6 × 6 chessboard has black squares, the lack of an obvious pattern makes it much harder to remember where they are. The image was generated by tossing a coin and marking a square with a squiggle if the coin landed heads. Mathematically the coin is equally likely to produce the chessboard pattern of squiggles, alternating between heads and tails, as to produce the random arrangement of squiggles. Yet the chessboard’s pattern makes it much easier to remember.

By identifying a pattern in an image you can write down a recipe to reproduce the picture. In mathematics that recipe is what is called an algorithm. This idea of the size of the algorithm needed to memorise an image is a powerful measure of the randomness contained inside the image. The chessboard pattern is very ordered. Its generating algorithm is small. The image created by the toss of a coin probably requires an algorithm that is no shorter than one recording the contents of each of the 36 individual squares in the grid.

You will find that a photograph with an obvious story in the picture has a JPEG that is much smaller than the original, while an image with random pixels fails to get smaller when you try to compress it using the JPEG algorithm because there are no patterns to help.

From humans to machines, anyone or anything that is memorising something is applying a distinctly mathematical side of their brain. Memory requires finding patterns, connections, associations and logic in the data that we are trying to store. Pattern is the shortcut to a good memory.

Climbing the stairs

Let’s return to the question I posed at the start of the chapter. How many ways are there to climb a flight of 10 steps if you use a combination of one or two steps? There are several ways to go about this. One way would be just to randomly start writing down different possibilities. Clearly an unsystematic approach will inevitably miss some options and it will also take time to record them all. Is there a better strategy?

A slightly more systematic attack would be to say: let’s start with just 1-steps. There is just one way to achieve this: 1111111111. What if I next allow one 2-step among the 1-steps. This means making a total of nine steps: eight 1-steps and one 2-step, and you can choose at which point to make the 2-step. There are nine different places you could do the 2-step.

This is looking a promising strategy. Next I could consider the combinations with two 2-steps threaded through six 1-steps. A total of eight steps up the stairs this time. But I am going to have to calculate how many ways there are to choose which of the eight steps gets a 2-step. I could put one 2-step in eight different locations, and the second 2-step in the remaining seven open positions. So that looks like 8 × 7 different possibilities. But I need to be careful because I’ve counted things twice. I could have chosen the first 2-step in position 1 and the second in position 2 or I could have done it the other way round. The result would be the same. So the total number of possibilities now is 8 × 7/2 = 28. There is actually a mathematical name for this number. It is ‘8 choose 2’ and it is denoted by:

More generally the way of choosing two numbers from N + 1 numbers is given by the formula ½N(N + 1), which is the same formula Gauss came up with for the triangular numbers. There’s that wheel we invented coming up again! There is a way to translate the question of choosing two numbers from N + 1 into the challenge of calculating the triangular numbers. I will explain in Chapter 3 how changing one problem into another can often be a great shortcut to solving the problem.

These tools for calculating choices, called binomial coefficients, were actually some of the formulas that Gauss and his classroom assistant Bartels pored over in their algebra books at school together.

But to solve this puzzle, next I will have to calculate how to choose three locations out of a choice of seven to place our three 2-steps up the staircase. Although this seems like a good systematic way to build up the possibilities, it is going to require us coming up with formulas for choosing how to put in an increasing number of 2-steps into the climb to the top of the stairs. It’s beginning to look like a hard slog heading along this path. It doesn’t really feel like a shortcut.

So here’s a better one exploiting what I have shown you in this chapter. With puzzles like this, I find a very powerful strategy is to consider a smaller number of steps and to see if there is a pattern in the way that the numbers are falling out.

Here are the possibilities for staircases with 1, 2, 3, 4 and 5 steps, which can be worked out quickly by hand:

1 step: 1

2 steps: 11 or 2

3 steps: 111 or 12 or 21

4 steps: 1111 or 112 or 121 or 211 or 22

5 steps: 11111 or 1112 or 1121 or 1211 or 2111 or 122 or 212 or 221

So the number of possibilities is going 1, 2, 3, 5, 8 … Now you might already have spotted a pattern. You get the next number by adding the two previous numbers together. You might even know a name for these numbers. They are the Fibonacci numbers! Named after the twelfth-century mathematician who discovered that they are the key to the way things grow in the natural world. Petals on flowers, pine cones, shells, populations of rabbits. The numbers all seemed to follow the same pattern.

Fibonacci discovered that nature was using a simple algorithm in order to grow things. The rule of adding the two previous numbers together was nature’s shortcut for building complex structures like a shell or a fir-cone or a flower. Each organism just uses the two last things it built as ingredients for the next move.

1.4 How to use the Fibonacci numbers to grow a spiral.

Using a pattern to evolve structures is a key shortcut for nature. Take, for example, the way nature builds a virus. Viruses come in very symmetrical structures. This is because symmetry requires a simple algorithm to make the structure. If a virus is in the shape of symmetrical dice the DNA that replicates the molecule only has to create several copies of the same protein that will make up the faces and the same rule can then be used across the virus to build its structure. No strange instructions for different faces. A pattern means building the virus is fast and efficient, making it potentially so deadly.

But are we really sure just from this small amount of data that this Fibonacci rule is the secret to climbing the stairs?

Actually the rule explains exactly how to work out the next possibilities with 6 steps on the staircase. Take all the possible steps up to the fourth step and then add a 2-step on the end. Or all the possible routes to the fifth step and add a 1-step on to these. This gives all the ways up to the sixth step. It is a combination of the two previous numbers in the sequence.

The answer to the puzzle is to calculate the tenth number in the sequence.

1, 2, 3, 5, 8, 13, 21, 34, 55, 89

There are 89 different routes. The pattern is the shortcut to knowing how many ways to get to the top of the staircase. And the pattern will help crack this conundrum even if there are 100 or 1000 steps.

Although these numbers are named after Fibonacci, he wasn’t the first to discover them. Indian musicians were. Tabla players have long been interested in showing off the different rhythms they can make on their drum. As they explored the different sorts of rhythms that they could make from long and short beats, it led them to the Fibonacci numbers.

If the long beat is twice the length of the short beat, the number of rhythms the tabla player can make using them is the same answer as climbing the staircase. Each 1-step corresponds to a short beat, each 2-step a long beat. So the number of possible rhythms is given by the Fibonacci rule. And the rule even gives the tabla player an algorithm to construct them from the previous shorter rhythms.

There is something exciting here about seeing the same pattern explaining such different things. For Fibonacci it was the way nature grows things. For the Indian tabla players, the pattern generated rhythms. The pattern also explains the number of ways to climb a staircase in 1s and 2s. There are even some analysts in the financial sector who believe that these numbers can be used to predict when a stock that is falling will eventually bottom out and start climbing again. This financial pattern is somewhat controversial and certainly doesn’t hold universally but it has led some investors to make the correct call. It is this power of revealing the underlying structure behind different façades that can be so powerful as a shortcut. One pattern can solve a multitude of very different-looking challenges. It is often worth checking when faced with a new problem if it might be an old problem in a new disguise that you have already found a way to solve.

Connecting shortcuts

I can’t resist adding a little coda to this story because it makes use of the earlier hard work. My initial strategy for calculating the number of routes to the top of the staircase led me to the question of how to choose 3 things from a group of 7 objects. Mathematicians have actually found a clever way to shortcut calculating all those choices. It’s something called Pascal’s triangle (although like Fibonacci, it turns out that Pascal was beaten to the discovery, this time by the ancient Chinese).

1.5 Pascal’s triangle.

The triangle has a rule similar to the Fibonacci rule but you build the numbers in the layers below by adding the two numbers above. The table is easy to build using this rule, but the great fact is that it contains all the choice numbers I was after. Suppose I run a pizza restaurant and I want to boast about the number of different pizzas I have on offer. If I want to know the number of ways of choosing 3 toppings from a choice of 7 different toppings, I go to the (3 + 1)th number in the (7 + 1)th row: 35. That is my shortcut to knowing that there are 35 different pizzas I can make. In general to choose m things from n things you go to the (m + 1)th number in the (n + 1)th row. But because these choice numbers were one way to solve our staircase problem, this means that the Fibonacci numbers are actually hiding inside Pascal’s triangle. Add up numbers in diagonal lines through the triangle and the Fibonacci numbers appear.

1.6 The Fibonacci numbers, triangular numbers and powers of 2 inside Pascal’s triangle.

This kind of connection is one of the things that I love about mathematics. Who would have thought that the Fibonacci numbers were hiding inside Pascal’s triangle? Yet by looking at the puzzle in two different ways, I’ve found a secret tunnel, a shortcut, that links these two seemingly different corners of the mathematical world! And look how the triangular numbers and also the powers of 2 are all hidden inside this triangle. The triangular numbers are sitting along one of the diagonals through the triangle while you get the powers of 2 by adding all the numbers up in each of the rows. Mathematics is full of these strange tunnels providing shortcuts that we can exploit to change one thing into another.

Finding patterns in data is not just about cute problems like ways to climb stairs. It is key to predicting the way the universe will evolve, as Gauss discovered when he predicted the course of Ceres. It is crucial for understanding climate change. It is central to giving businesses an edge as they try to navigate the uncertainties of the future. It might even give us some inkling into the evolution of human history. In this incredibly data-rich age one exabyte (10^18) of data is created on the internet every day. That’s a lot of numbers to explore. But spot a pattern and you’ve got a shortcut to navigating this vast digital landscape.

The pattern shortcut is all about identifying an underlying rule or algorithm that is the key to generating the data you want to understand. It is the sort of shortcut that keeps working for you even if the problem scales seemingly out of control. The staircase might get bigger and bigger but the shortcut still gets me the answer.

But patterns don’t just apply to numbers. Many parts of life have patterns that we can exploit to transfer understanding from one realm to another. Understanding patterns in music, for example, is a crucial part of mastering a musical instrument. For Natalie Clein, international cellist, a pattern in music can help her to make predictions of where a piece might be heading before she’s read the notes on the page.

Later I will get the chance to talk to Susie Orbach about shortcuts in therapy. It turns out that she exploits a lot of patterns in human behaviour. She can take patterns learned from the history of a previous patient to be able to help a new patient that enters her practice. But people are a little bit messier and more individual than numbers, and these patterns Orbach will reveal need to be treated carefully. They are at their best when the world is turned into numbers, something our digital world is doing more and more. Our digital footprints are increasingly transforming our human behaviour into numbers. Find the pattern in the numbers and you have a shortcut to be able to predict what the human might do next.

Shortcut to the shortcut

Spotting a pattern is an amazing shortcut for navigating your way into the future. Find a pattern in the stock prices and you might gain an edge when it comes to investing. Wherever you’ve got numbers, examine the data for any hidden patterns. And it isn’t just numbers that have patterns. People do too. Spot the pattern of shots that your opponent in tennis tends to follow and you’ll be ready for the passing shot next time it comes. Understand the pattern in people’s eating habits and your restaurant can cater for its customers without excessive wastage of unwanted food. Sniffing out patterns has been a fundamental shortcut for the human species ever since we started taking our first steps out of the savannah.

Pitstop: Music

A few years ago I decided to learn to play the cello. But it is taking me longer than I had hoped, so I am keen to sniff out any cunning shortcuts that might help me. If mathematics is the science of patterns, music is the art of patterns. Could exploiting these patterns be the key?

The cello isn’t the first instrument I learned to play. The same year that Mr Bailson had shared the story of the young Gauss, my music teacher at my comprehensive school had asked the class whether anyone wanted to learn a musical instrument. Three of us put up our hands. At the end of the lesson the teacher led us into the music storeroom. It was pretty bare except for three trumpets stacked on top of one another. So the three of us ended up playing the trumpet.

I don’t regret the choice. The trumpet is a wonderfully flexible instrument. I cut my teeth playing for the local town band, played in the county orchestra, even tried my hand at a bit of jazz. But as I sat silently counting bars rest in the orchestra waiting for the next trumpet entry, I would stare at the cellos in front who seemed to be playing all the time. I must admit I was a little envious.

Now an adult, I decided that I would buy a cello with a bit of money that my godmother left me in her will. I would use what was left over to get some lessons. But I was slightly concerned whether as an adult I’d be up to learning a new instrument. As a child, the time it took to learn an instrument didn’t bother me. I was at school and we had years of learning ahead of us. But as adults we have fewer years ahead of us and so become much more impatient. I wanted to be able to play the cello now, not in seven years’ time. Was there any shortcut to learning an instrument?

Malcolm Gladwell’s book Outliers popularised the theory that to become an expert in anything requires putting in a minimum of 10,000 hours’ practice. It controversially proposed that this might be enough to become internationally recognised in your field, although the team that produced the original research said that this was a misinterpretation of their work. But was there really no way I could shortcut the 10,000 hours of practice before I could play Bach’s cello suites on the stage? An hour a day would mean over 27 years of practice!

I decided to seek the advice of Natalie Clein, one of my favourite celloists of all time. Clein first came to international attention as one of the youngest winners of the prestigious BBC Young Musician of the Year competition in 1994 when she performed Elgar’s Cello Concerto. What had been her trajectory to international fame?

Natalie started playing the cello at the age of six, but didn’t get serious until a few years later. ‘By fourteen or fifteen,’ she told me, ‘I was trying to do four to five hours a day. There are some who do much more. There are kids out there doing about eight hours a day practice when they are sixteen. There are colleagues from places like Russia or the Far East where they’re put into this disciplined mode of hard work much earlier than we are in the west.’

This level of discipline, Clein explained, was necessary to achieve the motor memory and control that mastering an instrument requires: ‘There’s certainly a minimum number of hours you need to put in when you’re learning an instrument, three or four hours a day in your teenage years, which you have to cover, because you physically just don’t get the motor mechanics otherwise.’ Take Jascha Heifetz, for example. Heifetz was one of the greatest violinists of all time. Famously, he practised scales every morning for most of his life, thousands of hours in total, just on scales.

In this way, cellists are similar to athletes. You can’t run a marathon or win a 100 metre sprint without putting in the hours to physically train your body. This aspect of tuning the body and mind so that it can play passages at speed requires brute repetition. I know from my own practice that it’s only by repeating a passage over and over in such a way that the body almost knows what to do without the need to engage my brain that I can play certain pieces.

But Clein was keen to stress that hard work alone is not good enough. ‘It’s repetition of the right thing,’ she says. ‘It’s all very well doing 10,000 hours, but it really has to be the right kind of 10,000 hours. You can’t simply do them. What I tell my students is that you have to have mind, body and soul involved in those 10,000 hours.’

Practising diligently may not seem like a shortcut, but it is. How often do we waste time doing something because we are doing it the wrong way or not maximising the effort we are making or just missing the point of putting in so many hours?

When considering what makes for effective practice, you will often hear people talk about flow. Flow is a term coined by the Hungarian psychologist Mihaly Csikszentmihalyi in 1990 to describe the psychological state in which we become fully immersed in a task. As he wrote, ‘The best moments in our lives are not the passive, receptive, relaxing times … The best moments usually occur if a person’s body or mind is stretched to its limits in a voluntary effort to accomplish something difficult and worthwhile.’ Flow lives at the meeting point of extreme skill and great challenge. If you don’t have the skills and you try something too challenging, you end up in a state of anxiety. If something is too easy for you given your skill set, you are likely to border on boredom. But if you have both the skills and the right challenge, you can reach this point of flow or ‘being in the zone’. We’d all love to get to that state and many people have written guides to achieving flow: meditation, flow soundtracks, dietary supplements, mental flow triggers, caffeine.

But Clein is sceptical of quick fixes. ‘You can’t shortcut your way to the flow,’ she says. ‘You have to learn the rules in order to be allowed to break them, and it’s the moment of breaking them that you find this liberation that takes you to the flow in some way. It’s the discipline that gets you to a state of inspiration.’

Although there are no shortcuts through the physical training of becoming an instrumentalist, I do think that performers spend so much time practising scales and arpeggios because these provide a shortcut when they are performing. If you see a pattern of notes on the page that corresponds to a scale or an arpeggio, you don’t have to read each note. Instead you can tap into the shortcut you’ve spent hours learning.

If there are no shortcuts to achieving the physical fine motor skills to play at a high level, perhaps there still may be shortcuts to learning a new piece. Clein pointed me to the work of musical analyst Henrich Schenker. As it happened, I’d come across Schenker before, in a different context. Computer scientists have used his work to try to program AI to compose convincing music. The goal in Schenkerian analysis is to identify a deep structure underlying a piece of music, called its Ursatz, which is a bit like the pattern underlying a sequence of numbers. AI music generation tries to reverse this process and make music by starting with the Ursatz and then fleshing it out. But for Clein, this analysis provides a more efficient way to navigate a piece of music she is learning.

‘He likes to reduce and reduce and reduce to the simplest possible formula to understand a piece,’ she says. ‘You could say that’s a shortcut to understanding the structure of a piece of music. It’s seeing the macro rather than the micro.’

Pattern, it turns out, is part of the musician’s toolkit for navigating the complexities of a piece of music. I wondered whether this is a useful shortcut to memorising a piece of music? Identifying the underlying structure of a sequence of numbers allows me to shortcut the need to rely on repetition to memorise something. For Clein, memorising a concerto just comes from the discipline of having practised something over and over until it becomes motor memory. But for others, patterns may play a bigger role. As Clein told me, ‘I have a friend, Vadym Kholodenko, who’s a kind of genius. I’ve seen him sight read a piece in the afternoon that he’d heard once or twice before and then play it in the concert in the evening and it was really much better than other people had played it after working at it for three months. He sees the big shapes and has an overriding confidence that he can do it and then the rest of the gaps are filled in. He definitely sees the macro and believes in the macro more than the micro, that’s for sure.’

My cello teacher taught me another interesting shortcut when learning a new piece. There are often multiple ways to play a passage on the cello because it’s possible to play the same note on different strings. Frequently the first and most obvious way to play a note is inefficient, and you end up having to jump around the instrument. But if you think more strategically you can find alternative ways to play the passage that mean you don’t have to move your hand wildly up and down. Working out how to play a piece can become something of a puzzle: what is the most efficient way to place your fingers on the strings so that you can play it easily?

Clein agrees: ‘It can be very creative. I don’t think anyone taught me but I thought it would be a very good idea myself to be able to use my thumb quite a lot. That’s been really helpful to me. There are a couple of cellists that do that, starting with the great cellist Daniil Shafran. It was something that I thought I’d made up but I hadn’t. It’s all about problem solving. The more burning the problem is the more creative the solution can be.’

But despite these helpful ways to navigate music, the bottom line for Clein is that there are no shortcuts to what she does: ‘To become a good professional cellist, especially somebody who has to play solo works and be exposed and feel that you’re under the spotlight for your craft, there’s just no way. There’s no shortcut. And that’s what I love about it. Pablo Casals famously practised all his life and somebody asked him when he was ninety-five, “Maestro, why do you keep practising?”, and he replied, “Because I feel finally I’m getting better. I’m improving.” I think that’s what keeps you going. It’s a lot of hard work and it continues to be. You have to be interested in the work to keep it going through a lifetime. You don’t ever reach a summit.’

For many experts, this is why shortcuts aren’t a real concern. As Clein told me, ‘The idea of a shortcut is sort of attractive in the short term but in the longer term, it’s not. I think if there were lots of shortcuts maybe we’d be less attracted by the challenge.’

I recognise that tension between the desire to reach your goal and the ease with which it can be realised. If it’s too easy it loses its satisfaction. And yet I don’t just want mindless slog. For me the most satisfying shortcuts are precisely those that emerge after I’ve been stuck for some time wondering how I’m going to reach my destination. The adrenaline release that accompanies the moment you see a cunning way through is a drug that I’ve become quite addicted to in my journey to perfect my mathematical performance. But when it comes to the cello, although exploiting patterns can help, I realise there’s just no shortcut around hard work.

2

The Calculated Shortcut

You are a grocer and you want to be able to measure all weights from 1 kg to 40 kg using a set of balance scales. What is the smallest number of weights you need to be able to do this and what are their values?

Finding the right shorthand to capture an idea is a powerful tool for speeding up thought. I take for granted the way that I can capture the concept of a million with seven symbols: 1000000. But bundled up in those seven symbols is a whole history of fascinating shortcuts for how to navigate numbers and to calculate efficiently. Throughout history and even today if you are in business or building or banking, you can get an edge if you know some faster, more efficient way to calculate the answer before your competitor. In this chapter I want to share with you some cunning ways that we have found to navigate numbers and calculations. Interestingly, these shortcuts can still be powerful strategies even when there are no numbers involved.

People often think that as a research mathematician I must be doing long division to a lot of decimal places. Surely the calculator has to have put me out of a job by now? This misconception of the mathematician as a super-calculator is a common one. But it doesn’t mean that calculation is not part of the job description. A lot of cunning mathematics has started from the challenge of finding clever ways to do arithmetic, like Carl Friedrich Gauss’s childhood shortcut. There is a whole history of shortcuts that humans have discovered while trying to more effectively do calculations. Even the calculator we reach for today has been programmed using some of the clever shortcuts mathematicians have come up with over the years.

We tend to think of the computer as all powerful, able to do anything. But computers have their limits too. Take Gauss’s challenge of adding up the numbers to 100. Sure, a computer will have no trouble with that. But there will be a number that is too big even for the computer. If you ask it to add up all the numbers up to that point, it too will grind to a halt. In general the computer still relies on us humans to come up with the shortcut that, when implemented in the computer’s code, will allow it to get further faster. I shall reveal in this chapter a rather startling use of a seemingly abstruse bit of mathematics called imaginary numbers that provided a crucial shortcut for getting computers to do a whole range of tasks, including landing planes quickly enough that they don’t crash mid-flight.

Shortcut to counting

The very way that we write our numbers can determine whether a calculation is easy or ends up being a complicated slog prone to errors. It was an important moment for human progress when we realised that good symbols for complex ideas was a shortcut to better thinking. Historically it seems that each civilisation realised that writing and recording spoken words was a powerful way to preserve, communicate and manipulate new ideas. And every development of a new script for language has generally been accompanied by a clever new way to think about how to record the concept of number. But it is the civilisations that found the better ways to write numbers that found they had a shortcut to faster and more efficient calculations and control of the data.

One of the earliest shortcuts that mathematicians discovered was the power of the place-value system. If you are counting something like sheep or days, the first way you might go about it is to make a mark for each sheep or day. This seems to be how the first humans counted. There are bones dating back 40,000 years with notches down the side that it is believed are examples of our first attempts to count.

This is already an impressive moment. The abstract concept of number is beginning to emerge. Archaeologists don’t know exactly what these notches were counting but there is an understanding that the number of them and the number of sheep or days, or whatever was being counted, have something in common. The trouble is that trying to distinguish between 17 or 18 notches on a bone can be pretty tricky. You have to do the work of counting all over again. At some point in nearly every culture a bright spark has hit on the idea of creating a shorthand for all these notches that is easier to read.

When I was living in Guatemala a few years ago I was intrigued to find a strange series of dots and dashes on the banknotes. I asked our neighbour whether it was some weird Morse code hidden in the local money. She explained that it was indeed code but code for the numbers that each banknote represented. The dots and dashes were shorthand for the way number was represented in Mayan culture. The Mayans recognised that the human brain has difficulty distinguishing between anything beyond four notches. So instead of recording more and more dots on the page, when they hit five they put a line through the four dots, just like a prisoner counting off the days to their release. In this way a line became shorthand for the number five.

But what if you want to count even further? The ancient Egyptians came up with an impressive list of hieroglyphs to denote different powers of 10. They represented the number ten by a hobble for cattle (a device that limits their movement), a hundred by a coil of rope, a thousand by a water lily, ten thousand by a bent finger, one hundred thousand a frog and finally a million is a picture of a man on his knees with his arms in the air looking like he’s just won the lottery.

This is a clever shorthand. Rather than representing a million by a million notches on a bone, the Egyptian scribe could simply draw on the papyrus the figure of the kneeling man. This facility to record large numbers efficiently was one ingredient in the rise of Egypt as a powerful civilisation that could tax its citizens and build cities effectively.

But there is still something rather inefficient about the Egyptian system. If a scribe wanted to record the number 9,999,999, they would need 63 figures. Add one more to the total and someone had to come up with yet another little picture to represent 10,000,000. Look how with our modern system of numbers we get away with recording a large number like 9,999,999 in just seven symbols, and using just ten different symbols (0, 1, 2, … 9) we can go as far as we want. The key is the place-value system, an extraordinary shortcut that three different cultures came up with at separate points in history.

The first to come up with this shortcut was a rival civilisation to the Egyptians: the Babylonians. Interestingly theirs was not a culture that worked with powers of 10 like the Egyptians did or we do today. They worked with powers of 60. They had numbers all the way up to 59 before they felt the need to regroup. They wrote the numbers from 1 to 59 from just two symbols: representing 1 and representing 10. But it meant that the number 59 needed a collection of fourteen symbols.

At first sight this seems far from efficient. But their choice of 60 had a very different sort of shortcut embedded in it. This is a result of the high divisibility of the number. Because 60 can be divided in so many different ways – as 2 × 30 or 3 × 20 or 4 × 15 or 5 × 12 or 6 × 10 – it gave merchants who were using the system many possibilities for how they could divide their merchandise. And the high divisibility of 60 is the reason that we ended up using it for counting time. Sixty minutes in the hour and 60 seconds in the minute have their origins in ancient Babylon.

The Babylonians’ real breakthrough, however, was counting beyond 59. One option was to start creating new symbols, like the Egyptians. But the Babylonians had a different idea. The meaning of a symbol would change according to its position relative to other symbols. In our modern system, the number 111 has the same symbol repeated three times, and the brilliance of this shorthand is that reading from right to left, the first 1 represents 1 but the second 1 represents ten and the third 1 represents a hundred. Each time you add another number to the left the value goes up by a factor of ten.

For the Babylonians, however, because they were working with a base of 60 not 10, each time you moved to the left the value went up by a multiple of 60. So 111 in Babylonian is 1 × 60^2 + 1 × 60 + 1 = 3661. This was an exceptionally powerful shortcut. Using the two symbols and , you could represent numbers as big as you wanted. But you couldn’t represent every number. This required the introduction of a new symbol. What if you wanted to record the number 3601? This meant indicating that there are no 60s. A symbol for nothing was needed. In Babylonian cuneiform the absence of a particular power of 60 was represented by two little notches:

The Mayans too discovered this shortcut to writing big numbers. They already had the symbol for 5. A line. Three lines could denote 15. Three lines and four dots denoted 19. But then the Mayans decided that things were getting too cluttered. And so the next positions in the number denote powers of 20. So 111 in Mayan represents 1 × 20^2 + 20 + 1 = 4041. They too realised the need for recording nothing in some places and the symbol of a shell was used.

The Mayans were great astronomers and kept track of huge swathes of time. This efficient number system exploiting the position of a symbol allowed them to talk about astronomically large numbers without the need for a huge list of symbols.

But there was still something missing in both the Babylonian and Mayan systems: a symbol for nothing. This was the revolutionary step taken by the third culture to invent the place-value system: the Indians.

The numbers that we use today are often referred to as Arabic numerals but this is a mistake. At least it is not the full story. The Arabs learned about the system used by Hindu scribes and brought it to Europe. The numbers should really be referred to as the Hindu-Arabic numerals. The Hindu numerals use symbols from 1 to 9 and each place increases by a factor of 10 as you move left through a number. They also had the symbol for nothing. The zero.

When Europeans were shown this idea they couldn’t get it. Why would you need a symbol if there is nothing to count? But for Indians nothing, the void, was a very important philosophical concept and so they were happy to name it or number it.

In Europe they were still using Roman numerals and the abacus to do calculations. But using the abacus required skill and expertise. So calculation was not something that was available to the common citizen. Calculation allowed the establishment to maintain power. There was no record of the calculation done by an abacus. There was just the result. The system was open to abuse by those in power.

This is why the establishment tried to ban the incoming numerals from the east. They would give the common citizen access to calculation and the ability to record those calculations. The introduction of this shortcut to navigating numbers is probably as significant as the invention of the printing press. It brought maths to the masses.

Black mathematical magic

Today the computer and calculator are our shortcut to calculation. But those who are over fifty will remember another tool that was taught as an aid to shortcutting complicated arithmetic: the book of log tables. For centuries, this was the go-to shortcut for any merchant, navigator, banker or engineer. It was the tool that would give them an edge over any competitor who tried to do calculations directly.

The power of the logarithm was unlocked by the Scottish mathematician John Napier. I would have loved to have met Napier, not just because he came up with this clever calculating shortcut, but because he sounds like a crazy character. Born in 1550, Napier was fascinated by theology and the occult. He would walk around his estate accompanied by a black spider that he kept in a little cage. His neighbours believed he was in league with the devil. When he threatened to imprison one of their pigeons for eating his grain, the neighbour decided to call his bluff believing it impossible to catch the birds. The next morning he was shocked to see the pigeons sitting passively in the field and Napier wandering round putting them into sacks. Had they been bewitched? It turns out that Napier had drugged them by lacing peas with brandy.

Napier capitalised on the local belief that he was a sorcerer. To catch a thief among his staff he told them that his black rooster could identify the criminal. One by one, the staff had to enter a room and stroke the bird. Napier claimed it would cry out when touched by the thief. When all the staff had visited the rooster Napier asked them to show him their hands. All but one had soot on their hands. Napier had covered the rooster with it knowing that only the thief would be too afraid to stroke the bird.

In addition to all his theological studies, Napier had a fascination with mathematics. But his interest in numbers was nothing more than a hobby and he lamented the fact that he didn’t have sufficient time to carry out his calculations given all his theological studies. But then he came up with a clever strategy to sidestep the long calculations that he was trying to wade through.

As he wrote in the book he published about his shortcut:

Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.

What Napier discovered was a way to turn the complicated job of multiplying two large numbers together into the much simpler task of adding two numbers together. Which of these could you do quicker by hand?

379472 × 565331

or

5.579179 + 5.752303

The key to this magical transformation is the logarithm function. A function is like a little mathematical machine that takes one number as input and then manipulates the number according to the internal rules of the function to output a new number. The logarithm function takes a number and outputs the number that you would need to raise 10 in order to get the number you started with. For example, if I input 100, the logarithm function outputs the number 2 because if I raise 10 to the power of 2 I get 100. If I input 1 million into the logarithm function, the output is 6 because 10 to the power of 6 is a million.

The logarithm function is a bit trickier when it comes to inputting numbers that are not obviously powers of 10. For example, to get the number 379472, I need to raise 10 to the power of 5.579179. To get the number 565331, I raise 10 to the power of 5.752303. So, as with many shortcuts, a lot of work would need to be done in advance to be able to implement the shortcut. Napier spent many hours preparing tables to be able to look up the logarithm of a number, but once the tables had been prepared the shortcut came into its own.

Because if you have two powers of 10, for example 10a and 10b, and you want to multiply them together, the answer is very simple. It is 10a + b. You just add the powers. This means that instead of doing the hard work of multiplying 379472 × 565331, I can add the logarithms 5.579179 + 5.752303 = 11.331482 and use the tables Napier prepared to calculate 1011.331482.

This idea of using tables of calculations to speed up arithmetic wasn’t new. Indeed it seems that some of the cuneiform tablets of the ancient Babylonians were used in a similar fashion. They exploited another formula to work out big multiplications. If I’ve got two large numbers A and B, the algebraic relation

A × B = ¼ × {(A + B)2 − (A−B)2}

changed the problem into subtracting square numbers. Although this algebraic notation would not appear until the ninth century, the Babylonians understood this connection between squares and products which gave them a shortcut to calculating the product of A and B. Instead of calculating the squares you simply looked them up on one of the tablets of squares that had previously been calculated by a scribe.

Napier described the shortcut he’d come up with in his book A Description of the Wonderful Table of Logarithms. And as the book’s ideas spread it did indeed instil wonder in its readers. Oxford mathematician Henry Briggs, who was the first to hold the prestigious Savilian Professorship in Geometry at New College, where my own professorship is attached, was so taken by the power of Napier’s logarithms that he journeyed four days to visit Napier in Scotland, writing: ‘I never saw a book which pleased me better or made me more wonder.’

For centuries these tables would provide scientists and mathematicians with a shortcut to complicated calculations. The great French mathematician and astronomer Pierre-Simon Laplace declared 200 years later that logarithms, ‘by shortening the labours, doubled the life of the astronomer and spares him the errors and disgust inseparable from long calculations’.

Laplace captures here the essential quality of a good shortcut: it frees up the mind to dedicate its energies to more interesting pursuits. But it was the advent of machines that truly liberated the scientist from the tedium of calculation.

Mechanical calculators

One of the first to realise the power of machines as a shortcut to calculation was the great seventeenth-century mathematician Gottfried Leibniz: ‘It is unworthy of excellent men to lose hours like slaves in the labour of calculation which would safely be relegated to anyone else if machines were used.’

Leibniz got his idea for the machine that he would eventually build after encountering a pedometer: ‘When I saw an instrument with whose help one could count one’s steps without thinking, the idea immediately came to me that all of arithmetic could be achieved through a similar type of device.’

The pedometer made use of the simple idea that once a cog with ten teeth had clicked through one rotation it could be connected to another cog that would be clicked on one step to record ten steps. The place-value system in cogs. Called the Stepped Reckoner, Leibniz’s calculating machine was capable of addition, multiplication and even division. But physically realising his ideas turned out to be a challenge: ‘If only a craftsman could execute the instrument as I had thought the model,’ he wrote.

He took a wooden prototype to London to give a demonstration to the fellows of the Royal Society. Robert Hooke, who already had a reputation as a cantankerous character, was deeply unimpressed and after having taken the machine to pieces declared that he could make a much simpler and more efficient contraption. Leibniz was not deterred and eventually managed to employ a skilled clockmaker to build a machine that could achieve the calculated shortcut that he had promised.

Leibniz had an even grander vision. He didn’t just want to mechanise arithmetic but all thought. He wanted to reduce philosophical argument to a mathematical language that could itself be implemented on a machine. He envisioned a time when if two philosophers disagreed on some idea they could simply turn to the machine to sort out their differences and discover who was right.

On a visit to Leibniz’s home town of Hanover, I was lucky enough to see one of his machines. It is a thing of beauty, and we are lucky to have it. For some years, this original machine lay buried in an attic in Göttingen, Gauss’s university, and was only rediscovered in 1879 when workers were trying to fix a leak in the roof of the building and came across it hidden in a corner.

Leibniz’s machine is the beginning of what would eventually lead to the calculators and computers of today. But that’s not to say that there aren’t limits to the power of the computer. These days we tend to think that computers are so good at doing calculations quickly that there are really no limits to what they can do. As Time magazine reported in 1984: ‘Put the right kind of software into a computer and it will do whatever you want it to do.’ But computers have limits. And even they sometimes need the human programmer to come up with a cunning shortcut to avoid calculations requiring the lifetime of the universe for the computer to execute.

One of the most intriguing shortcuts that computers have tapped into is that of exploiting a new sort of number, one that seems to have nothing to do with the practical world of computing: an imaginary number.

Through the mathematical looking glass

Can you solve the equation x^2 = 4? You probably had no trouble coming up with the answer x = 2 because if you square 2 you get 4. If you were smart you might also have come up with a second answer because x = −2 also works. That’s because when you square a negative number the answer is positive. So −2 squared is also 4.

That equation was quite simple. But what if I give you the following to solve:

x^2 − 5x + 6 = 0

This probably sent a shiver down a good number of readers’ backs because this is one of the quadratic equations – equations with an x squared term – that students have to learn to solve at school. In fact the ancient Babylonians had already come up with a general algorithmic procedure that would unlock the answer. Although they didn’t yet have the algebraic language to express their ideas, in modern terms if you want to find the solutions to a general quadratic equation

ax^2 + bx + c = 0

there is a formula that finds the answers:

So in the case of the equation x^2 − 5x + 6 = 0, we put a = 1, b = −5 and c = 6 into the formula and out come the answers x = 2 or x = 3.

It is during the Babylonian period that the power of mathematics to shortcut hard work began to emerge. Before the discovery of this formula, each quadratic equation would have been solved by hand. Each time the scribes were reinventing the wheel not recognising that although the numbers were different, what they were doing each time was the same. But at some point a scribe recognised that there was a general algorithmic procedure that worked whatever the numbers.

This is the moment that mathematics begins. It is the art of seeing the pattern that underlies all these infinitely many equations. Instead of a potentially infinite amount of work, the pattern reveals that there is essentially only one act of labour needed. Learn the algorithm or formula to solve the equations and you’ve got a shortcut to solve infinitely many different equations. With the birth of mathematics in the Babylonian era we witness why mathematics is really the art of the shortcut.

But does this shortcut solve all possible quadratic equations?

How about the challenge of solving the equation x^2 = −4? For many centuries this equation was regarded as unsolvable. After all, the numbers that we use for counting have the property that when you square them they are always positive. The Babylonian algorithm or formula wasn’t any help because you had to make sense of the square root of −4.

But in the middle of the sixteenth century something rather strange happened. In 1551 the Italian mathematician Rafael Bombelli was working on a project to drain the marshes of the Chiana Valley which belonged to the Papal States. Everything was going well until an interruption suddenly halted the work. With nothing to do, Bombelli decided to write a book on algebra. He’d become interested in exciting new formulas that had emerged to solve equations in a book he’d read by a fellow Italian, Gerolamo Cardano.

The Babylonians had come up with the formula to solve the quadratic equations. But what about cubic equations like x^3 − 15x − 4 = 0? Several decades earlier a number of mathematicians had announced that they had found formulas to solve these cubics. Rather than publishing in scholarly journals, mathematicians at the time enjoyed sparring with each other in public mathematical face-offs. I have this wonderful image of heading to the local square on a Saturday afternoon to cheer on your local mathematician in the latest battle of the boffins. There was one mathematician whose formula was clearly superior to all the others on the block. The name of this mathematical champion was Niccolo Fontana, better known by his nickname Tartaglia. He was understandably loath to give away the secret to his success but eventually he was persuaded to explain his formula to Cardano on the proviso that Cardano not publish it.

Cardano resisted for several years but eventually couldn’t hold back. The formula appeared in all its glory in his famous book Ars Magna published in 1545. When Bombelli read Cardano’s book and applied the formula to the equation x^3 − 15x − 4 = 0 something rather odd happened. At some point the formula demanded that he take the square root of −121. Bombelli knew how to take the square root of 121. That was simple. The answer was 11. But what was the square root of −121?

It wasn’t the first time that mathematicians had hit this strange need to take the square roots of negative numbers but usually, at this point, they gave up. Cardano had hit the same issue and had abandoned his calculations. There were no such numbers. But Bombelli held his nerve. He kept working with the formula that was in Cardano’s book, just leaving this strange imaginary number in the formula. As if by magic the numbers cancelled each other out and he was left with the answer that x = 4. Sure enough when he fed that back into the equation it worked.

In order to reach the final destination of x = 4, Bombelli had needed to take a journey through this world of imaginary numbers. It was like stepping through some magic mirror to find a strange new land beyond, where a pathway led to another portal back to the land of normal numbers and the destination you wanted. But without stepping into this imaginary world no path existed to a solution. He began to speculate that this wasn’t just some trick but that maybe these numbers on the other side of the mirror really did exist after all. It’s just that mathematicians needed to have the courage to admit them into their world of numbers.

Bombelli’s text led to the discovery of imaginary numbers. The most basic number, the square root of −1, was eventually given a name: i. The i stood for imaginary, a derogatory term coined some years later by the French philosopher and mathematician René Descartes who was far from enamoured by these strange elusive numbers.

And yet Bombelli had revealed their power. In his book he produced a complete analysis of how to manipulate imaginary numbers. If you wanted to solve these cubic equations, you could take a shortcut to the answer provided you were prepared to step through the looking glass into the world of imaginary numbers. Mathematicians eventually began to refer to them as complex numbers, in contrast to the real numbers we all grew up on.

Leibniz was impressed by Bombelli’s persistence, declaring him an outstanding master of the analytical art: ‘Thus we have an engineer, Bombelli, making practical use of complex numbers perhaps because they gave him useful results, while Cardano found the square roots of negative numbers useless. Bombelli is the first to give a treatment of any complex numbers … It is remarkable how thorough he is in his presentation of the laws of calculation of complex numbers.’

For centuries, mathematicians remained very suspicious of these numbers. If you want the square root of 2, although it’s a number whose decimal expansion is infinite, you still felt you could see this number on a ruler. It was somewhere between 1.4 and 1.5. But where was the square root of −1? You couldn’t see that on a ruler. It was my hero Carl Friedrich Gauss who eventually came up with a way to see imaginary numbers.

Before Gauss, the numbers mathematicians had been using were depicted running along a horizontal line with negative numbers going off to the left, positive numbers to the right. Gauss took the inspired decision to head off in a new direction. These new numbers ran vertically up the page. In Gauss’s picture, numbers were no longer one-dimensional but two. This new map proved extremely powerful. Its geometry reflected the algebraic way that these numbers behaved. As I will explain in Chapter 5, a good diagram can be an amazing shortcut to explain complex ideas.

Gauss had discovered his picture of these numbers while proving an extraordinary fact about them. If you take any equation, however complicated, made up of powers of x, not just cubes, these imaginary numbers can always be used to find a solution. You don’t have to make up new numbers. The imaginary numbers were already powerful enough to solve all equations. This great breakthrough that Gauss made is now called the fundamental theorem of algebra.

Gauss’s map became a fantastic shortcut to navigate this strange new world of imaginary numbers, but weirdly Gauss kept his two-dimensional picture a secret. It was later rediscovered independently by two amateur mathematicians: first by a Dane called Caspar Wessel and then a Swiss by the name of Jean Argand. Today the picture is known as the Argand diagram. Credit is rarely fairly attributed.

The French mathematician Paul Painlevé later wrote in his book Analyse des travaux scientifiques:

The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.

As well as being a mathematician, Painlevé also served as prime minister of France. His first stint in 1917 lasted only nine weeks but required him to deal with the impact of the Russian Revolution and the entry of the USA into the First World War, as well as quelling a mutiny in the French army.

Even if I’m not explicitly using complex numbers in my work, I often tap into their philosophy. Shortcuts like these are a bit like the wormholes that science fiction writers like to create to get from one side of the universe to the other. In any setting it’s worth exploring whether a looking glass might be hidden somewhere that will get you to your goal.

In my mathematical research I’m trying to understand all the symmetries that it is possible to construct. But strangely the way I’ve found to tackle this challenge is to create a new object, called a zeta function, that has its origins in a completely different area of mathematics. And yet it’s given me a view on my research that I would never have had if I’d stuck to the world of symmetry. As I will explain in our next pitstop with entrepreneur Brent Hoberman, the arrival of the internet provided a fantastic looking-glass world through which to step and cut out the middle man in many different commercial transactions.

Sometimes that wormhole to help find a way through to a solution can be about simply changing the terrain you are trekking through. When I am stuck on a mathematical problem I often listen to a piece of music or practise the cello – it’s a way to let my mind wander. Frequently when I come back to my desk the view of the problem has strangely altered. Using music, taking myself into a completely different environment, is like being allowed into the world of imaginary numbers and seeing if, as Painlevé suggested, the path here is shorter to my desired destination. It’s worth experimenting with what alternative paths might be out there that can help you to access a sneaky trapdoor to a new way of thinking.

Today the world of imaginary numbers is key to understanding a whole range of concepts that it would be almost impossible to understand without this shortcut through the looking glass. Quantum physics – the physics of the very small – really only makes sense if encoded in these imaginary numbers. Alternating currents in electronics are most easily manipulated if you describe them using the square root of −1. Another striking example of the shortcut that these numbers provides can be found inside the computers that help land planes in airports across the globe.

BA 107 … you are clear for landing

I was lucky enough some years ago to be given access to the air traffic control tower at one of the UK’s major airports. The screens with mini icons of planes dancing around them made it look like a wild computer game. But I quickly became aware that the operators had the lives of many thousands of people in their hands. I was told to be very quiet as I looked on! But when I got a chance to talk to one of the controllers after he’d finished his shift I was absolutely amazed to find that the system they used to land the planes used imaginary numbers to speed up the calculations involved in the radar tracking the aircraft as they came in.

It was the German physicist Heinrich Hertz who first discovered that radio waves were reflected by metal objects. He made the discovery during his experiments in 1877 to prove the existence of the electromagnetic waves and he is honoured in the name of the unit that measures how fast a wave vibrates.

But it was a compatriot of Hertz’s who realised the practical possibilities this scientific discovery offered. Christian Hülsmeyer obtained patents in Germany and Britain for an electromagnetic device that he thought could help a ship detect the presence of other ships if visibility was affected by fog. It is said that he was motivated to create such a device after witnessing the grief of a mother who lost her son in a collision of two ships at sea.

He demonstrated his invention in an experiment he conducted from a bridge across the Rhine on 18 May 1904. The device would pick up the presence of a boat coming down the river once it was within a three-kilometre radius. But his equipment was a discovery ahead of his time, in part because he hadn’t implemented the mathematics to be able to detect how far away the boat was and in which direction. For some years the idea would be one reserved for the science fiction of the likes of Jules Verne. Its real-world implementation would take decades, and a world war.

Who exactly invented radar, which stands for radio detection and ranging, is a tricky question. Its development by different countries was kept secret during the build-up to war because it was clear that it would give any country that successfully developed the idea an edge in detecting incoming planes. But certainly the Scottish physicist Robert Watson-Watt was one of the pioneers of the technology. He’d been asked to comment on rumours of a German death ray based on radio waves. He was quickly dismissive of the idea but it led him to explore what might be possible with the technology. His demonstration of how the mathematics could be combined with radio signals to track incoming planes led to the establishment of a system of radar stations to detect planes approaching London from the North Sea. His radar network is generally credited with giving the Royal Air Force a crucial edge in the Battle of Britain.

If you are tracking an incoming plane, both in war and in peacetime, speed is of the essence. Finding a shortcut to calculating its location based on the radio waves bouncing back off the aircraft would be crucial. The basic calculation involved is one of trigonometry (a shortcut that I will explain in Chapter 4). The shape of the wave forms that are being transmitted and then detected is described by the mathematics of sines and cosines. It turned out that the calculations involved are incredibly tricky and time consuming. But here is where imaginary numbers came to the rescue.

The great eighteenth-century Swiss mathematician Leonhard Euler had discovered that if you feed imaginary numbers into the exponential function – the simple function of raising a number to the power of x, like 2x – the result was rather curious. The output was a combination of wave functions that looks very much like the waves that would be used for radar. The connection is key to what many mathematicians rank as the most beautiful equation in history. Because one instance of this connection between waves and exponential functions produces an equation that links together five of the most important numbers in mathematical history: 0, 1, i (the square root of −1), π = 3.14159 … and e = 2.71828 … (perhaps the second most famous number after π in mathematics that will be introduced in more detail in Chapter 7):

eiπ + 1 = 0

Raise the number e to the power of i times π and magically (or mathematically) if you add 1 to the result everything cancels to give an answer of 0. It is a curious incidence of this connection that imaginary numbers provide between exponential functions and wave functions.

So instead of having to calculate using the complicated mathematics of wave functions, mathematicians realised you could simplify and speed up the calculations by gluing everything together using imaginary numbers. Using these strange numbers meant that the calculations simply became about exponential functions and could be implemented quickly and efficiently. Even with the extraordinary power of modern computers at their fingertips, air-traffic controllers are now exploiting this same shortcut through imaginary numbers to be able to detect and to help land planes at airports around the world. Without it, those planes would have crash-landed before the calculations locating them had been completed.

This is a very graphic illustration of Paul Painlevé’s thesis that ‘between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain’.

Binary and beyond

One of the other shortcuts that computers have exploited on the way to efficient calculating is to use a very economical way of writing numbers. As we have seen, the ten symbols of decimal numbers aren’t the only way to represent numbers. We can choose powers of any number to represent our numbers, not just 10s like in decimal. The Babylonians had symbols all the way from 0 to 59 and worked to a base of sixty. The Mayans had symbols for the numbers from 0 to 19, creating a number system using powers of twenty. The choice of 10 for our numbers was purely a quirk of our anatomy, given we have 10 fingers.

The Babylonian system could also relate to our anatomy. Our fingers have three knuckles. You can therefore use your thumb on your right hand to point at 1 of 12 knuckles on that hand. Once you have counted a batch of 12 knuckles you count or record this with the left hand, beginning a new count of 12 on the right. Given that you have 5 fingers on your left hand you can record 5 lots of 12 knuckles, giving you 60!

To represent the number 29 you would raise two fingers on your left hand and use the thumb on your right to point to the fifth knuckle (the middle knuckle of your second finger).

But a computer is limited to just one finger. Essentially they work on the principle of switches being on or off. They need a system that uses just two symbols: 0 for off and 1 for on. Using only these two symbols the computer can still represent every number. Instead of powers of ten, the places in the place-value system represent powers of two, using what is known as the binary number system. So the number 11011 represents

1 × 2^4 + 1 × 2^3 + 0 × 2^2 + 1 × 2 + 1 = 27

Given that we have found ways to translate conversations, pictures, music and books into digital, this shortcut using binary has translated the world around us into strings of 0s and 1s.

This idea of binary is also the key to cracking the puzzle that opened this chapter. How few weights can the grocer get away with to measure weights from 1 kg to 40 kg? The trick is not to think in binary but ternary or powers of three. The scales allow three settings: a weight on the right (+1), a weight on the left (−1) or no weight (0). By thinking in ternary it is possible to show that the grocer needs only four weights – 1 kg, 3 kg, 9 kg and 27 kg – to measure every possible weight between 1 kg and 40 kg.

For example, to measure a sack weighing 16 kg you need to put the sack in one pan along with the 3 kg and 9 kg weights. This will be balanced exactly by putting the 1 kg and 27 kg weight in the other pan. Instead of using 0, 1 and 2 to represent numbers, you are using the symbols −1, 0 and 1. So 16 is represented by

1(−1)(−1)1

which represents 1 unit, minus one lot of 3, minus one lot of 9, plus one lot of 27, making 27 − 9 − 3 + 1 = 16.

Whether it is numbers or some other complex idea, finding the best notation for representing the concept can be the shortcut to navigating your way through to a solution. The grocer who can think in ternary can get away with purchasing just four weights to do the job. The competitor who hasn’t understood this shortcut will find themselves wasting resources on unnecessary weights.

Shortcut to the shortcut

Finding good shorthand to represent complex concepts has been a crucial shortcut throughout history, not just when it comes to recording numbers. If you are taking notes during a lecture or meeting, you’ve probably already started to create shortcuts for key ideas that keep recurring. But might there be a better way to notate your ideas that make them easier to manipulate? Sometimes data in one form can be unilluminating, but change the way you are recording it and new insights emerge. Logarithmic graphs often tell one more about data than the original numbers, which is why for example earthquakes are measured using the logarithmic Richter scale. And keep an eye out for a looking glass, like those imaginary numbers, that might take you out of the world you’re stuck inside and give you an alternative world to access a shortcut to your destination.

Pitstop: Start-up

‘I used to say to my marketing directors, you’re really succeeding if you get arrested. None of them managed to do it.’

This is what Brent Hoberman, who founded the start-up incubator Founders Factory, told me during a recent visit. Hoberman (who, it must be said, has not yet been arrested) credits pushing the law to its limits for the success of his most famous venture lastminute.com, which he cofounded with Martha Lane Fox in 1998. Breaking the rules of the game is part of what Hoberman considers the ‘entrepreneurial mindset’ and is his shortcut to a successful business venture.

There is a wonderful sense of playfulness about the offices of the Founders Factory. Whiteboards covered in mad scribblings decorate the walls, not so dissimilar to the whiteboards found in maths departments all over the world. The open-plan nature of the space means that different start-ups are rubbing shoulders with one another, sharing ideas. Food, drink and games are available to stimulate ideas. But it is breaking the rules of the game that Hoberman believes is the best shortcut to the success of the ventures that are being cooked up in the Factory.

‘Many entrepreneurs historically broke the rules and then asked for forgiveness later,’ says Hoberman. ‘That was Uber’s journey and Airbnb’s journey. They’re both breaking the law. Why shouldn’t people not be able to rent out their own houses? And then society will look at that and say, actually yeah, why not? That was their shortcut.’

Breaking the law is a strategy that has served quite a few mathematicians well. The mathematical laws stated that if you square a number, it must be positive. But Rafael Bombelli had the nerve to start working with a number whose square is −1. By stepping outside the rules of the game, you get access to a whole slew of interesting new mathematics. The ancient Greek mathematician Euclid states that triangles have angles that add up to 180. But as we will see later on, mathematicians came up with new geometries where triangles break the Euclidean law. The key to breaking a law is that the benefits are worth the fracture.

As Brent Hoberman explained to me: ‘It’s almost about redefining what this thing is. The regulations may be outdated. It may be that regulations are so slow. Sometimes people are maybe dangerously redefining their own moral compass and saying the trade-off is worth it for society.’

The key to the success of lastminute.com is that it exploited the unused inventory of airlines, car rental companies and hotels, and then created bundles that were cheaper than buying à la carte. That idea first occurred to Hoberman in his student days when he was looking to treat his girlfriend to a fun weekend away. He would phone up hotels at the last minute asking how many suites they had left for the following night. If they said five or six, he knew they were unlikely to sell them all so he’d offer to take one at 70 per cent discount. ‘One out of three times it worked.’

He began to wonder why everyone wasn’t doing this. ‘They were all too British. The Brits don’t do that,’ he jokes. He remembers banking the experience as a student and realising that it could be done on an industrial scale. That was the genesis of lastminute.com. But discovering the unused inventory on an industrial scale required sailing close to the edges of the law. As Hoberman admits, lastminute.com technically broke the Computer Misuse of Information Act, which was potentially criminal.

But pushing legal limits is a shortcut that many start-ups have employed to get an edge over their competitors. Facebook became famous for its mantra ‘Move fast and break things’. As its CEO Mark Zuckerberg once said, ‘Unless you are breaking stuff, you aren’t moving fast enough.’ Richard Branson credits his early brush with the law in the 1970s for having sparked his business success, although in Branson’s case it was because he needed to pay back £60,000 for the tax fraud that he had committed in his early days selling records. The fine prompted Branson’s much more systematic approach to making money. ‘Incentives come in all shapes and sizes,’ he wrote, ‘but avoiding prison was the most persuasive incentive I’ve ever had.’

But as start-ups look towards disrupting highly regulated industries like health, it becomes more difficult to justify moving fast and breaking things. The health industry works under strict regulatory conditions for obvious reasons. To build trust in your idea requires working within these. The ethos of ‘do no harm’ outweighs the desire to disrupt. You don’t want to be breaking a patient on your way to a successful exit.

One of the other reasons for Hoberman’s success was that he took advantage of the amazing shortcut that the internet provided in those early days of the dot-com boom. Time and again it allowed one to cut out the middle men. In the case of lastminute.com it was the travel agent. Another of Hoberman’s ventures, made.com, exploited a similar shortcut. The site’s idea was to give consumers access to designer furniture without having to pay designer prices. Hoberman’s co-founder Ning Li had had his eye on a £3000 sofa, but he coincidentally discovered a friend from school had taken charge of the factory that manufactured the sofa. They were making it for £250. It sparked the idea of connecting consumers with manufacturers and cutting out the costly middle man. As Ning says, ‘There is an elitist mentality in the furniture industry that only customers that can afford £3000 are entitled to have a trendy, well-made sofa. But there shouldn’t be any reason for that.’ The internet allowed the company to shortcut the supply chain.

Hoberman also recognises another important shortcut when it came to setting up companies like lastminute.com and made.com: ‘Ignorance. I would never have started lastminute.com if I knew how hard it was going to be. You don’t want to know too much. Ignorance helps you think differently.’

Hoberman’s philosophy reminds me of a character in one of my favourite operas. In Wagner’s Ring cycle the young Siegfried, who does not know fear, successfully kills the dragon Fafnir and takes the ring it was guarding. He eventually learns what fear is when he meets a woman for the first time!

Not knowing fear is I believe one of the reasons that the young are perhaps so successful at cracking big unsolved mathematical challenges. Many of us learn to fear a mathematical beast like the Riemann hypothesis, our great unresolved problem about prime numbers, such that we think it crazy to try to tackle such a difficult problem. If generations of mathematicians have failed, what can I offer? The dragon remains unslain. You need a bit of ignorance mixed with a bit of arrogance: not to be frightened by the problem’s history and to have the self belief that, why couldn’t it be me who cracks this great unsolved mystery?

Hoberman also believes that perfectionism can be another killer for success. This has been Amazon’s philosophy: don’t create a glittering palace and go to the consumer ‘ta-da!’, but get the basic castle built and let the consumer move in and tell you what is needed to improve it. If your product is 70 per cent ready for launch, launch it and correct things on the fly. If you wait for it to be 99 per cent ready, it will be too late. This philosophy does have its limits. Once other companies began relying on Facebook’s platform, for example, it became costlier to just let things crash. Companies might stop using your platform if it is too unreliable. In 2014, Zuckerberg introduced a new philosophy: ‘Move fast with stable infrastructure.’ ‘It may not be quite as catchy as “move fast and break things”,’ Zuckerberg said with a smirk. ‘But it’s how we operate now.’

Perfectionism is considered essential when it comes to mathematics. Most mathematicians believe there is no point even publishing a proof that is 99 per cent complete because that last 1 per cent can be deadly. But perhaps we mathematicians are too obsessed with perfectionism. Maybe it is worth sharing ideas that aren’t complete rather than sitting on them and keeping them to yourself. Isaac Newton and to a certain extent Carl Friedrich Gauss held back progress because they were nervous of sharing incomplete and potentially heretical ideas.

Changing this ethos in the scientific research community is at the heart of the Chan Zuckerberg Initiative set up by the Facebook founder with his wife Dr Priscilla Chan. The whole point of CZI is to foster better networks between different research groups, which they believe could solve some of the medical challenges that are currently being held back by a fear of sharing research in progress.

Brent Hoberman has gone on to become a big investor in new start-ups but he still thinks that perfectionism is dangerous when it comes to knowing which companies to back.

‘Instinct I think is another shortcut,’ he says. ‘We take shortcuts when we invest in companies. Our best decisions are probably made after a five- or ten-minute meeting. Johannes Reck of GetYourGuide is now a business worth well over a billion. I met him and after ten minutes I said to my colleagues, “You have to come in and meet him tonight.” Because this guy’s got something special. Similarly with alan.eu, a successful health company in France. I could tell the guy behind it was a genius. I don’t need more. Many of my best friends that I tried to bring into that company over-analysed it.’

It is clear from our conversation that Hoberman is a fan of exploiting whatever shortcut will get him to another successful exit.

‘I think shortcuts are brilliant and I chastise my kids when they don’t think of shortcuts,’ he says. ‘Often you’ll see people queuing. There are three queues and everyone just stands in the first one. If you moved to the third one, three metres further away, you’d save ten minutes. But people don’t do that. People aren’t thinking, How do you work out how to get to the front of the queue or find another queue or launch another one? Life is like a series of those sorts of decisions and you should always be trying to look for the shortcut.’

3

The Language Shortcut

During the festive season one of the songs I enjoy singing is ‘The Twelve Days of Christmas’. ‘On the first day of Christmas my true love sent to me … a partridge in a pear tree.’ On each subsequent day you get the presents of the previous day plus an extra number of presents:

First day: 1 partridge

Second day: 1 partridge + 2 turtle doves

Third day: 1 partridge + 2 turtle doves + 3 French hens

Etc.

So by the twelfth day of Christmas, how many presents in total had my true love sent to me?

One of the most powerful shortcuts I’ve discovered in my time as a mathematician is that of finding the right language to talk about a problem. Very often the issue will have been couched in a language that obscures what is going on. By finding an alternative way of putting it and translating the puzzle into a new idiom, the solution suddenly becomes much clearer. A change of language can help to pick out strange correlations in a company’s sales data that the numbers are obscuring. Much of life is a game, but translating the game into one you know how to win can give you an amazing edge. And one of the most exciting revelations of my apprenticeship as a mathematician was discovering how a dictionary that changes geometry into numbers can provide a shortcut to hyperspace – a multi-dimensional universe that I have been exploring ever since as a professional mathematician.

There are an increasing number of concepts in science and beyond that don’t even seem to exist unless you find the right language to describe them. The notion of emergent phenomena, that qualities emerge from constituent parts, is one such example. Wetness of water, for example, is hard to capture if you are talking about individual molecules of H_2O. Although science seems to imply that you can reduce everything to the behaviour of such fundamental particles and the equations that determine their behaviour, this language is often totally inadequate to describe phenomena. The migration of a flock of birds can’t be captured by equations for the movement of the atoms that make up the birds. Macroeconomics is rarely comprehensible if you stick to the language of microeconomics. To understand the effect of an increase in interest rates on inflation is not possible in the language of the individual goods themselves even if the microeconomic changes are the cause of the macro phenomenon. Even our ideas of free will and consciousness cannot really be captured by talking about neurons and synapses.

Finding a different language to talk about your emotional state can fundamentally change the way you feel. Instead of saying ‘I am sad’, which seems to identify you in a rigid formula equating you with sadness, you could change this to ‘Sadness is with me’ and suddenly there is the chance for the sadness to move on. As the nineteenth-century American psychologist William James wrote: ‘The greatest discovery of my generation is that human beings can alter their lives by altering their attitudes of mind.’ But the power of language doesn’t only affect the personal. Language plays a crucial role in the social construction of reality. A society can make things appear by naming them. The concept of a nation state is as much conjured out of language as it is by geography or a collection of people.

Changing language sometimes means you can express ideas that are elusive in one language but somehow are possible to articulate in another. The fact that German nouns have gender allows one to play games with language that just don’t work in English. The poet Heinrich Heine wrote of the love of a snowy pine tree for a sunburned Oriental palm. In German a pine tree is masculine and a palm tree feminine but this nuance is lost in translation to English. Sometimes things get lost the other way. In English you can talk about ‘his car and her car’ but Google Translate will mash this up in French to read ‘sa voiture et sa voiture’ because the car’s gender overrides the gender of the owners. Russian has a different word for every possible type of snow and rainstorm imaginable. Some languages have only five words for colours while English has many. As I’ve stressed, pattern is an important concept for me, yet when I try to translate the word into French it turns out that there isn’t a word that captures the many facets that it embodies in English.

The significance of differences between languages also fascinated my hero Carl Friedrich Gauss. At school his teachers had been very impressed by his command of Latin and his lightning mastery of the classics. Indeed, rather than mathematics, Gauss almost chose philology – the study of the history of languages – as the course he embarked on, funded by the Duke of Brunswick.

My own journey to become a mathematician followed a not so dissimilar course. When I was younger I wanted to be a spy and thought languages would be an important skill to communicate with my fellow agents across the world. At school I signed up for all the languages my comprehensive taught: French, German, Latin. I even started following a Russian course that was being taught on the BBC. But I was not as successful as Gauss at picking up these new languages. It was all irregular verbs and strange spellings. I became very despondent as my dream for a career in espionage slipped away.

It was when my teacher, Mr Bailson, gave me a copy of a book called The Language of Mathematics that I began to understand that mathematics was also a language. I think he saw that I was craving a language without irregular verbs where everything made perfect sense, but he also realised that I wouldn’t be able to resist how potent this language was at describing the world around me. In this book I discovered that mathematical equations could tell the story of the planets as they traversed the night sky, that symmetry could explain the shape of the bubble, beehive or flower, that number was key to musical harmony. If you wanted to describe the universe, it wasn’t German or Russian or English you needed, but mathematics. The Language of Mathematics also taught me that mathematics is not just one language but many and that it is very good at creating dictionaries that change one into another so that in the new language shortcuts appear.

The history of mathematics has been punctuated by brilliant moments such as this.

Mathematical grammar

Implicit in the explanation of many of the patterns that I’ve shown you so far is an amazing mathematical shortcut: algebra. The trick of algebra is to move from the particular to the general. It means that I don’t keep on having to carve out a new path every time I consider a different case. Instead of taking each particular number in turn, I can let the letter x stand in to mean any number.

Let me do a little trick with you: think of a number. Double the number. Add 14. Divide the number you’ve got by 2. Subtract the number you started with. I’ll guarantee that the number you’re now thinking about is 7. We used this little trick at the beginning of a play that I advised on called ‘A Disappearing Number’, about the collaboration between Indian mathematician Srinivasa Ramanujan and Cambridge mathematician G. H. Hardy. It always amazed me the gasp that this trick elicited each night from the audience, as if we’d magically read their minds. What had happened, of course, was not magic but mathematics. The key to understanding how you’ve been mathematically manipulated is the idea of algebra.

Algebra is the grammar that underlies the way that numbers work. A bit like a code for running a program, algebra will work whatever numbers you feed into the program.

Algebra was developed by the director of the House of Wisdom in Baghdad, a man called Muhammad ibn Musa al-Khwarizmi. Founded in 810, the House of Wisdom was the foremost intellectual centre of its time and attracted scholars from around the world to study astronomy, medicine, chemistry, zoology, geography, alchemy, astrology and mathematics. The Muslim scholars collected and translated many ancient texts, effectively saving them for posterity. Without their intervention, we may never have known about the ancient cultures of Greece, Egypt, Babylonia and India. However, the scholars at the House of Wisdom weren’t content with translating other people’s mathematics. They wanted to create a mathematics of their own. It was this desire for new knowledge that led to the creation of the language of algebra.

You can probably spot algebraic patterns on your own, even if you don’t know you’re doing algebra. As a kid learning my multiplication tables I started to spot some curious patterns that are hiding underneath these calculations. For example, ask yourself what 5 × 5 is. Then look at 4 × 6. What’s the connection between the two answers? Now take 6 × 6 followed by 5 × 7. Then 7 × 7 followed by 6 × 8. Hopefully you’ve spotted that the second answer is always 1 less than the first.

For me, spotting patterns like these turned the tedium of learning times tables into something slightly more interesting. These patterns helped me to shortcut the rote learning that was often expected. But does this pattern always persist? If I take a number and square it, will it always be one more than taking the numbers either side and multiplying them together?

I’ve used words to try to describe this pattern but in the ninth century in Iraq the new mathematical language of algebra was created that could do it more clearly. Let x be any number. Then, if you square x, it will be 1 more than (x − 1) times (x + 1). Or written as an algebraic formula:

x^2 = (x − 1)(x + 1) + 1

This algebraic language also allowed mathematicians to show why this pattern will persist whatever numbers you choose. Expand (x − 1)(x + 1) and you get x^2 − x + x − 1 = x^2 − 1. Add 1 to this and you’ve just got x^2.

The same approach of letting x stand for any choice of number is the key to the simple magic trick that got you to the number 7. The trick is to translate the instructions into algebra.

Think of a number: x.

Double it: 2x

Add 14: 2x + 14

Divide by 2: x + 7

Subtract the number you first thought of: x + 7 − x = 7

And you’re now thinking of the number 7.

The point is that it works whatever number you first thought of, even if you were being clever and thinking of an imaginary number! Here’s another trick I learned from a mathematical magician friend of mine, Arthur Benjamin. Algebra is key to understanding why the trick works. Throw two dice. Multiply the two numbers you get. Multiply the numbers on the bottom of each die. Then multiply the top of die 1 by the bottom of die 2. Then multiply the bottom of die 1 by the top of die 2. Finally add up the four numbers you’ve got. The answer is always 49. What Benjamin has exploited here is the nice fact that the top and bottom numbers on a die always add up to 7. Combine this with some algebra and you always get the answer 49, which is 7 squared.

x × y + (7 − x) × (7 − y) + x × (7 − y) + (7 − x) × y = 7 × 7 = 49

But it wasn’t only magic tricks that algebra allowed. It unleashed a huge wave of new discovery. Instead of having just words, mathematicians now understood the grammar that enabled us to put those words together. It gave us the language to describe how the universe was working.

As Leibniz said of the power of algebra: ‘This method spares the work of the mind and the imagination, in which we must economise above all. It enables us to reason with small cost in effort, by using letters in place of things in order to lighten the load of the imagination.’

Lighting the dark labyrinth

One person who realised the power of this language to decode nature was the sixteenth-century Italian scientist Galileo Galilei. He once famously wrote: ‘The Universe cannot be understood without first learning to comprehend the language and know the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to understand a word; without these one is wandering in a dark labyrinth.’

One of the stories of the universe that he wanted to be able to read was the challenge of understanding how objects fell to Earth. Was there a rule to the way things dropped to the ground or flew through the air? Gathering data from an object dropped from a high building was tricky, as the objects generally fell too fast. Galileo had a clever idea to slow down the experiment so that he could gather the data he needed. Instead of dropping things, he would examine the way a ball rolled down a hill. This was slow enough for him to be able to record how far the ball had rolled after each second.

The incline needed to be smooth enough that the ball wouldn’t be slowed by friction. Galileo wanted to approximate as closely as possible a ball falling through space. Once he’d rigged up a smooth surface and started recording the distance the ball would travel each second, he discovered a very simple pattern emerging. If after one second the ball had travelled 1 unit of distance, in the next second it covered 3 units of distance. Then 5 units of distance in the following second. Each subsequent second the ball was gaining in speed and covering more ground but the distance it covered was simply following the odd numbers.

When Galileo considered the total distance travelled over a period of time, the secret of the way things fall to Earth revealed itself.

Total distance travelled after 1 second = 1 unit

Total distance travelled after 2 seconds = 1 + 3 units = 4 units

Total distance travelled after 3 seconds = 1 + 3 + 5 units = 9 units

Total distance travelled after 4 seconds = 1 + 3 + 5 + 7 units = 16 units

Have you spotted the pattern? The total distance is always a square number. But why do odd numbers have anything to do with square numbers? We can find out how by turning numbers into geometry.

3.1 Connecting square numbers and odd numbers.

To grow bigger and bigger squares, I’ve got to wrap the next odd number in the sequence around the previous square. Suddenly the connection between squares and odd numbers is obvious. This way of looking at things geometrically rather than arithmetically is a powerful shortcut.

Galileo was now able to cook up a formula to express the total distance that a ball would fall to Earth: after t seconds, the total distance covered was proportional to the square of t. The fundamental square law of gravity had revealed itself. The discovery of this equation ultimately led to our ability to calculate where a ball would land once it was fired from a cannon and to predict the trajectory of the planets as they orbited the Sun.

On the nth day of Christmas

The trick of using a clever geometric way to show the connection between odd numbers and square numbers can also be used as a shortcut to solving this chapter’s puzzle. To work out how many presents I get from my true love over Christmas, I could go the long route by adding up lots of turtle doves and lords a-leaping. But the shortcut is to change the problem from arithmetic into geometry. Let me start by showing how a geometric perspective can help to get a hold on the number of presents I get each day. The daily count is simply the triangular numbers that we encountered in the pattern chapter. I’ve already explained how Gauss cracked these numbers by pairing them up.

But there is another way to shortcut the hard work, which is to view the challenge geometrically. Arrange the presents in a triangle with the partridge at the top. Counting presents making up a triangle is a little tricky. What if I stick two triangles together? Then I get a shape that is a rectangle. But counting things in a rectangle is easy: it’s just base times the height. The triangle is half this.

This geometric shortcut to the solution is essentially Gauss’s trick of pairing numbers dressed up in a slightly different guise. But the geometric perspective allows me to create a simple formula to calculate any number in this sequence. If I want the nth triangular number I put two triangles of presents together which create a rectangle of dimensions n × (n + 1). Now just divide by 2 to get the number of presents in the triangle: ½ × n × (n + 1).

But what is the total number of presents I get after each day? Here is a running total starting on the first day:

1, 4, 10, 20, 35, 56 …

You get the next number by adding the triangular numbers up in sequence. So to get the seventh, add the seventh triangular number on to the previous number. The seventh triangular number is 28 so the seventh number in this sequence is 56 + 28 = 84. But is there a clever shortcut to get to the twelfth number, the total number of presents over the whole of Christmas, without having to add triangular numbers in sequence?

The trick again is to change numbers into geometry. Think of all the presents coming in boxes of equal size, then instead of a triangle I can stack the boxes I’ve received into a triangular-based pyramid. At the top is a box with a single partridge in a pear tree inside. The next layer down has three boxes: one containing a partridge, the other two holding turtle doves. Each time the day’s new presents arrive I add them to the bottom of the pyramid. Is there any way to understand how many boxes are in the pyramid now that I’ve changed the number into a shape?

Remarkably there is. Just as I put two triangles together to make a rectangle, it is possible to put 6 pyramids of the same size together to make a rectangular stack of boxes. (To make this work you have to do a little shifting around of the way the presents are stacked in each pyramid.) If the pyramid has n layers, the rectangular structure has dimensions n × (n + 1) × (n + 2). But I’ve got 6 pyramids making up the structure. The formula for the number of presents in each individual pyramid is

1/6 × n × (n + 1) × (n + 2)

So how many presents did I get from my true love by the twelfth day of Christmas? Put n = 12 into this formula and you get 1/6 × 12 × 13 × 14 = 364. That’s a present for every day of the year except one!

3.2 Six pyramids make a cuboid.

Descartes’s dictionary

I’ve always loved the way that a picture can make you see something that the numbers obscured. But one needs to be careful. There are times when the eye deceives. Take the following picture.

3.3 Rearrange the pieces to create an extra square.

It looks like I’ve rearranged the pieces that make up the square into a nice rectangle. But hold on. The area of the square is 64 whereas the rectangle has an area of 65. Where did the extra bit come from? What the picture makes it hard to see is that the diagonal across the second shape is not truly straight. The edges on the pieces don’t line up exactly, leaving just enough room to accommodate one more unit square. As Descartes famously said: ‘Sense perceptions are sense deceptions.’ After I’d seen this trick I think I didn’t really trust my eyes ever again. I’m only really happy if I can truly explain a pattern or connection through the language of algebra. What if a similar sneaky trick is going on with those odd numbers I wrapped round the squares?

To reveal these kinds of visual tricks it can be useful to reverse the shortcut and turn geometry into numbers. Descartes was one of the mathematicians to come up with the idea of a dictionary that translates between numbers and geometry. This dictionary was one of the other great linguistic discoveries, in addition to algebra, that allowed one to find shortcuts to navigating the universe.

We are in fact all very accustomed to using this dictionary when we see a map or GPS. Using a grid that has been layered across a city or country means that I can identify any point in the landscape: two numbers will pinpoint where on the grid the point is. GPS uses a grid in which the horizontal axis is the Equator and the vertical axis is the line of longitude running through Greenwich.

For example, if I want to visit the house where Descartes was born, located in a town called Descartes (so named after his death, rather than an extraordinary coincidence), the following GPS coordinates will get me there: latitude 46.9726497 and longitude 0.7000201. Every position on the planet can be turned into two such numbers. The geometry of the planet translated into two numbers.

In his book La Géometrie Descartes introduced this powerful idea of using coordinates to describe geometry. Now called Cartesian coordinates in honour of the man who suggested this translation, these numbers could be used to locate geometric positions not just on the surface of the planet but in any image. Descartes’s dictionary unleashed a way to translate between geometry and algebra.

The power of this translation comes to the fore when I want to describe how something might be moving through space. Throw a ball and I can use two numbers to describe the ball’s height above the ground when it is a given distance from the person who threw it. There is a mathematical equation that binds these two numbers together. Suppose x is the distance that the ball has travelled horizontally. Let v be the ball’s speed in the vertical direction when I launched it and u the horizontal speed. If y is the height above the ground, these ingredients will give a formula for that height:

y = (v/u)x − (g/2u^2)x^2

The letter g stands for a number that is called the gravitational constant. It determines on each planet how hard the ball gets pulled down to the ground by the effect of gravity.

However fast or high one launches the ball, the equation works the same. One has only to change the values of u and v, which are like dials I can twiddle to change the shape of the trajectory. Spotting this pattern that underlies the way all balls fly through the air gives one access to predictions about where the ball will land. The equation is a quadratic equation in x. If you are a football player and you want to know where to stand in order for an incoming ball to land on your head to knock it into the back of the opponent’s net, you will need to know how to solve this equation for x. As I explained in the last chapter, the ancient Babylonians found an algorithm for it two thousand years ago.

But it isn’t just the trajectory of balls that these quadratic equations describe. If you look at the price of goods as supply and demand vary, quite often this can be described by the same sort of equations. Knowing how to find the equilibrium point in economics, when a good is priced at the point where supply equals demand, becomes possible once the numbers are described by the equations. The company that fails to use the language of equations to map its data will, as Galileo said, be wandering around in a dark labyrinth while their competitor is raking in the profits.

If you have a set of data points, it is useful to look for the equation that might bind them together. Uncover that and you potentially have an amazing shortcut to predicting what might happen next.

It is extraordinary how universal these patterns can be. In the case of throwing a ball, it doesn’t matter who throws the ball, how they throw it, or where they throw it. Change the ball and the equation still has the same general form.

But one has to be careful with fitting equations to data. If you take the population numbers for the United States for the last century, they can be nicely approximated by a quadratic equation like the one used to trace the trajectory of a ball. But if you use a more complicated equation with powers of x up to x^10, you can get an exact fit to the data. That would seem to encourage one to believe the more complex formula to be a better predictor. The only trouble is that this equation predicts that the population of the United States will plummet to zero in the middle of October 2028. Or maybe the equation really knows something we don’t.

This story acts as a word of caution to those who think that we can do science just by exploiting the power of big data. Data can suggest patterns but we still need to combine this with analytic thought to see why the pattern should be dictated by a particular equation. Galileo’s discovery of a quadratic rule behind gravity was subsequently explained thanks to Isaac Newton’s theoretical analysis, which revealed why quadratics were the right equations to use.

The shortcut to hyperspace

The idea of turning geometry into numbers not only allows us to navigate our three-dimensional universe more efficiently. It also provides that portal to worlds we’ll never see. One of the most exciting moments in my mathematical journey through the art of the shortcut was the discovery that I could study hyper-dimensional space. The day that I first read about the power of this language to make a cube in four dimensions is still etched on my memory.

This explained how a spaceship could get from one end of the universe to the other by taking a shortcut through the fourth dimension. It solved the challenge of how the universe can be finite but without walls. It even allowed one to unravel knots that are impossible to untangle in three dimensions.

But the dictionary enables more than just travelling in space. By mapping data into higher-dimensional worlds, hidden structure appears. When you draw a graph of data, you are looking at a two-dimensional shadow of something that should really be plotted in hyper-dimensional space. Such a shortcut could well reveal subtleties that these 2D shadows are obscuring. So buckle up as I take you on our journey into hyperspace.

To get to the fourth dimension you need to start in the second. Let’s say I want to describe a square in terms of Descartes’s dictionary of coordinates: I can say that it is a shape with four vertices located at the points (0,0), (1,0), (0,1) and (1,1). You can see that the flat two-dimensional world needs just two coordinates to locate each position, but if I also want to include the height above sea-level I could add a third coordinate. I will need this third coordinate too, if I want to describe a three-dimensional cube using coordinates. A cube’s eight vertices can be described by the points (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1) and finally the extremal point at (1,1,1).

3.4 Making a hypercube using coordinates.

On one side, Descartes’s dictionary has shapes and geometry and on the other it has numbers and coordinates. The trouble is that the visual side runs out of options if I try to go beyond three-dimensional shapes, since there isn’t a fourth physical dimension. It was the great nineteenth-century German mathematician Bernhard Riemann, a student of Gauss’s in Göttingen, who recognised this beautiful fact about Descartes’s dictionary: the other side of the dictionary keeps going.

To describe a four-dimensional object I just add a fourth coordinate to keep track of how far you are moving in this new direction. Although I can never physically build a four-dimensional cube, I can still describe it precisely by using numbers. It has 16 vertices starting at (0,0,0,0), then extending to points at (1,0,0,0), (0,1,0,0) and stretching out to the furthest point at (1,1,1,1). The numbers are a code to describe the shape. And using this code I can explore the shape without ever having to physically see it.

And it doesn’t stop there. You can move into five, six or even higher dimensions and build hypercubes in these worlds. For example, a hypercube in N dimensions will have 2N vertices. Out of each of these vertices there will be N edges emerging, each of which I am counting twice. So the N-dimensional cube has N × 2N − 1 edges.

The taste I got of that four-dimensional cube whetted my appetite for discovering more shapes in this strange multi-dimensional universe. It became my passion to carve out new symmetrical objects there. If, for instance, you have ever visited the beautiful Alhambra Palace in Granada, you’ll have (hopefully) enjoyed the wonderful symmetrical games that the artists played on its walls. But is it possible to understand these symmetries? The shortcut for me to understanding at first sight something that appears very visual is to turn symmetry into language.

The creation of a new language for understanding symmetry, called group theory, emerged at the beginning of the nineteenth century. And it was the brainchild of an extraordinary young man: the French revolutionary Évariste Galois. His life, alas, was cut tragically short before he could fully realise the potential of his discovery. He was shot dead, aged twenty, in a duel over love and politics.

Although two walls in the Alhambra may be adorned with very different pictures, the mathematics of symmetry is able to articulate that these two walls have identical symmetries. This is the power of Galois’s new language.

Symmetry can be described as the actions that I do to an object which leave it looking like it did before I took any action. What Galois understood was that the essential characteristic of symmetry was the interaction between individual symmetries. That if you gave the symmetries names there was a kind of underlying grammar to all of them. This grammar was the shortcut to unlocking their world. The pictures would disappear and in their place appeared a kind of algebra that expressed the way that symmetries interact.

With group theory, mathematicians at the end of the nineteenth century were able to prove that there are only 17 different symmetrical types of design that it is possible to draw on the walls of the Alhambra or anywhere else. My own research continues this journey into hyperspace. I am trying to understand the number of ways I might tile an Alhambra in multi-dimensional space. It’s a building made from language not bricks.

It is possible to glimpse these surreal shapes in our mundane three-dimensional world. La Grande Arche at La Défense in Paris, built by the Danish architect Johan Otto von Spreckelsen, is actually a shadow of a four-dimensional cube, a cube inside a cube. Salvador Dalí in his painting Crucifixion (Corpus Hypercubus) depicts Christ crucified on the three-dimensional net of a four-dimensional cube.

There is even a computer game that promises to give players an experience of living in a four-dimensional universe. Called Miegakure, it is the brainchild of designer Marc ten Bosch who has been making this hyper game for over a decade. Players faced with a wall that seems to prevent them advancing in the 3D environment on the screen can turn towards a fourth dimension and by moving in this new direction find a parallel world that offers a shortcut round the wall. The game sounds extraordinary and I can’t wait for its release but I suspect part of the long delay in its progress has been down to the sheer complexity of being a 3D-minded developer trying to weave together all these 4D worlds.

Winning at games

I’m a big fan of games, not just 4D crazy games. I love collecting games on my travels around the world. But it always amazes me how, despite games from different corners of the globe looking very unlike one another, often they are the same game dressed up in a different costume. Which led me to realise that many games become much simpler to play if you can change them into another, apparently different game.

Many of life’s challenges are basically games in disguise. A potential cooperation between two rival companies can frequently turn out to be an example of a game called the prisoner’s dilemma. A three-way rivalry can hide a game of rock paper scissors. If you’ve watched the film A Beautiful Mind, you might remember the moment when one of the inventors of game theory, John Nash, played by Russell Crowe, turns the challenge of getting the beautiful woman in a bar into a game. But games have rules that mathematics is very good at navigating. One of the great shortcuts that mathematics has discovered for winning a game is to turn it into something completely different, at which point a strategy for winning becomes much more transparent.

One of my favourite examples is a game called 15. Each player must choose numbers in turn from 1 to 9 with the aim of getting three numbers that add to 15. Once a number is taken the other player can’t use it. You have to get 15 with exactly three numbers. For example 1 + 9 + 5. You can’t pick 6 + 9. It’s quite a tricky game to play because you have to keep track of the different ways of getting 15 with your numbers and also to stop your opponent getting to 15 before you. It’s worth playing a round with a friend to see how hard it can be to keep abreast of the different possibilities.

The shortcut to this game is to turn it into a completely different game that is easy to play: noughts and crosses, or tic-tac-toe. Except you are going to play the game on a magic square.

The magic square has the property that the sum of the numbers in every row, column or diagonal is 15. If you play noughts and crosses on this square, you are in fact playing the game of 15. But the geometry of the game of noughts and crosses is much simpler to keep track of than the arithmetic of adding numbers up to 15.

Overleaf is another game that becomes easy to play once you spot the right way to look at it. Consider the following map of cities connected by roads. The roads consist of all the straight lines in the map (and so can have 2, 3 or 4 cities on them).

3.5 Network of roads.

You take it in turn to claim roads. The first person to claim three roads through one city wins. Again it’s worth playing to get a feel for any possible strategies. But this is in fact noughts and crosses again in disguise. If you label the roads with the following numbers, you are once again playing tic-tac-toe on the magic square.

3.6 Network of roads labelled with a magic square.

Another classic game that becomes transparent to play once translated into another language is the game of Nim. There are three piles of beans. Players take it in turns to remove any number of beans from one pile. The winner is the last person to take a bean. You can start with any number of beans in the three piles.

Suppose for example that the three piles consist of 4, 5 and 6 beans. Is there a strategy to help you win the game? The trick is to turn the numbers of beans in each pile into binary. Recall from the previous chapter binary numbers are built from powers of 2 rather than the powers of 10 used in decimal numbers. So in binary 100 represents 4 because it is one lot of 2^2. Similarly 5 = 2^2 + 1 is 101 in binary and 6 = 2^2 + 2 is 110. Now there is a strange rule for adding these numbers together which is going to help you to see if you are in a winning position or not. You add the numbers in columns but use the rule that 1 + 1 = 0. So

100

101

110

——

111

Your strategy must be to remove beans from one pile so that this sum changes into 000. It turns out that this is always possible. For example, if I take the 5 pile and remove 3 beans there are 2 beans left. 2 in binary is 010. Do the sum again and we get 000:

100

010

110

——

000

And the great thing is that whatever your opponent does next they have to change this total to something with some 1s appearing. If there are any 1s, they definitely haven’t won the game yet. But you always have a strategy to reset the sum to 000. At some point that will result in you genuinely having removed all the beans from the table and winning the game.

The language of binary numbers translates this game into one that you can always win, even if the number of beans or the number of piles changes. As long as you learn your binary numbers. And if at the outset the sum is already a string of 0s, make sure you offer to go second. Otherwise go first and make a move that sets the sum to 0.

This strategy of using the language of binary numbers to keep track of the state of the game turns out to help one solve a whole slew of similar games. Try playing the following one called turning turtles. A line of turtles is laid out randomly with some turtles on their feet and some on their back. (If you don’t have turtles at home, you can use coins. Heads are turtles on their feet, tails are turtles on their back.) You take it in turns to turn a turtle onto its back (or a coin from heads to tails). In addition, if desired, you can also turn over one turtle (or coin) to the left of the turtle you put on its back. This turtle or coin can be in either state, on its feet or back (heads or tails). For example, consider the sequence of n = 13 coins:

THTTHTTTHHTHT

One possible move in this position is to turn the coin in place 9 from heads to tails, and also the coin in place 4 from tails to heads.

The player to turn the last turtle onto its back (or coin from heads to tails) wins. At first sight this looks nothing like the game of Nim, but it’s actually the same game in disguise.

The number of turtles still on their feet corresponds to the number of piles and the position of the turtle numbered from the left is the number of items in each pile. In the case of the set-up with 13 coins we have 5 piles with 2, 5, 9, 10 and 12 stones in each pile. Turning the turtle on its back (or to tails) in position 9 and also turning the turtle in position 4 on to its feet is the same as taking 5 stones from the pile with 9 stones. Using the language of binary numbers that wins you Nim now translates into a strategy for turning turtles, a game that at first sight looked totally unrelated.

Although you may never be faced with a game of turning turtles, the philosophy at the heart of winning this game is worth remembering. Faced with any challenge, is there a way to change it into something you already know how to play? Is there a dictionary that you can use that translates the challenge into a language that makes the solution more obvious? You might be stuck in a language where there is a wall in the way. But shift into a parallel world by changing that language and a shortcut might just appear that allows you to sneak round the other side of the wall.

Shortcut to the shortcut

If a problem looks intractable, try to find a dictionary that will translate the description into another language that might reveal the solution more easily. If your new DIY enthusiasm isn’t being matched by the results you are producing, perhaps you need to change the pictures you are drawing into numbers to see if the measurements can tell you why things aren’t fitting together as you hoped. If that business plan packed with tables of numbers just isn’t communicating its impact, see if a picture or graph helps people to see your vision. Could a clever bit of algebra save you hours as you enter your company’s finances into yet another spreadsheet? Is your tussle with a competitor in reality a game in disguise that you already know a strategy to win? The message of this chapter is to look for the right language that helps you to think better.

Pitstop: Memory

Although I’ve successfully learned the language of mathematics, it has always frustrated me that I couldn’t master those more unpredictable languages like French or Russian that I’d tried to learn in hopes of becoming a spy. Although Gauss too left his love of languages behind to pursue a career in mathematics, he did actually return to the challenge of learning new languages in later life, such as Sanskrit and Russian. By the age of sixty-four, after two years of study, he had mastered Russian well enough to read Pushkin in the original.

Inspired by Gauss’s example, I’ve decided to revisit my attempts at learning Russian. One of the problems I have is simply that of remembering strange new words. Spotting patterns is my shortcut to memory. But what if there are no patterns? I wanted to know if there might be alternative shortcuts that others exploit. Who better to ask than Ed Cooke, a grandmaster of memory and founder of a new venture for learning languages called Memrise?

To earn the title of Grand Master of Memory, one must be able to memorise a 1000-digit number in one hour. The next hour you are faced with memorising the order of ten packs of cards. And finally you get two minutes to memorise a single pack. In truth it sounded like a pretty useless skill to try to acquire but I realised that if you could manage that then remembering a list of Russian words should be a triviality.

Given that the 1000-digit number is chosen randomly, my strategy of looking for a pattern won’t be much help. So what was Cooke’s shortcut to 1000 randomly selected digits? It turns out to be something called a memory palace.

‘The shortcut is to transform something which is difficult to remember into a proxy that is easier to remember,’ says Cooke. ‘We remember what is sensory, what is visual, what is tactile, what evokes an emotion. So that’s what you want to do, to transform into something like this which recruits your first brain power.

‘To remember a 1000-digit number what I do is to arrange a lot of images around a space and every image stands for a number. If I’m trying to recall a number like 7831809720, that’s normally a very difficult thing to remember because they’re just numbers and they all sound pretty much the same and don’t have any meaning. But in my mind 78 is a guy who used to bully me at school and hung me upside down from one leg in my boxer shorts over a staircase, which was a very memorable moment. Much more memorable than the number 78.’

Each two-digit number is turned into a character. In Cooke’s private language, 31 is Claudia Schiffer ‘dressed in memorable yellow underwear from the Citroën advert’. This addition of an extra bit of colour to the image is important. ‘The more vivid and bizarre the image the better the memory.’ The number 80 is a friend who has a very amusing face. 97 is the cricketer Andrew Flintoff. 20 is Cooke’s father.

‘I stashed this dictionary of numbers around the age of eighteen so it’s become a fossilised form of my teenage imagination, my humours, beautiful people I’d read about in magazines, my family, my best friends,’ he says.

Although Cooke is right that for most people one number looks much like another, as a mathematician who spends more and more time wandering around the world of numbers, one starts to get to know the special characteristics of each one. They begin to have their own personalities. It was said of Ramanujan, the great Indian mathematician, that he knew every number like his own personal friend. His collaborator Hardy once visited him in hospital when he was ill and, stuck for any comforting words to offer the mathematician, Hardy recalled that his taxi had had a rather uninteresting number: 1729. Ramanujan replied immediately, ‘Not so Hardy, it is a very interesting number. It is the smallest number which can be written as the sum of two cubes in two different ways.’ 1729 = 12^3 + 1^3 = 9^3 + 10^3. But most people don’t have this kind of intimate emotional relationship with numbers. Claudia Schiffer in yellow underwear is probably more memorable than sums of cubes.

But how does Cooke use this cast of characters to remember 1000 digits? The key is to put those characters in space. ‘If you want to make very, very long chains of information of things, you need a backbone on which to project your images and it so happens that we have an extraordinary potency of memory for space. Mammals developed an incredible capacity to navigate and remember an incredible repertoire of spaces. Even if we don’t think so, we’re all really good at this. Just after wandering around an elaborate building for a few minutes we can memorise its layout. So we can use this powerful skill as a shortcut to piggyback our images representing our numbers. This is called building a memory palace.’

A memory palace isn’t just a story, but a story that moves through space. This last part is key. ‘The advantage of a memory palace over a simple story is that stories are more vulnerable to breaks in the chain. You’re also giving yourself the extra burden of making narrative logic rather than just exploiting pure spatial location and therefore it’s slightly more taxing on the imaginative mind.’

I’d watched Cooke building such a palace a few years ago. We’d both taken part in the Serpentine Gallery’s Memory Marathon, a weekend exploring the concept of memory, and I remember him taking the audience on an amazing walk around the grounds of the gallery, using what he saw to create a memory palace that the audience employed to remember all the presidents of the United States. Each name was translated into a powerfully vivid image. For example, President John Adams became the image of Adam and Eve balancing on top of a toilet. ‘John’ being slang for a toilet. These images were then placed at locations around the park. To remember the presidents, the audience just had to mentally resurrect the walk, something that our brains seem very adept at doing, and then use the absurd images located at various points on the walk to recall the presidents.

Using spatial memory appears to be an amazing shortcut to remembering very long sequences, whether it’s numbers, presidents or whatever list you might be trying to commit to memory. It’s a fantastic hack, because remembering things by rote seems to scale exponentially in difficulty. The first 10 are easy, the next 10 are more difficult, and more than 100 are almost impossible. But as Cooke explained to me: ‘The absolutely extraordinary thing about spatial memory is that it seems to scale linearly in difficulty. I can remember a pack of cards in about a minute, perhaps two minutes if I just want to check it over. The thing is that this scales linearly so that in an hour I’d be able to remember 30 packs of cards.’

When I suggest that for my readers, perhaps remembering a pack of cards wouldn’t really be a skill they were desperate to acquire, Cooke was keen to stress that the cards are irrelevant. The tactic works whatever you are trying to remember. He explained to me that when he is giving talks without notes, he uses exactly the same strategy. Turn the talk into a journey around a familiar place like your house and place the points you want to make in each of the rooms. As you deliver your presentation you will find you can much more easily remember the speech by navigating your way around the memory palace you have built in your mind.

‘In the memory palace, as one embarks on the journey the scene of the action is constantly moving forward so that the danger of the memories interfering with each other is thereby diminished because you have a new context in which to excite the new memory.’

This technique of translating two-digit numbers into visual images is also key to an extraordinary feat of calculation that my magician friend Arthur Benjamin can do. Benjamin has trained himself to be able to multiply two 6-digit numbers in his head. One of the tricks he used was to exploit a bit of algebra to break 6-digit numbers into pieces that could be multiplied together separately. But to be able to continue the calculation, he needs to bank these numbers in his memory and recall them later for use.

What Benjamin found was that if he tried simply to remember the number, it interfered with his calculations. It’s as if number memory was happening in the same place as the calculations. So instead he came up with a special code that translated numbers into words. Memorising the words seemed to happen in a part of the brain that wasn’t disrupted by further numerical calculations and so the words could be recalled and translated back into numbers when needed.

My conversation with Ed Cooke took place during the UK lockdown caused by the Covid-19 crisis and Cooke recalls that it was another medical lockdown, when as a teenager he found himself in hospital for three months without much to do, that he embarked on his mission to become a Master of Memory. ‘Partly the motivation was the pleasure of extending a craft to its logical conclusion. As a student my party trick was to remember long numbers and packs of cards in bars in an attempt to win bottles of champagne. And I began boasting to my house mates that I thought I might be one of the world’s fastest card memorisers. And they’d retorted “Shut up, Ed! Go and prove it” and that’s what took me to these memory championships.’

Memory palaces might be good for memorising strings of digits or delivering speeches without notes, but what about my dream to learn Russian. Is this what Cooke is employing in his company Memrise, his initiative to learn languages? Will I finally find the secret shortcut to learning new languages?

‘Repetition and testing,’ says Cooke. ‘By repeating stuff we prove to our brains it’s worth remembering. Important things tend to repeat. Testing is crucial because memories are movement of the mind and those movements become more consolidated the more you practise them.’

Those didn’t sound like shortcuts, if I’m honest. But Cooke had more.

‘The third thing is mnemonics. Say I’ve got this tricky Russian word “ostanovka” which means “bus stop”. How can I get my head around that word? Well, why don’t I relate it to words I know in my own language which will connect it together. So if we want to clock something in our mind, we have to weave it into the existing network of associations. So “Osta” sounds like “Austin”, the English car manufacturer. They have made enough cars, giving me “novka”, so I will take the “bus”, giving me “bus stop”.’

That sounds more promising. It seems clear that the repetition and testing part means I’m not going to be able to learn Russian in an hour. But mnemonics might actually be a shortcut to remembering a list of Russian words that previously weren’t sticking. Cooke also has one last tip for a shortcut to learning a language, which he learned from his grandmother.

‘The best way to learn language is between the sheets. If you are enchanted and highly motivated and paying a lot of attention and immersed, then you’re going to learn extremely fast.’

4

The Geometric Shortcut

There are ten people in Edinburgh and five in London. The distance between the two cities is 400 miles. Where should they meet so that the total distance travelled is smallest?

My concept of the shortcut for most of this book is an abstract mental shortening of the journey to my destination. But in this chapter I want to consider some real physical shortcuts. If you want to get from A to B in a physical landscape, understanding the underlying geometry of that landscape can help you to plot paths that get you to your destination faster, even if they look at first sight to be heading in the wrong direction.

And even if you’re not planning an actual journey, sometimes the challenge you face can be translated into something geometric where a tunnel or a bypass in the geometry translates back to a shortcut in the original problem. For example, as I shall explain later, digital companies like Facebook and Google have used the way large groups of people can collectively find shortcuts through terrain and translated this philosophy into a way of finding shortcuts in the digital landscape that we stomp through on a daily basis.

Mapping physical shortcuts would also become a passion for Gauss in his later years. Although he had fallen in love with mathematics as a student through playing around with numbers, he also enjoyed the challenges of geometry. But this wasn’t just the abstract circles and triangles of Euclid. For a man who loved the abstract ideas of mathematics, it is rather strange to find Gauss signing up in his forties to the very practical task of preparing a land survey of the Kingdom of Hanover for the local government. As Gauss once wrote, ‘All the measurements in the world are not worth one theorem by which the science of eternal truth is genuinely advanced.’ The work that he engaged in was not the exact and beautiful theory of numbers that he’d been drawn to at school but lots of messy and inaccurate measurements that were full of errors due to faulty equipment or human fallibility. The map he ultimately produced of Hanover was not a particularly accurate one by all accounts.

But the time he spent out measuring the state of Hanover did lead to the discovery of revolutionary new sorts of geometry.

Getting from A to B

In 1492 Christopher Columbus famously set sail to find a shortcut to the Indies. Traditional trade routes required a long treacherous journey across land that limited the goods that you could bring in any one trip. Traders were keen to find a route by boat. Some believed that there was a course round Africa although others thought that the Indian sea was landlocked and unreachable this way. Even if there was a way around, many thought that it would take too long. Columbus believed that if he headed west he would hit China and India from the other side and hence establish a smoother route to bring back the spices and silks that Europe was trading with the east.

He’d sat down and done the maths. He believed that to get from the Canary Islands to the East Indies would mean travelling just 68 degrees west. This, he believed, was a distance of just over 3000 nautical miles. Definitely a shortcut if you considered that, to sail from London, around Africa, to the Arabian Gulf was 11,300 nautical miles. Unfortunately Columbus made several crucial mistakes in his mathematics which meant that he grossly underestimated the true distance that he would need to travel if he was to go the other way.

Estimates of the circumference of the Earth had been made since ancient times. In 240 BC the Greek mathematician Eratosthenes had calculated it to be approximately 250,000 stadia. How long is a stadion? This is one of the problems with calculating distances. What unit of measurement are you using as your standard? In Eratosthenes’ time the unit was a stadion, which was the length of an athletic stadium. The trouble is that Greek stadiums were 185 metres long, while Egyptian stadiums where Eratosthenes lived and worked were shorter at 157.5 metres. Giving Eratosthenes the benefit of the doubt and taking the Egyptian length gives a measurement that is just 2 per cent off the planet’s actual circumference of 40,075 km.

But Columbus took a more recent estimate by the medieval Persian geographer Abu al-Abbas Ahmad ibn Muhammad ibn Kathir al-Farghani, known in the west as Alfraganus. Columbus assumed the mile that Alfraganus had used in his calculation was the Roman mile with a length of 4856 feet. In reality Alfraganus was using an Arabic mile, which was much longer at 7091 feet!

Fortunately for Columbus, rather than being stuck in the middle of the ocean only halfway to his destination and running out of food and supplies, he stumbled on a small island in the Bahamas that he named San Salvador. He didn’t realise his mistake for some time and referred to the inhabitants of the island as Indians, assuming that he’d reached the East Indies.

The true shortcut to the east turned out to be one that would be physically carved by humans. During his time in Egypt, Napoleon was already flirting with the idea of cutting a canal between the Mediterranean and the Red Sea. But due to some more faulty calculations it was believed that the Red Sea was actually ten metres higher than the Mediterranean. To avoid flooding the countries that bordered the Med, a complex system of locks would need to be built. The idea ultimately proved too expensive a proposition for the French state.

Once it was realised that the levels of the seas were in fact the same, the idea for a canal soon gathered momentum. The shortcut was finally opened on 17 November 1869. Although the Suez Canal was under French control, it was a British ship that ended up being the first to pass through. The night before the opening, under cover of darkness and without lights, the captain of HMS Newport piloted his ship through the flotilla waiting to enter the canal and managed to berth it at the front of the line. When everyone awoke to celebrate the opening, they found HMS Newport in the way of the route to the Red Sea. The only way to get ships through was to let the British ship through first. Although officially reprimanded by the British navy, the captain of HMS Newport was privately congratulated by the Admiralty for the publicity coup.

The Suez Canal shortened the distance from London to the Arabian Gulf by 8900 kilometres, reducing the journey by 43 per cent. The importance of this shortcut can be judged by the number of times that it has been fought over, most famously when Egyptian president Gamal Abdel Nasser seized the canal from British control in 1956 precipitating the Suez crisis. Today 7.5 per cent of the world’s shipping passes through the canal and earns the Suez Canal Authority, owned by the Egyptian government, $5 billion a year.

An equally important shortcut that spared ships having to navigate round South America’s Cape Horn was opened in 1914. The Panama Canal that connects the Atlantic Ocean to the Pacific Ocean actually does have several locks that ships must pass through. Not because the sea levels are different on either side but because it turned out to be too expensive to dig so deep: instead ships cross an artificial lake in the journey across Panama.

Around the world

Given that the first circumnavigation of the Earth didn’t occur until the beginning of the sixteenth century, how did Eratosthenes manage to measure the circumference of the planet in 240 BC so accurately? He obviously couldn’t wrap the planet in a measuring tape. What he did instead was to measure a small distance on the Earth’s surface and then used some clever maths to shortcut having to measure the whole thing.

Eratosthenes was the librarian at the great Library of Alexandria and made fascinating contributions to science in a number of fields from mathematics to astronomy, geography to music. But despite his innovative work, his contemporaries were rather dismissive of his abilities and gave him the nickname Beta to indicate that he wasn’t in the top rank of thinkers.

One of his clever ideas was a systematic method for producing a list of prime numbers. To find the primes in a list of numbers from 1 to 100, Eratosthenes proposed the following algorithm: take the number 2 and strike out all the subsequent numbers that are multiples of 2. This can be done simply by moving through the table in steps of 2 deleting the numbers that you encounter. Move to the next number after 2 that hasn’t been struck off. This is clearly the number 3. Now strike out all the multiples of 3 by stepping through the table in steps of 3 systematically deleting all the numbers you encounter. At this point the method begins to come into its own. The next number you haven’t struck off the list is 5. Repeat the method that we’ve applied to the previous numbers: move through the table in steps of 5, knocking out all the numbers you encounter.

This is the key to the algorithm: move to the next number not struck off previously and then knock out the numbers that are a multiple of this new number by moving through the table in steps corresponding to the new number. If you do this systematically, by the time you knock out multiples of 7 you’ll have produced a table of primes up to 100.

It’s a clever algorithm. It shortcuts having to think very much. It is perfect for a computer to implement. The trouble is that it very quickly becomes a slow method for churning out primes. It’s a shortcut to thinking because you can behave like a machine to produce the list. But it isn’t the sort of shortcut that I want to celebrate in this book. What I want is a clever strategy to sniff out the primes.

However, I’m going to give Eratosthenes high marks for his calculation for the circumference of the Earth, because it is inspired. He’d heard about a well in the city of Swenet above which on one day each year the Sun shone directly overhead. On the summer solstice at midday the Sun would shine straight down on the well, casting no shadows on its sides. Today Swenet is known as Aswan and lies not far from the Tropic of Cancer, which is the line of latitude marking the furthest point north, at 23.4 degrees, that the Sun can be found directly overhead.

Eratosthenes realised that he could use this information about the location of the Sun to do an experiment on this particular day to give him his calculation of the circumference of the Earth. Although it saved him having to lay a tape measure round the whole planet, the experiment would require some walking. On the summer solstice he erected a pole in Alexandria, which he believed to be due north of Swenet. It’s actually 2 degrees of longitude off, but it’s the spirit of his experiment that I am applauding if not the total accuracy.

On the day, while the Sun was directly overhead in Swenet casting no shadow in the well there, it caused a shadow to be cast by the pole in Alexandria. Measuring the length of the shadow and that of the stick, Eratosthenes could construct a triangle with the same ratio and measure the angle in it. That would tell him how far round the circumference of the Earth Alexandria was from Swenet. The angle he measured was 7.2 degrees, or 1/50th of a complete circle. Now he just needed to know the physical distance from Alexandria to Swenet.

Rather than walking there himself, he employed a professional measurer, called a bematist, who would walk in a straight line between the two cities – any deviation would mess up the calculation. The result was recorded in a larger unit of measurement: the stadion. Alexandria proved to be 5000 stadia north of Swenet. If this was 1/50th of a complete journey round the globe that made the circumference of the Earth 250,000 stadia. Today we’re not sure exactly how many steps the surveyor that Eratosthenes had hired was using to measure his stadia, but as I explained above it was a quite amazingly good measurement. Using a bit of mathematical geometry, he’d shortcut the need to hire someone to walk all the way round the planet.

The word geometry has its origins in this experiment because if we break down the name it is Greek for measuring the Earth: ‘geo’ = earth and ‘metry’ = measuring.

Trigonometry: a shortcut to the heavens

The ancient Greeks weren’t just using their maths to measure the Earth. They realised it could be used to measure the heavens too. And the essential tool that made this possible was not a telescope or a sophisticated tape measure but the mathematics of trigonometry.

There is already a hint of this tool at work in Eratosthenes’ calculation. Trigonometry is the mathematics of triangles, explaining the relationship between the angles in the triangles and the lengths of the sides. This mathematics gave the mathematicians of antiquity an extraordinary shortcut to measuring the cosmos without ever leaving the comfort of the surface of the Earth.

For example, already in the third century BC Aristarchus of Samos had used trigonometry to work out the distance from the Earth to the Sun relative to the distance from the Earth to the Moon. To do this, all he had to do was to measure the angle between the Moon to Earth to Sun – the three points of a triangle – on the day that the Moon is half full. This is when the angle from the Earth to Moon to Sun is exactly 90 degrees (see below). Then by constructing the triangle made up of the angle he had measured, he could calculate the ratio of the distance from the Earth to the Moon compared with the Earth to the Sun because the ratio would be the same for this smaller triangle that he’d drawn. The clever realisation is that it doesn’t matter how big or small the triangle is, the ratio stays the same. This ratio is in fact called the cosine of the angle that Aristarchus had measured.

To calculate the actual distance, rather than the ratios of the distances, you needed an angle and one length. It was Hipparchus, traditionally credited as the founder of trigonometry, who would discover a cunning way to tease out the actual distances between the Earth and the Moon and the Sun. He exploited a series of solar and lunar eclipses, in particular the solar eclipse observed on 14 March 190 BC.

4.1 Using triangles to measure the solar system.

Like Eratosthenes, Hipparchus used observations at two different locations on the Earth. At Hellespont the eclipse was observed as total, but in Alexandria it was a partial eclipse, the Moon obscuring only four-fifths of the Sun. Just as Eratosthenes had, Hipparchus now had a distance that he could measure on Earth. Combining the distance between the two cities with the angles he measured of the eclipse, he was able to use trigonometry to calculate how far the Moon was from the Earth.

The power of this trigonometrical shortcut was extraordinary. It led Hipparchus to start preparing the first examples of trigonometrical tables. In these you could take an angle and if you formed a right-angle triangle with this angle at one corner, they would tell you the relative lengths of the triangle with respect to each other. Even here mathematicians discovered shortcuts that meant they didn’t have to draw out loads of triangles and start doing measurements of lengths and angles.

Take an equilateral triangle, where all sides are equal and all angles are 60 degrees. Now drop a line from one corner dividing that angle into 30 degrees and creating an angle at the base of 90 degrees. The cosine of 60 is the ratio of the two lines making up this angle in the newly constructed right-angled triangle. This is easily read off as ½ because the adjacent length of this new triangle is half the length of the original side of the equilateral triangle.

4.2 The cosine of 60.

But mathematicians discovered a cunning formula that relates cosines in one triangle to cosines in a triangle with half the angle and this gives us a tool to calculate more.

Cos(x)2 = ½ + ½Cos(2x)

Using these shortcuts, tables for cosines of many angles could be drawn up. And it was these tables that became the most effective measuring tool for exploring the night sky. They were also key to shortcuts of measurement on Earth. Gauss would have used them as he conducted his survey of Hanover. Even today surveyors use this mathematical shortcut to measurement.

For example, if you want to work out the height of a tree, it is quite difficult to lay measuring sticks out from the base to the top. Instead, a surveyor will walk a distance from the tree and measure the angle from the ground to the top of the tree. Combining this with the much simpler measurement of the distance from the surveyor to the base of the tree and looking up the tangent (which expresses the ratio of the two short sides of the triangle, in this case the height of the tree and the distance from the base to the surveyor), the surveyor can find out the height of the tree without climbing any ladders.

A rather beautiful demonstration of the shortcutting power of trigonometry is in the measurement of the metre. You might think it rather a curious task to measure a metre given that this is a unit of measurement. But the tale begins with the first definition of what a metre actually meant.

Measuring the metre

Ever since the first ancient civilisations began building the first cities, we’ve needed units of measurement to help coordinate construction. The earliest forms date back to the ancient Egyptians who used parts of the body as units. A cubit was the length from your elbow to the tip of your middle finger. The same use of body parts is evident in pre-metric measurements. A foot is obvious. The word for inch and thumb is the same in many European languages. A yard relates closely to a human pace. Rather intriguingly, the unit of measurement known as a rod used to measure land in Saxon times was defined as the total length of the left feet of the first 16 men to leave church on a Sunday morning. But given that we all come in different shapes and sizes, such measurements will vary from one person to another.

King Henry I tried to resolve this problem by insisting that the king’s body be used to standardise the units. He decreed that the yard should be the distance from the tip of the king’s nose to the end of his outstretched thumb. But there are clearly problems with this given that the yard potentially changes length each time a new monarch assumes the throne.

The leaders of the French Revolution believed that a more egalitarian system of measurement should be put in place that everyone could access. Galileo had proved that the swing of a pendulum depends on its length, not its weight or amplitude. It was initially proposed that a metre be the length of a pendulum that takes two seconds to swing back and forth. But it turned out that the swing also depended on the strength of gravity, which varies at different locations around the world.

It was decided instead that a metre be defined as one ten-millionth of the distance from the pole to the Equator. Although in principle anyone has access to measuring this distance, the impracticality of using the definition quickly became apparent. Two scientists, Pierre Méchain and Jean-Baptiste Delambre, were charged with measuring the distance from the pole to the Equator and returning to Paris with their discovery of the metre. But just as Eratosthenes realised that you don’t have to measure the whole distance, the two scientists decided to measure the distance from Dunkirk to Barcelona, two cities roughly on the same line of longitude. Then, as Eratosthenes had done, they could scale up their calculation to get the distance from the Equator to the pole.

Delambre started out from Dunkirk in the north while Méchain was responsible for the southern section, setting off from Barcelona. They agreed to meet at the midway point of the city of Rodez in southern France. But how did they make their calculations? For a start they needed a standard length that they would both use to do the measurements. But even then, they couldn’t be laying out these lengths along the whole route from Dunkirk to Barcelona.

Here is where the power of trigonometry and triangles kicks in. Starting at the top of the church tower in Dunkirk, Delambre looked across the countryside for two other high spots that could form the other points of a triangle. He would need to measure the distance from the church tower to one of these points. That hard work couldn’t be avoided. But from then on he could use the measurement of two angles in the triangle to calculate the lengths of the other two sides of the triangle. To measure angles, he used a piece of equipment called the Borda repeating circle. This consisted of two telescopes mounted on a shared axis with a scale to measure the angle between the two. Delambre pointed the telescopes at the two high points that he had identified from the top of the church tower and simply read off the angle between the telescopes.

By moving to one of the other high points in the triangle, he could get the second angle. Then trigonometry takes over to give the lengths of the two missing sides. But here was the really clever thing. One of these sides, which he now knew the length of, would become the side of a new triangle made from picking another high point that he could see from the two high points that he’d identified from the church in Dunkirk. In this new triangle he already had one distance. So it just meant measuring two angles with his Borda repeating circle and he could calculate the missing distances in the new triangle.

4.3 Knowing the distance between A and B and the angles a and b, trigonometry allows you to calculate the distances from C to A and B.

This was a brilliant shortcut. By rigging up triangles all the way from Dunkirk to Barcelona, the scientists needed only to measure one length in a single triangle and from that point on the angles would do the rest. The science of triangulation is an extraordinary shortcut for surveying land. Measuring angles can be done from the comfort of the high points marking out the corners of the triangles. No need to pace out the distance or to lay down rulers.

But climbing high points and looking through telescopes wasn’t without its dangers. This was not an ideal time to be surveying the land with telescopes and strange equipment. A revolution was raging. The two scientists had to endure many attacks from locals suspicious of these two spying from the tops of towers and trees as they measured their way across France. At Belle Assise, north of Paris, Delambre was arrested on suspicion of being a spy. Why else would he be scaling towers carrying such strange equipment? He tried to explain he was measuring the size of the Earth for the Academy of Sciences, but a drunk militiaman interrupted, ‘There is no more ’Cademy. We’re all equal now. You’ll come with us.’ Eventually after seven years, Delambre and Méchain returned triumphant to Paris with the metre.

A platinum rod was cast whose length corresponded to their calculation and from 1799 the standard metre resided in the archives in France. But it suffered the same problem in some sense as Henry I’s yard. Despite its universal definition, it was still easier for scientists to journey to France to get a copy of the metre for measurement rather than setting out on their own measurement of the distance from the pole to the Equator.

From London to Edinburgh

When Delambre and Méchain were deciding where to meet up, it was obvious that the halfway point between Dunkirk and Barcelona made sense. But what about the 15 people in our puzzle that opened the chapter? Where should our team of 15 people meet if 5 are in London, 10 are in Edinburgh and they want to minimise the total journey? Rather strangely, they should all meet in Edinburgh. At first sight, you might think that given a 2 to 1 split in the group they should meet two-thirds of the way from London to Edinburgh. But for every mile that you meet outside Edinburgh, the Scottish group walks an extra 10 in total but only saves the English group 5.

More generally, if our 15 people are scattered at random points along a line from London to Edinburgh, the shortcut is for them all to head to the position of the person in the middle, the eighth person you encounter heading from London (or Edinburgh). On the same principle, for every mile away from person number 8 you meet, one group saves 7 miles, the other group has an additional 7 miles more to walk (so these even out) but person number 8 adds one more mile to the total.

What about an even more general setting of 15 people scattered around New York, a city of avenues and streets laid out in a grid? Scanning from east to west, you should meet on the avenue that the eighth person you encounter is standing on. But now choose the street where the eighth person is standing when you scan from south to north. Note this will in general be a different person from the eighth when scanned east–west.

This kind of analysis is essential if you are trying to find the optimal place for a network exchange for internet cables and you want to minimise the amount of cable you’re using. But there is another curious strategy for finding shortcuts through physical and digital spaces that has been exploited throughout history and even in today’s technological landscape.

Desire paths

The explorers of the fifteenth century were after geometric shortcuts that could get them efficiently from one side of the world to the other. In our everyday lives we are often on the lookout for the clever shortcuts that might get us to our destination quicker. In my nearest park in London the urban planners laid out a criss-cross of tarmacked paths to guide the local inhabitants from one side to the other. On paper it probably looked a perfectly good layout but the evidence from our park suggests otherwise. In addition to the tarmac paths, you will find a dry earthen path cutting through the grass where people have decided there is a much quicker route from one side of the park to the other.

Urban planners often like tarmac paths that make nice right angles to each other but sticking in a diagonal that cuts out the need to walk round the right angle makes much more sense when you’re walking. Humans prefer the hypotenuse as their path from A to B. Time and again you’ll see these paths of flattened grass where people have taken the shortcut to their goal.

An interesting example of these diagonal shortcuts cutting through the right angles can be found in Manhattan. The layout of the roads in streets and avenues running parallel and perpendicular to each other is definitely the mark of human planning. But there is one street that, curiously, cuts diagonally across the grid of roads: Broadway. It runs from top left to bottom right across the right angles of Manhattan. It turns out that this is in fact an ancient shortcut that was used by indigenous travellers before Manhattan and the European settlers appeared on the scene. Broadway follows the Wickquasgeck Trail, which is believed to have been the shortest route between Native American settlements that existed at the time and avoided swamps and hills. When the European settlers arrived they kept this shortcut as a route across Manhattan. This path that was trampled down by the feet of travellers going from one side of the island to the other is now preserved in tarmac for the city’s cars and pedestrians to use.

These public-made shortcuts have a name: desire paths. Some people call them cow paths or elephant trails because often they would have been carved by the livestock being taken along them. J. M. Barrie, the creator of Peter Pan, describes them as paths that make themselves because there is never a moment where you see someone laying out the path. No one makes the conscious decision to flatten the grass and clear the way. They emerge gradually as if, as Barrie describes, making themselves.

Some of these desire paths are rather curious as they seem to be making the paths longer than they should be. They don’t look like shortcuts at all. But if you look closely you realise that this is a path created to avoid something. Often it isn’t too clear what that is. But dig a bit deeper into the local culture and you’ll probably find that some superstition is the key. For example, many people will not walk under a ladder as it is considered bad luck. They would prefer to go around the ladder. Ladders are usually not in place long enough for a permanent desire path to emerge but in Russia a similar superstition exists for two posts that are leaning against each other. Old street lights in Russia are often placed at the top of such posts and you will frequently find that a permanent desire path has emerged to avoid walking between the posts.

Some urban planners realised that they could use these shortcuts as a shortcut. Instead of planning the tarmacked paths in advance, and finding that people didn’t use them, planners had the clever idea of letting local inhabitants mark out the desire paths that got them to where they wanted to go and after the paths had emerged in this organic way, the planners could tarmac over them.

Michigan State University used the footfall of students to decide the paths through the new university buildings erected in 2011. From an aerial view the paths look like a crazy spaghetti mess of interweaving strands. Definitely not something that any designer would have chosen in advance. But having let the students’ feet do the talking, or walking, the final layout of paths has created a network that works for all the students trying to get to lectures across the campus.

The architect Rem Koolhaas used a similar strategy for his design of the campus of the Illinois Institute of Technology in Chicago.

Snowfall also provides an effective way to understand how pedestrians and drivers are using a city. After inhabitants have trampled down most of it, the patterns of remaining snow can give the municipality a chance to understand that part of the road or park that is not being used as a way to traverse the city. This can give urban planners the opportunity to use the land for something else, like a traffic island in a road or a place for a piece of urban art.

You’ll find this sort of shortcut used again and again in the commercial sector: get the public to generate the material from which you can then distil the value. In a way the collection and exploitation of our digital data by companies like Facebook, Amazon and Google are just an example of companies observing the digital desire paths that we trample down and then capitalising on those well-trodden shortcuts.

Twitter, for instance, did not introduce the idea of the hashtag in a top-down manner. It was something that the company started seeing users employ to classify their tweets. In fact the hashtag seems to have originated from user Chris Messina who first suggested them in August 2007. He wanted a way to shortcut finding other users interested in the same subjects as he was tweeting about. The hashtag provided a clever way to eavesdrop on conversations of interest. As more and more people followed Messina down this digital desire path, Twitter cottoned on to the shortcut that users had carved out and in 2009 it became an official Twitter path, tarmacked over if you will.

Geodesics

If you looked at a map of the world and marked out what you thought was the shortest path to fly from Madagascar to Las Vegas, your first intuition might be to draw a straight line joining the two places across the map. After all this looks like the desire path that you’d think people (or birds) might fly along. But that doesn’t take into account the curvature of the Earth. The true desire path, the path of shortest length, when considered on the surface of a sphere, is along a path that flies over the UK and then Greenland, far from that original straight line that you might have drawn across a flat map.

4.4 The fastest route from Madagascar to Las Vegas is via the UK.

On a sphere if you take two points the shortest path between them will be a line called a great circle. This is like a line of longitude that passes through two polar points. In fact if you took a line of longitude and moved it around the globe until it passed through the two points you were trying to join, this would be the great circle through them.

Once one starts to explore the implications of these shortcuts across the globe some rather curious features begin to emerge. For example, take three points like the North Pole, Quito in Ecuador and Nairobi in Kenya. The latter two cities are pretty nearly on the Equator. The shortest paths between these three points would mark out an example of a triangle on the Earth’s surface. Classically triangles in Euclid’s geometry have angles that add up to 180 degrees. But examine the sum of the angles in this triangle and they add up to much more than 180 degrees. After all the two angles at Quito and Nairobi are already each 90 degrees since the line of longitude from the pole meets the Equator at 90 degrees. The angle at the North Pole between the two lines of longitude running through these two cities is 115 degrees. So the sum of the angles in the triangle is 90 + 90 + 115 = 295 degrees.

4.5 On a sphere angles in triangles add up to more than 180 degrees.

There are also geometries where angles in triangles sum to less than 180. For instance, in something called an antisphere, which looks like a cone with curved sides, the shortest paths between two points on this surface will also trace out strange triangles where the sum of the angles is less than 180 degrees. These have what is called negative curvature, while spheres like the Earth have positive curvature. A flat geometry like the map I started with has curvature zero.

4.6 On an anti-sphere angles in triangles add up to less than 180 degrees.

The discovery of curved geometries was one of the exciting developments in mathematics made in the early nineteenth century. But the discovery caused something of a fight between three mathematicians who each claimed to have found the geometries first. The idea of these new geometries was floated publicly for the first time in the 1830s simultaneously by Russian mathematician Nikolai Ivanovich Lobachevsky and Hungarian János Bolyai. Bolyai’s father was particularly impressed with his son’s discovery and was keen to boast to a good friend of his: Carl Friedrich Gauss. But Gauss wrote back to Bolyai’s father with the rather stinging response:

If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years.

It turns out that indeed Gauss had discovered these curved geometries with strange shortcuts across the surfaces many years earlier while making his survey of Hanover. This had involved triangulating the land like Méchain and Delambre had done to measure the metre. Although the task had seemed at first sight a tedious chore for the great mathematician, it was the catalyst for deep theoretical insights. Gauss had speculated whether not just the surface of the Earth but the geometry of space itself might be curved. He decided to use some of his measurements of triangles to test whether light beams travelling between three hilltops around his home in Göttingen might create a triangle whose angles added up to something different from 180 degrees.

Light loves shortcuts. It always finds the shortest path between two points. So if the angles added up to something different from 180 degrees, it would mean that the light was following a curved path through space. Gauss was hoping to prove that three-dimensional space was actually curved just like the two-dimensional surface of the Earth. When he didn’t find any discrepancies he abandoned his ideas since these new curved geometries went against his belief that mathematics was there to describe the universe we see around us. He swore to secrecy the few friends with whom he had discussed his research.

We now know of course that Gauss was working on too small a scale to detect the curvature of space. It was Albert Einstein’s new theory of gravity and the geometry of space-time that would spark a renewed interest in testing out Gauss’s ideas.

Einstein discovered that the distance between two objects in space could vary according to who was observing them. If you were travelling near the speed of light, the distance would look shorter. Time also seemed to depend on the observer. The sequence of events could change according to how the observer was moving. Einstein’s big breakthrough was to realise that you needed to consider time and space together in a four-dimensional geometry made up of three spatial dimensions and one dimension of time. To measure distance in this new space-time geometry led to a shape that was curved.

Einstein’s insights redefined gravity not as a force as Newton had done, but a bending of the geometry of space-time. An object with large mass would warp the fabric of space. Rather than gravity being like a force pulling objects together, one could rethink the story. Gravity was the shortcuts that objects took through this geometry. The freefalling of an object was really just the object finding the shortest path through the geometry from one point to another.

Hence the planets orbiting the Sun are not to be thought of as objects pulled by a force like a string attached to them but simply a ball rolling down the side of this four-dimensional space-time geometry. It seemed a crazy idea but Einstein had a way to test it. Just like planets, light should also find the shortest path through space. If light was due to pass near an object of large mass, the theory would imply that the shortest path would be to take a detour, curving towards the object.

The British astronomer Arthur Eddington realised that there was a way to test this idea using a solar eclipse that was due to hit Earth in 1919. The theory predicted that the light from distant stars should be bent by the gravitational effect of the Sun. Eddington needed the eclipse to block out the glare of the Sun so that he could see the stars in the sky. The fact that the light did indeed seem to bend round objects of large mass confirmed that the shortest paths weren’t straight lines but curved as Einstein’s theory had predicted.

The bending and warping of space might also provide shortcuts across the universe to get around some long cuts implied by Einstein’s theory of relativity. He understood that the universe has a speed limit: the speed of light in a vacuum. Nothing can go faster than this. This causes a problem if you want to get from one side of the galaxy to the other. It’s going to take time. This is a major issue faced by many science fiction writers: how do you get your cast of characters from one venue to the other without wasting years in transit? Often the secret is to use a wormhole, a special solution to Einstein’s field equations that provides a speculative shortcut between different bits of the geometry of space-time. A wormhole is a bit like a tunnel through a mountain, except it’s one linking two points across the universe that ordinarily would take you millions of years to traverse.

4.7 There’s the long way around the universe from A to B or the shortcut through the wormhole.

So Gauss’s idea of light taking a shortcut through curved paths as it travelled from one hilltop in Göttingen to the other was correct. It’s just that to see the effect one needed to work on a far grander scale, surveying not Hanover but our galaxy. To his credit, Einstein always acknowledged that the mathematicians of the nineteenth century had created the geometry that allowed him to discover relativity: ‘The importance of C. F. Gauss for the development of modern physical theory and especially for the mathematical fundament of the theory of relativity is overwhelming,’ he wrote. ‘Indeed … I do not hesitate to confess that to a certain extent pleasure may be found by absorbing ourselves in questions of pure geometry.’

Shortcut to the shortcut

If you’re planning a trip from A to B, it’s often worth remembering the way light finds the quickest path: sometimes it pays to take a detour because the route is faster even if the path is longer. Measurements around the house can sometimes be tricky because you can’t get a tape measure in place. But perhaps an angle is measurable. Sines and cosines were always meant to be fantastic shortcuts to measuring not just the night sky or the Earth’s surface but anything that might at first seem inaccessible. The urban planners’ strategy of letting the crowd search out the shortcut is a tactic that might apply not just to getting from one side of the park to the other. Letting the public guide you to an optimal solution is a potential shortcut that saves you from doing all the work on your own.

Pitstop: Journeys

I love walking. The slow pace allows me to experience the landscape and natural world in a way that often gets overlooked in our fast-paced lives. The walk is not about getting from A to B. Frequently it is about getting from A to A but enjoying the long way round to get back to the beginning. When he was younger, my son found this absurd. We set out on a walk for a day in the countryside. After half a mile my son suddenly noticed a path breaking off from the one we were on across a field. On the other side he saw our house. ‘Dad! I’ve found a shortcut! Look, we just need to take this path and we’ll be home.’

But for me walking is also a kind of shortcut. Three miles an hour seems to be the perfect speed for thinking. As Jean-Jacques Rousseau wrote in Confessions: ‘I can only meditate when I am walking, when I stop, I cease to think; my mind only works with my legs.’ Walking is my shortcut to mathematical revelation, the necessary detour that I need to make in order to allow my subconscious to explore a problem in a new way.

Robert Macfarlane talks about the connection between walking and thinking in his book The Old Ways. He describes how, while walking through the Norwegian countryside, Ludwig Wittgenstein made a major breakthrough in his work: ‘It seems to me that I had given birth to new thoughts within me,’ wrote the philosopher. But it is the word that Wittgenstein chose to describe those thoughts that, as Macfarlane points out, is so revealing. Wittgenstein used ‘Denkbewegungen’, which literally translates as ‘thought-ways’. Macfarlane describes them as ‘ideas that have been brought into being by means of motion along a path (weg)’.

Macfarlane loves making journeys, being in the landscape, trekking and travelling. His books are a beautiful eulogy to the journey on foot. So I was really keen to talk with him about his relationship to the idea of the shortcut. Is it possible that we might miss something by always looking for shortcuts?

‘I can get the funicular railway up to the top of Cairn Gorm in my most beloved mountain range in north-east Scotland and will feel that’s the shortest cut to the top but it will bring almost zero fulfilment and pleasure,’ he says. ‘But I’ll come to the top of that same mountain over a two-day walk and it will be one of the most extraordinary places I’ve ever been.’

Macfarlane tells me about the Scottish mystic and mountaineer W. H. Murray whose writing captures the power of being in these places: ‘When his spirit is burdened or lightened the natural movement of a man’s heart is to lift upward.’ Murray wrote these words on toilet paper he collected while incarcerated in a prisoner of war camp during the Second World War. Unable to journey in his body, he walked in his mind across the Scottish Highlands. Another of Macfarlane’s heroes is the modernist writer and poet Nan Shepherd.

‘Nan Shepherd writes at the end of The Living Mountain in the 1940s about how these moments of being as she calls them, echoing Woolf and Wordsworth and others, are produced only when “walking thus, hour after hour, the senses keyed, one walks the flesh transparent”. It’s the most amazing phrase,’ says Macfarlane. ‘These hills have no business with haste, I think is how she puts it. So the shortcut is absolutely antithetical to revelation in that model.’

But Macfarlane reminds me that many of these paths that we walk for pleasure today were first trodden in Neolithic times because they were shortcuts. Living under scarce conditions, people had to balance energy expenditures, resources and so on. They were not likely to pass up a shorter route if they found one, whether or not it offered the same kinds of contemplative opportunities as a longer one.

But not always. As Macfarlane points out, sometimes Neolithic cultures expended excessive resources in projects that weren’t simply about the utility of survival. To this point, he tells me a beautiful story about hand-axes mined in Little Langdale in Cumbria in the Lake District that reveals that not all Neolithic paths were about shortcuts to efficiency: ‘There were perfectly good hand-axe rock exposures at low levels in that valley so they could have made use of those to get the tools that they wanted. But it’s clear that they chose to ascend to much higher and more difficult ground on a crag called Gimmer Crag.’

I was curious as to why they would go to the hard place to get the same rock that they could have got from the easy place.

‘There’s an aura of place that remains with the object once the object’s been detached from a place,’ he says. ‘So there are reasons why the long cuts as well as short cuts might have been taken prehistorically.’

Then Macfarlane turns the tables on me. Are there examples of the long cut in mathematics being unusually productive?

Conjectures are, I think, one example. A conjecture is like the mountain peak. I don’t want to look up the answer in the back of the book. That’s like taking the funicular to the top of Cairn Gorm. The satisfaction of arrival depends on the days, nay years, that I’ve taken to reach that peak. But on the other hand, I don’t want to just slog through boring landscape for the sake of it. There are some walks that just feel like hard work.

There is a strange, delicate tension in mathematics between things being so easy that they become boring and things being so complex that understanding what is going on becomes impossible. In his book Adventure, Mystery and Romance, John Cawelti characterises the quality of this tension in literature but it applies equally to mathematics: ‘If we seek order and security the result is likely to be boredom and sameness. But rejecting order for the sake of change and novelty brings danger and uncertainty … the history of culture can be interpreted as a dynamic tension between the quest for order and the flight from ennui.’

Sometimes the fact that you have to take the long road to get to a peak is part of the joy. Fermat’s Last Theorem took 350 years of work by generations of mathematicians, making journeys into strange and esoteric lands before we found a way to reach the destination. But those detours and long cuts are part of the joy of the proof. We were forced to discover fascinating new mathematical lands that might have been left untouched if we hadn’t been pushed around the impassable mathematical bogs we came up against.

It’s interesting to consider whether if the proof had been short and perhaps rather trivial, the value that we would have assigned Fermat’s Last Theorem would have been so much less. Great unsolved conjectures, like the Riemann hypothesis, get their aura from the challenge they offer and the work we need to put in to solve them. We talk about great conjectures being like climbing Everest. If it wasn’t tough to reach the peak, we wouldn’t perhaps value the achievement of reaching the solution as much.

I try to describe to Macfarlane that I think that what I enjoy in mathematics is not so much slogging across a moor but being stuck in front of a mountain, searching for a way through and then that extraordinary buzz of discovering a chink, a tunnel, the shortcut that gets me through.

‘I’m watching your hands describe what you have to do and what you look like was a climber,’ he says. ‘You look like a rock climber not a walker and I’m talking here about gymnastic climbing, which is sort of distinct from mountaineering which is distinct again from hill walking.’

Was the challenge of rock climbing something Macfarlane enjoyed?

‘I was very bad, but very into rock climbing for some years of my life and climbers talk of the crux in a climb,’ he says. ‘Every great climb will have a crux move. It sounds as though that’s very similar to the process you were describing where you’ll work at a problem. They’re called bouldering problems. You’ll start with the easy stuff and you’ll do it again and again and you’ll reach the crux and then you’ll fall off. It’ll chuck you and you can’t quite make that dynamic leap. And then when you do it, the few times that I did it, it’s absolutely thrilling. It is a problem-solving activity.’

I do recognise that sense of frustration followed by elation that overcoming a mathematical bouldering problem can instil. Before our meeting I had just been watching Free Solo, a film that documents Alex Honnold’s extraordinary scaling of El Capitan in Yosemite National Park without any ropes. The climb has about eight cruxes, it is the Riemann hypothesis of climbs. The hardest is simply called ‘the boulder problem’ which is a difficult sequence across thin handholds, some no wider than a pencil, that are far apart. It requires a bizarre karate kick to get across a near-vertical wall. It is a moment that if he fails, he falls to his death. He doesn’t have the luxury of being able to be chucked off again and again. One of the things that struck me about the climb is that the shortest path to the top is definitely not a straight line. The route that Honnold takes often has to descend in the middle of the journey, moving away from the final destination, in order to find a way to the top that is climbable. Geodesics in climbing are definitely quite strange lines that twist and turn up the mountain face.

I wondered what it is that determines which route you take to the top of a mountain. The fastest? The most scenic? The toughest? There are 18 named routes to the top of Everest, of which several have never been climbed. The vast majority of climbers use two routes: the South Col and the North Col. George Mallory died trying to climb the North Col. He would talk of a ‘beautiful line’. The beautiful line isn’t necessarily the hardest route, but is rather celebrated for its beauty. This is interesting in light of the way that mathematicians also talk about beautiful proofs. What is the quality that makes a route beautiful? As Macfarlane puts it, ‘The beauty is typically a function of a kind of continuity of movement or line itself. So you don’t necessarily need to traverse left and then pick up the next ridge line or whatever. It will also be to do with the nature of the rock, if the rock is very non friable and firm. It’ll be literally just an elegance of the line you would draw in the air if you were describing it. And then there’s danger as well. The beautiful line unites all of those. And then there is a hardest line, the tiger line. And then there’s what’s called the diretissima, the most direct line.’ The term stems from the Italian climber Emilio Comici, who said: ‘I wish some day to make a route, and from the summit let fall a drop of water, and this is where my route will have gone.’ This same route also represents what’s known as the fall line, the most perfect downhill gradient on a slope, the line water would take if it had a free run.

This has been the key to some of the shortcuts that Macfarlane has taken to get off the mountain quickly when dangerous weather conditions or the night might be closing in: ‘When you have to get off a mountain fast because you’ve got bad weather coming or particularly in case you’re going to be benighted then you start to look for the fall line because in theory that’s the shortest cut to the lowest ground, which is probably where safety and shelter is going to be.’

But you also have to consider the hazards that lie on the fall line: ‘The fall line might take you over a crag, and I know you don’t want to do that. I can think of lots of occasions where I’ve had to move fast down and I’m busy balancing the fall line off against other hazard assessments that I’m carrying out. That’s led me to some good decisions and some bad decisions. The shortcut can be a wonder but also a peril.’

I wondered if there were any particular occasions when this shortcut came to the rescue.

‘One of the best fall lines I’ve ever taken was when I surfed a small avalanche,’ he says. ‘We were descending from a Scottish mountain and time was running late and we came to a steep snow slope and it was obviously ground that we wouldn’t be able to cover if snow wasn’t on it. But the snow kind of evens it out and solves some of the problems under foot and it’s soft snow like big granular sugar. So it wasn’t going to avalanche in a big problematic way.’

I must admit it still sounded terrifying. Avalanches aren’t generally things you want to encounter on a mountainside.

‘We could see that it would run us out more or less safely down around 200 feet. So we just lay face first on the slope and then let it take us. It deposited us safely but wet, 200 vertical feet below where we started. It was brilliant. That was a good bit of risk assessment. It was one of the most exhilarating shortcuts I’ve ever taken.’

5

The Diagram Shortcut

Which song used in Quentin Tarantino’s film Reservoir Dogs is represented by the following diagram:

5.1 Venn That Tune.

If a picture is, as they say, worth a thousand words, then perhaps it is truly the ultimate shortcut. Leonardo da Vinci seemed to think so anyway: ‘A poet would be overcome by sleep and hunger before being able to describe with words what a painter is able to depict in an instant.’ While the written word is a relatively new invention, humans have been developing the ability to interpret meaning in a visual image ever since we evolved as a species. Twitter revealed for example that tweets involving images or videos are three times more likely to be engaged with than those containing only text, which perhaps explains why more visually oriented social media apps like Instagram are increasingly becoming the platform of choice for businesses trying to deliver content quickly and effectively. A well-designed image can be a remarkable shortcut to convey your message much more effectively than the words you might use.

In mathematics, too, sometimes a picture will communicate an idea where equations have failed. The square root of −1 was for centuries regarded as a strange aberration by mathematicians. Ultimately it was Gauss’s picture of imaginary numbers, depicting them as a two-dimensional map, that brought these numbers into the mainstream. But it was shortly before Gauss’s death in 1855 that the political power of a picture to represent numbers was truly revealed.

The rose diagram

When Florence Nightingale arrived at the hospital in Scutari in Turkey in November 1854 she was horrified by what she found. The Crimean War had been raging for a year and the hospital was responsible for tending to British troops injured in the conflict. The building was sitting on a cesspool, it was unhygienic, with no proper sanitation. The place was sordid and overcrowded.

Immediately, Nightingale set about improving conditions, setting up a laundry, improving supplies and providing nutritious food. But it didn’t work. Despite her best efforts, the death rate continued to rise. The sick and injured received assiduous care from Nightingale and other nurses, like Mary Seacole, but nursing wasn’t enough. Then, after months of fighting this losing battle, two men arrived on the scene: Dr John Sutherland, a cholera specialist, and Robert Rawlinson, a sanitary engineer. After some exploration, they found the underlying problem: the water system. It was choked with dead animals and human excrement was leaking from the privies into the water tanks. Rawlinson and Sutherland flushed the whole thing out. And they started to see a difference.

As a result of the Sanitary Commission, as they were called, all the military hospitals rapidly improved. Within a month, deaths from infectious diseases had halved. Within a year, they had fallen by 98 per cent, from well over 2500 in January 1855 to 42 in January 1856.

After the war ended, Nightingale reflected on what she’d experienced over the previous eighteen months. She accepted that, in war, lives are lost during battle. But what she couldn’t accept was the far higher number of deaths due to disease. She felt desperate about the losses: 18,000 men had died, and many of them could have been saved. The challenge before her was how to make lasting improvements in army hospitals so that such a tragedy would never happen again. But she knew that persuading the establishment of the urgency of radical reform wouldn’t be easy.

Nightingale managed to secure an audience with Queen Victoria and her advisers. She impressed on them the need for an inquiry into why so many soldiers had died in hospital. The queen and the government were not keen on any further investigation into the war, but Nightingale’s reputation was by now legendary. So the government decided to ask her to write a confidential report that would be presented to a new royal commission. She wanted to help out, but what should she write? And, more importantly, how could she show the horror and tragedy that she’d seen unfold at Scutari?

Nightingale was afraid that the government would ignore her figures, and realised the essential facts, her clarion call to action, had to smack them in the eye. So she created a diagram, now called the rose diagram, to distil the message behind the numbers.

5.2 Florence Nightingale’s rose diagram.

The diagram consisted of two roses. On the right-hand rose, she depicted the battle year 1854–5, with the soldiers’ deaths shown for each month, according to the cause of death. And on the left, a smaller diagram shows that of 1855–6. What matters is the area of each colour. The central area she coloured in red (depicted in dark grey in our version) and represented death from wounds, the black was death from other causes, such as frostbite or accidents. But the staggering number of deaths from infectious diseases like dysentery and typhus appeared as large blue rose petals, bursting out from the centre (depicted in light grey here).

Nightingale does not provide the number of deaths, but the areas of blue nonetheless create a disturbing impression. As the winter of 1854 advances, they get steadily bigger – until, in January 1855, over 2500 men are dying in a single month. But the second rose shows that things didn’t need to be like this. The far smaller blue region in the second diagram revealed that improved sanitation in the hospitals was the trigger for a dramatic drop in the numbers of lives lost due to infectious diseases.

It was this diagram, rather than all the words written in the report, that forced the British establishment to recognise that army medical practice was killing off thousands of soldiers unnecessarily. Its strikingly visual appeal won over both hearts and minds, and set in motion a process of reform that would change medical care for ever.

The diagram is meant to grab the eye first, and engage the brain second. Nightingale wrote that the diagram should ‘affect thro’ the Eyes what we fail to convey to the brains of the public through their word-proof ears’. It provides a shortcut to the message hidden in the numbers.

I recently learned from Ian Lipkin, professor of epidemiology at Columbia University, of a more modern version of the power of visuals to persuade governments of health risks. For many years he has been advising governments on responses to pandemics. But he told me that his first attempt to explain to the US government about the potential impact of a pandemic was met by stony silence. His thorough report, seven hundred pages long, had probably gone unread. So he prepared a highly condensed version. Still silence. Eventually he realised that he needed to change the medium. Instead of words in a report, he made a movie: Contagion. Starring Matt Damon and Gwyneth Paltrow, the film’s visual impact of a virus killing so many people startled the US government into action, just as Florence Nightingale’s rose diagram had done in Victorian times.

Nightingale’s rose diagram illustrates the power of representing a complex problem visually to provide a shortcut to understanding. It was not the first such diagram. In fact she probably took some inspiration for the picture from the work of William Playfair. His book The Commercial and Political Atlas published in 1786 contains 44 graphs. Most of them plot time against some other value, in the familiar x/y form. But one is slightly different. It’s a very early form of bar chart recording exports and imports from Scotland: not as a graph but bars representing each of the values. This is the kind of diagram that Nightingale might have seen and considered herself.

Playfair believed that our brains had evolved to decode certain messages much more accurately when seen in pictures: ‘Of all the senses, the eye gives the liveliest and most accurate idea of whatever is susceptible of being represented to it; and when proportion between different quantities is the object then the eye has an incalculable superiority.’

Today in this very visual age we are bombarded by pictorial representations of numbers. Diagrams to decode the secrets hidden in data are a powerful political and commercial tool. But just as a good diagram can provide a shortcut to understanding, a bad one can completely mislead.

Some news organisations are notorious for abusing diagrams to communicate a political message. Look at the bar chart overleaf. It was used to illustrate the apparent disastrous effect on taxes if US president George W. Bush’s tax cuts expired. The difference looks enormous. Until you notice that the vertical axis does not start at 0 but 34. Redraw the picture with the axis starting at 0 and the difference becomes much smaller.

5.3 Two different perspectives on the effect of tax cuts.

Another classic misuse of bar charts is the following:

5.4 A misleading diagram of company sales.

The graph is meant to show the dominance of Company C over Company B or A. But it is just the height of the icon which is important in recording the data. Yet by also scaling up the icon’s width, the company has completely exaggerated its importance. Although Company C’s sales are 5 times bigger than Company A’s, it would require 25 times the icon of Company A to cover the icon of Company C.

In some ways Florence Nightingale’s rose diagram missed a trick here. She created it so that the rose’s area corresponded to the numbers, but because the petal’s area expands in every direction she ended up downgrading the impact. If she had replaced the rose with a bar chart, the heights of the sections corresponding to the blue regions would have been even starker in contrast to the rest.

Map making

A map is probably the perfect example of a diagrammatic shortcut. It is not a replica of the land it is mapping. For a start the whole point is that it is a scaled-down version of the land being surveyed. Even then, many features have to be thrown away. But map the landscape well, choosing the essential features to include and discarding what is not necessary, and you have an amazing shortcut to finding your way.

I’ve always enjoyed the story that Lewis Carroll tells in his last novel Sylvie and Bruno Concluded of a country that didn’t appreciate the importance of discarding information when making maps. They prided themselves on how accurate their maps were:

‘We actually made a map of the country, on the scale of a mile to the mile.’

‘Have you used it much?’

‘It has never been spread out, yet. The farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well.’

As Carroll humorously points out, maps need to make choices about what to leave out.

Some of the first maps made by humans are maps of the heavens, not the Earth. The caves at Lascaux in south-western France map the configurations of the Pleiades, stars that were often used to mark the beginning of a yearly cycle. One of the first maps of the Earth is a clay tablet that was carved by a Babylonian scribe maybe as long ago as 2500 BC. It shows a river valley between two hills. The hills are depicted by semi-circles, rivers by lines, cities by circles and it also includes directions for which way to orient the map.

The Babylonians are also responsible for the first attempt to map the whole world, which dates to 600 BC. The map is very much a symbolic rather than literal attempt. It depicts a circular shape surrounded by water, the Babylonian view of the lie of the land.

But once it was recognised that the Earth was spherical rather than flat, creating a two-dimensional map of a spherical surface became an interesting challenge for cartographers. It is the sixteenth-century Dutch cartographer Gerardus Mercator who is generally credited with finding a clever solution.

Because this was the age of exploration of the planet by sea, Mercator’s principal goal was to create a map that would help sailors get from one point on the planet to another. The key tool for navigation was a compass. The easiest way to get from A to B was to know a fixed compass direction to head in such that if the boat kept to that you would arrive at your destination.

Such lines have a constant angle to the lines of longitude running north–south. They are called rhumb lines and drawn on a globe can be seen to spiral into the North Pole.

They are not the shortest paths from A to B but if you are more concerned about not being thrown off course, these were by far the best paths to take.

Mercator’s map has the wonderful property of turning these curved paths into straight lines. If you wanted to find the correct angle to set course from A to B, you had only to draw a straight line on Mercator’s map between the two and the angle of the line to the lines of longitude running north would be what you’d need to maintain as you sailed across the ocean.

5.5 A rhumb line maintains a constant angle to the lines of longitude.

This projection of the sphere onto the rectangle is called a conformal mapping because it preserves angles. It can be achieved by doing the following. Imagine the Earth is a balloon, with wet ink all over its surface. Wrap a cylinder round the Earth so that the cylinder touches the Equator. Now start to inflate the Earth, so that gradually its surface begins to come into contact with the cylinder, the ink printing out a map of the surface as the balloon expands.

Unwrap the cylinder and you have your map. It’s not possible to map the poles in this way so the top will be a line of latitude close to the pole. The effect of this map is to stretch out the lines of latitude as you head north or south from the Equator. It was a fantastic tool for those at sea, which was clearly Mercator’s aim given that he gave the map the title ‘New and more complete representation of the terrestrial globe properly adapted for use in navigation’.

While angles between lines on the globe were preserved in the map, geographical area and distance were not. And this has had huge political implications. Because of how useful the map was, for centuries it became the accepted view of what the planet looked like. But the map greatly inflates the importance of countries that are far from the Equator, those like the Netherlands and Great Britain. For example, if you draw a circle on the Equator and a circle of the same size over Greenland, you will find that when these are mapped using Mercator’s projection, the second circle grows ten times in size. Africa, for example, is made to look about the same size as Greenland when in reality it is 14 times larger.

The Mercator map fell foul of post-colonial politics and an alternative called the Gall-Peters map was adopted by UNESCO. It is the map widely used in British schools but in the United States it was only as recently as 2017 that this map replaced the Mercator map in classrooms, in the Boston school system. Many other US school districts have not followed suit. For many the shrinking in size of the USA does not chime well with US citizens’ view of their country’s place in the world.

The truth is that any map is going to make compromises. In fact Gauss made this discovery when he was investigating the nature of the curvature of different geometries. In a discovery he called his Theorema Egregium, or his Remarkable Theorem, he proved that a flat map cannot be wrapped round a spherical globe without distances being distorted. Something has to be compromised in any map of the planet. The areas may be right in the Gall-Peters map but the shapes of the countries are not. Africa looks twice as long as it is wide while in reality it is more square-shaped.

Of course, most maps have always put the northern hemisphere at the top and the southern at the bottom. But a sphere is symmetrical so there is no reason not to have oriented the map the other way round. Again the choice reflects the fact that it was residents of the northern hemisphere who were drawing up the maps.

Australian resident Stuart McArthur decided to counter this northern bias in maps by producing one with the southern hemisphere at the top. When you see it for the first time it is quite a shock. It just doesn’t look right. And yet this is simply a reflection of how much we’ve got used to Mercator’s vision of the planet.

Maps are all about what it is you are trying to achieve. Is this a shortcut to navigation? A shortcut to understanding land size? Most try to preserve some geometric feature. Perhaps distances on the map correspond to distances on the planet. Or angles between lines are the same. But sometimes a good map throws all these things away and just preserves the most important features of how to get from A to B.

One of the maps I love most and use on a daily basis is the map of the London Underground. A physical map showing the geographic locations and routes on the Underground is not a very helpful picture to negotiate your way round the city. Instead Harry Beck’s iconic map published in 1933 isolates the way the network is connected together while ignoring physical dimensions. The map was so revolutionary that initially it was rejected by the company running the Underground. The trouble was that they were losing money because Londoners were not using the system. When they tried to find out why, they discovered that people just couldn’t navigate round the network. The maps the company had produced were trying to replicate the geography of the city but it resulted in a rather cramped, tangled mess of lines that people found difficult to read.

Beck had seen the problem and had decided that you needed to abandon geographical accuracy. Instead he pushed and pulled the lines around, straightening them out, making them cross at clean angles, stretching stations apart. It may have helped that Beck had a background in electronics because the map resembles something closer to the layout of an electronic circuit board than a train map.

Having realised that they needed a better map that passengers could use to navigate the system, the company eventually decided to adopt Beck’s proposal. Three-quarters of a million copies of the map were printed off and distributed to passengers. The map has become an international icon. It has inspired works of art: Simon Patterson’s reworking of it hangs in the Tate Modern in London with station names replaced by those of engineers, philosophers, explorers, planets, journalists, footballers, musicians, film actors, saints, Italian artists, sinologues (Chinese scholars), comedians and ‘Louis’ (French kings). J. K. Rowling even gave Professor Dumbledore a scar on his left knee in the shape of the map, a nod to the fact that she had her best ideas for the Harry Potter series of books while sitting on trains.

The power of the London Underground map is that it is not a geographic map but focuses instead on the more important quality of how to get from A to B. The fact that the same length of line was used to represent the connection between Covent Garden and Leicester Square as between King’s Cross and Caledonian Road did not mean the distances are the same. For a commuter, knowing there is such a connection is much more important than knowing the distance between the stations.

This is an example of a new way at looking at the world that was introduced in the mid nineteenth century. Here the exact distances between objects are not important and often it is how they are connected together that is key to the identity of the shape. Gauss was one of the first to start considering how the properties of surfaces might depend less on their physical geometry and more on how points on them were connected. Although he never published his ideas, they were the inspiration for the work of Johann Benedict Listing who in 1847 first used the name topology to describe this new way of seeing the world. We shall see in Chapter 9 how topological maps can be a very handy shortcut to finding your way around networks, not just on the London Underground.

But diagrams need not be restricted to showing the physical connections between locations in London. Very effective use has been made of maps that replace Underground stations with mental ideas. Called mind maps, their purpose is to tease out interesting connections between different ideas that you might be exploring. Mind maps have for years been the staple diet of students trying to cram for exams because they help to create an integrated story of a subject that in words can feel too difficult to navigate. In some ways they tap into Ed Cooke’s memory palaces. The mind map can turn a hotchpotch of ideas into a physical journey that you can navigate on the page.

These diagrams nonetheless have a long history. Newton made doodles in his notebooks showing a sort of mind map he used as an undergraduate in Cambridge to reveal his thoughts on how different philosophical questions might be interrelated. The point is that these maps hope to disrupt the rather linear manner in which a textbook might present ideas and try instead to mimic the more multi-dimensional manner in which we process ideas in our minds.

Mapping the big and the small

As Leonardo articulated, the visual world can describe things that will forever transcend the written word. A single image can convey the simple underlying pattern hidden by the complexity of words or equations. But a diagram is more than just a physical representation of what we see with our eyes. The power of a diagram is to crystallise a new way of seeing the world. Often it requires throwing away information, focusing on what is essential, as Lewis Carroll’s humorous unscaled map illustrated. At other times it changes a scientific idea into a visual language, providing a new map in which the mathematics of geometry takes over and helps us to navigate the science at hand.

The Polish mathematician and astronomer Nicolaus Copernicus certainly understood the power of a good picture. In his great opus De revolutionibus orbium coelestium published shortly before his death in 1543, Copernicus took 405 pages of words, numbers and equations to explain his heliocentric theory. But it is the diagram that he drew at the beginning of the book that captures in a simple image his revolutionary new idea: that the Sun is at the centre of the solar system, not the Earth.

5.6 Copernicus’ diagram of a Sun-centred solar system.

His picture encapsulates some of the essential elements of the best diagrams. The concentric circles are not meant to describe the precise orbits of the planets. Copernicus knew they weren’t circles. The uniform distances between the circles aren’t meant to tell you how far the planets are from the Sun or from each other. Rather this picture conveys the simple yet shocking idea that we aren’t at the centre of things. It transformed our view of our place in the universe.

Today cosmologists use diagrams to chart the entire universe over its 13.8 billion years of history, diagrams to capture the workings of massive black holes, diagrams to navigate the complexities of four-dimensional space-time. The power of a diagram to provide a shortcut to the vastness of the universe is perhaps the only way we could manage to conceive of our place in something that at first sight seems impossibly large.

But diagrams can also work like magnifying glasses allowing us to see the very small. Go into any chemistry lab and you will see written up on the whiteboards letters connected by single, double and sometimes triple lines depicting the bonds between the atoms that the letters represent. These are the diagrams that tell chemists how atoms are put together to make up the molecular world.

5.7 Molecular diagrams.

A picture of methane shows a C with four lines emerging from the central C with Hs at each end to depict a molecule of CH_4: 1 atom of carbon and 4 atoms of hydrogen. The colourless flammable gas ethylene C_2H_4 has a slightly different structure with a double line between the Cs and 4 Hs attached to them. How molecules might react and change can be navigated by using these diagrams. A double bond is often accompanied by the molecule being more reactive than those with a single bond. In chemistry we get so used to manipulating these diagrams that it is easy to forget they are a shortcut to the extraordinary reactions happening on a scale that is hard for a microscope to pick up. But these diagrams can also lead to the discovery of new structures hidden inside the molecular world.

As illustrated by the methane molecule, carbon likes to have 4 lines emerging from it. Hydrogen will have only 1 line. So it was something of a mystery when the benzene molecule was first extracted by Michael Faraday in 1825 and found to be made from 6 carbon atoms and 6 hydrogen atoms. If you try to make a diagram of its structure, the numbers just don’t seem to work out. It appears impossible to cover the 6 greedy carbon atoms each with 4 arms emerging using just 6 single-armed hydrogen atoms. It was August Kekulé, a German chemist working in London, who finally cracked the mystery.

‘One fine summer evening I was returning by the last bus riding outside as usual, through the deserted streets of the city,’ he wrote. ‘I fell into a reverie, and lo, the atoms were gambolling before my eyes … The cry of the conductor: “Clapham Road” awakened me from my dreaming; but I spent a part of the night in putting on paper at least sketches of these dream forms.’

Yet the structure of benzene remained elusive. He worked many late nights trying to make sense of these diagrams until another dream finally revealed the secret.

‘I turned my chair to the fire and dozed,’ wrote Kekulé. ‘Again the atoms were gamboling before my eyes … long rows sometimes more closely fitted together all twining and twisting in snake-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke.’

5.8 The ring structure of benzene.

He had it. The way to use up the carbon arms was to put the atoms in a ring. They were shaking hands with each other and using just one arm each to shake the hand of a hydrogen atom. The discovery of the benzene ring and similar ring structures in other molecules was to guide the development of a new field of chemistry. It turns out that many molecules with such ring structures are aromatic. For example, if you swap one of the hydrogen atoms for another carbon atom attached to an oxygen atom and two hydrogen atoms, the resulting molecule smells of almonds. Change that to a slightly longer molecule consisting of a string of three carbon atoms, an oxygen and three hydrogen atoms and the smell changes to cinnamon.

These molecules are simple enough that their structure can be depicted in a 2D diagram. But more complex molecules like haemoglobin are much more challenging to capture in a picture. The biochemist John Kendrew had successfully pieced together the structure of this protein’s crystal by using a large number of 2D x-rays, work for which he went on to receive the Nobel Prize in 1962. It was an extraordinary feat: the molecule consists of over 2600 atoms (and that is still quite small for a protein molecule). Although Kendrew had managed to make a picture of the structure in 1957, he decided he needed the help of a master draughtsman to really capture his discovery. He turned to Irving Geis, a trained architect and fine artist. After six months of work poring over Kendrew’s papers and models, Geis produced a watercolour image that appeared in the June 1961 edition of Scientific American. The stunning image made Geis famous, but is almost too complex to provide a shortcut to really navigating the molecule’s properties.

Probably the ultimate molecular challenge came with trying to picture DNA. As I’ve stressed, the secret of a good diagram is often to throw away information. When Francis Crick and James Watson wrote their paper for the journal Nature explaining the double helix structure of DNA, they could have drawn an incredibly complicated image of it with the full molecular description. But the essence of the discovery was the two strands that make up DNA and explain how the molecule enables our genes to be passed down the generations. They famously announced the success in the pub in Cambridge that they used to drink at. When Crick rushed home to say he’d found the secret of life, his wife Odile was rather dismissive. ‘He was always coming home and saying things like that,’ she said.

Interestingly Odile, who was a trained professional artist, played a major role in bringing the news to the world’s attention because she created the diagram that appeared in their Nature paper. Crick had given her a sketch of the sort of thing he wanted but he didn’t have the artistic skills to reveal the important message in their discovery. Odile had studied in Vienna in the thirties and had gone on to St Martin’s in London and the Royal College of Art. She did occasional portraits of her husband, but most of her work was focused on the female nude. Molecular structures weren’t really her thing.

But as Francis Crick explained the discovery with the aid of his rather cluttered sketch, Odile got the point and turned his vague impressions into a memorable image, the power of which she probably didn’t realise at the time – because this double helix has become a symbol not just of DNA or biology or even of scientific discovery.

From the beginning, the double helix appealed to artists. Salvador Dalí was quick to add it to his palette of scientific metaphors. He called this his period of ‘nuclear mysticism’ and his use of DNA revealed a surprisingly conservative and religious aspect to his art.

However for me one of the most amazing uses of diagrams is the Feynman diagrams. They not only enabled us to see things that were impossible even for a microscope to pick up, but also to shortcut the need for some extraordinarily complex calculations.

While a chemist’s blackboard is covered in Cs, Hs and Os connected with lines, on the physicist’s blackboard you will probably find diagrams representing the interactions of the fundamental particles that make up the chemists’ atoms. These dynamic diagrams show the evolution over time of what might happen, for example, when an electron and a positron interact.

5.9 Feynman diagram of the interaction between an electron and a positron.

The physicist Richard Feynman dreamed up the diagrams as a way to keep track of the hugely complex calculations he was making to understand these particles. He first revealed his discovery of this diagrammatic shortcut in a meeting of theoretical physicists at the Pocono Manor Inn in rural Pennsylvania in the spring of 1948.

At the closed meeting to discuss the theory of quantum electrodynamics or QED, explaining how light and matter interact, a young wunderkind from Harvard, Julian Schwinger, had spent the day explaining his complex mathematical approach to QED. The marathon all-day lecture had been punctuated with a few breaks for coffee and lunch, and by the end of it the audience’s brains were probably fried. It perhaps explains why when Feynman stood up at the end of the day to explain his approach and started drawing pictures on the board, people were initially perplexed by how these diagrams could help to do calculations. Indeed some of the big guns who sat through the lecture such as Paul Dirac and Niels Bohr were flummoxed by Feynman’s pictures, concluding that the young American simply did not understand quantum mechanics.

Feynman left the meeting disappointed and depressed. But the diagrams were eventually rescued by one of the other big names of physics, Freeman Dyson, who understood that they were in fact equivalent to the complex mathematical calculations that Schwinger was doing. It wasn’t until Dyson explained the insight during a lecture that the physics community started to take these pictures seriously. The articles that Dyson wrote subsequently offered a how-to guide, including step-by-step instructions for drawing the diagrams and how they were to be translated into their associated mathematical expressions.

Today these diagrams concocted by Feynman are the first port of call for any theoretical physicist trying to tease out what happens when particles interact. They are an amazing diagrammatic shortcut to the interplay happening within the very foundations of the physical universe. No experiment has ever captured a quark in isolation and yet on the blackboard these diagrams give us a way to navigate the evolution of such an elementary particle as it interacts with its environment.

My colleague in Oxford, Roger Penrose, has developed a similarly powerful visual shortcut to some of the most complex ideas in fundamental physics. His theory of twistors proposed in 1967 is an attempt to unify quantum physics – the physics of the very small – with gravity, a physics generally of the very large. It is a hugely mathematical theory and, for Penrose, the best way to navigate the complex mathematics was to draw pictures. Fortunately he is quite an adept artist in his own right having had some interesting interactions with the Dutch visual artist M. C. Escher. Penrose’s artistic skills probably helped him to create the diagrams that are the best shortcut to coping with the complex mathematics of his theory.

Although introduced in the late sixties, Penrose’s ideas have recently become mainstream thanks to new work connecting his theory with current thinking. One of the diagrams that has emerged out of this new approach, given the name the amplituhedron, has provided an amazing shortcut to understanding the physics of the interaction of 8 gluons, the particles that glue quarks together via the strong force. Even using Feynman diagrams, the same calculation would have taken roughly 500 pages of algebra.

‘The degree of efficiency is mind-boggling,’ commented Jacob Bourjaily, a theoretical physicist at Harvard University and one of the researchers who developed the new idea. ‘You can easily do, on paper, computations that were infeasible even with a computer before.’

Venn diagrams

You might recognise the sort of diagram that was used in the puzzle I set at the beginning of the chapter. Called Venn diagrams, they are an effective visual way to organise information. Each circle represents a concept and the regions created by the circles intersecting or not intersecting give the different logical possibilities for how the concepts relate to one another. For example, take the idea of a number being (a) a prime number, (b) a Fibonacci number and (c) an even number. We can distribute the numbers from 1 to 21 according to how many of these categories they satisfy.

5.10 The Venn diagram of prime numbers, Fibonacci numbers and even numbers.

The Venn diagram is a clever pictorial way to represent the different possibilities. In this case it reveals that 2 is the only even prime (mathematicians think it’s funny to say 2 is an odd prime because it’s the only even prime). There are no numbers that are even and prime but not Fibonacci numbers.

These diagrams are named after the English mathematician John Venn, who introduced them in 1880 in a paper he wrote entitled ‘On the Diagrammatic and Mechanical Representation of Propositions and Reasonings’. The diagrams were meant to help to navigate the logical language that his contemporary George Boole was developing. As well as diagrams, Venn also specialised in making bowling machines for cricketers to practise their batting. The Australian cricket team asked to try out the machine on a visit they made to Cambridge where Venn worked. They were rather shocked when it bowled out their captain four times in a row. But for Venn his diagrams were of more lasting importance.

‘I began at once somewhat more steady work on the subjects and books which I should have to lecture on,’ he wrote. ‘I now first hit upon the diagrammatical device of representing propositions by inclusive and exclusive circles. Of course the device was not new then, but it was so obviously representative of the way in which anyone, who approached the subject from the mathematical side, would attempt to visualise propositions, that it was forced upon me almost at once.’

Venn was right that the idea of using graphical images to represent logical possibilities was not new. Indeed there is evidence of the thirteenth-century philosopher Ramon Llull having created something similar. He used his diagrams to understand the relationship between different religious and philosophical attributes. They were meant as a debating tool for winning Muslims to the Christian faith through logic and reason.

But it’s Venn’s name that has stuck. Most often you will see the diagrams considering just three different categories. That’s because this seems the easiest diagram to draw on paper to represent all the possibilities. When you get to four different categories you have to work much harder to make your regions intersect in such a way that all logical possibilities are covered. For example, this isn’t good enough:

5.11 This diagram does not represent the Venn diagram of 4 sets.

There is nowhere representing being in region A and region D but not in the other two. Instead you need a diagram that looks like this:

5.12 The Venn diagram of 4 sets.

A seven-set Venn diagram is starting to miss the point about diagrams being aids to understanding:

5.13 The Venn diagram of 7 sets.

One of my favourite books has been Venn That Tune by Andrew Viner, who used Venn diagrams to capture song titles. The challenge at the beginning of this chapter is one of his. The diagram is a shortcut for ‘Stuck in the Middle with You’ by the British folk rock band Stealers Wheel.

Shortcut to the shortcut

How would you depict your message or data as a picture or diagram? There are many different genres that you can employ which might provide a shortcut to understanding. A simple graph showing the connection between your business’s profits at different times of year. A bar chart keeping track of the most popular dishes in a café. A Venn diagram to explain overlaps and differences in opinions between political parties. Or perhaps a network diagram like that of the London Underground to reveal connections between ideas that words are obscuring.

Pitstop: Economics

‘The most powerful tool in economics is not money, nor even algebra. It is a pencil. Because with a pencil you can redraw the world.’ These are the opening lines of Kate Raworth’s book Doughnut Economics in which she explains a new diagram that challenges the twentieth-century story of economics. It is a diagram in the shape of a doughnut.

5.14 The doughnut economic diagram.

I am a big fan of Raworth’s book, partly because the doughnut (or torus as we call the shape in mathematics) is one of my all-time favourite shapes. Not just because it is delicious to eat but because the mathematics of this shape is fascinating. Understanding its arithmetic was at the heart of proving Fermat’s Last Theorem. Its topology has been crucial to understanding the possible shape of the universe. But as I discovered in Raworth’s book, it is also key to a revolution in economics. So I was keen to talk to her about the genesis of this game-changing diagram as a shortcut to economic thinking.

Open any economics book, attend any economics lecture, watch any economics video and you’ll invariably see the same couple of diagrams appearing over and over again. One is a graph of growth that always shows a line bending upwards in an exponential way promising a future of seemingly unlimited production. The other is a graph with two straight lines or two curves crossing in an X shape, depicting supply and demand plotted against quantity and price. The demand curve shows how the cheaper the price, the more someone can buy. The supply curve shows that the amount a supplier will produce goes up if the price goes up. Placing the curves on top of each other is meant to reveal the economic equilibrium, the price at which quantity demanded equals quantity supplied.

These diagrams have been so powerful that they have led to the idea that economics is, at heart, only about supply and demand. But Raworth wanted to challenge this model. It was missing so much that was important to understanding global economics: the environment and human rights. As George Monbiot wrote in his book Out of the Wreckage, the best way to counter one story is with another story. Raworth’s philosophy is similar: ‘These old diagrams are like intellectual graffiti on the mind and like graffiti it’s very hard to scrub out,’ she says. ‘The best thing you can do is paint over it with something new.’

Raworth has always found visuals the best way to understand complexity. ‘At school I was discouraged from drawing pictures in the margins of books but now we understand that there are many forms of intelligence and visual intelligence is one of them. I loved reading Feynman as a teenager. His books were full of drawings. Maybe that told me early on that this is part of understanding even though other people said I was doodling.’

Raworth went on from school to study economics but felt that the subject didn’t really understand how human societies work. ‘I began to really be ashamed of the concepts I’d been taught.’

It was while doing jury service that Raworth came across a diagram by Herman Daly, an economist working at the World Bank, that sowed the seed for her economic insight. Daly wanted to challenge the assumption of limitless growth and so proposed that an outer circle be drawn around the economists’ graphs that should be labelled ‘The Environment’.

‘The power of a great diagram,’ says Raworth, ‘is that once you’ve seen it, you can’t unsee it. It allows you to make a mindset leap, a paradigm change.’

For years, Daly’s diagram sat in the back of Raworth’s mind until, while she was working at Oxfam, a diagram inspired by his idea sparked her diagrammatic epiphany. This second diagram changed her perspective on economics. It was environmental scientist Johan Rockström’s diagram of the nine planetary boundaries that represent the safe operating space for humanity. There was Daly’s circle but now there were big red regions radiating out from the middle, each representing things like the ozone layer, the water cycle, climate, ocean acidity. The trouble was that many of them were overshooting the circle.

‘I just had this visceral reaction,’ says Raworth. ‘This I realised is the beginning of twenty-first-century economics.’

But it wasn’t just a pretty picture. It was backed up by numbers. Economists usually measure everything by converting it into dollars. This is meant to be a clever shortcut that allows one to compare seemingly incompatible quantities. One number to rule them all. But Raworth is suspicious of this one-dimensional view. She compares it to trying to drive a car where all the information about speed, temperature, revs, how much fuel is left in the tank, are condensed into one dial. You would never drive that car.

‘You actually want the dashboard,’ she says. ‘Humans are very good at dashboards. We live in a complex system. Hiding the complexity isn’t going to give you a richer decision tool. It’s a dangerous shortcut.’

That was what made these new charts exciting. They weren’t using dollars as the sole metric. Instead they used multiple metrics: tonnes of carbon dioxide, tonnes of fertiliser use, ozone metrics. And yet, Raworth believed the diagram was still missing an important component: the human. ‘I was sitting in Oxfam surrounded by people who were responding to an emergency drought in the Sahara or campaigning for health and education for kids in India and thinking, Well, if there is an outer circle representing a limit of pressure that humanity could put on the planet then there’s also an inner limit which we’ve called for nearly 70 years human rights. The right of how much food everybody needs each day or how much water you need or the minimum conditions of housing or education to be part of society. If there’s this outer circle then I decided we needed to draw in an inner circle.’

At this point Raworth went up to the whiteboard of my office and drew a doodle of a doughnut with an outer ring for the environment and an inner ring for human rights.

Initially, Raworth kept the diagram to herself. Then at a meeting of Earth system scientists in 2011 discussing the nine planetary boundaries, someone turned to Raworth as the representative from Oxfam and said, ‘The problem with this framework of planetary boundaries is there’s no people in it.’

‘There was a big whiteboard on the wall,’ says Raworth, ‘I said, “Can I draw a picture?”’

She jumped up and drew the doughnut on the whiteboard and explained that just as we need an outer circle to bound the impact humans are having on our environment, we also need an inner circle that represents the minimum conditions for every human on our planet: food, water, health care, education, housing.

‘We need to use Earth’s resources to meet everyone’s needs but not use them so much that we go beyond the limits of the planet. We want to be in this space in between,’ she says, pointing to the doughnut. ‘I drew really quickly because I thought they’d say: yes, dear, sit down. But instead they responded excitedly that that’s the picture we’ve been missing all along and it’s not a circle it’s a doughnut.’

Raworth wrote up her diagram as a discussion paper for Oxfam and published it to immediate enthusiasm. ‘That was the moment when I was really struck by the power of images as a shortcut. If you take all the words that were in it – food, water, jobs, income, education, political voice, gender equality, climate change, ocean acidification, ozone layer depletion, biodiversity loss, chemical pollution – and wrote those things in a list, nobody would blink. But if you draw them in relation to each other in a pair of concentric circles, people say this is a paradigm change.’

As John Berger wrote in his classic 1972 book Ways of Seeing: ‘Seeing comes before words. The child looks and recognises before it speaks.’

For Raworth a diagram is a shortcut but it also encapsulates a world view. And this is its danger because it might simply be a shortcut to the way you see the world. It may indeed hide things that in your view are not important but which for others might be fundamental to their vision. If a company is only interested in short-term corporate profit, it might be happy with the exponential growth graph, but if you care about the environment then hiding the impact of growth on climate means the shortcut has been very selective in who has got quickly to their desired destination. It has put another group far from the goal they are after. Since a diagram throws away extraneous data, it borders on cutting corners. The corners you are cutting off, Raworth believes, are potentially a reflection of your world view. What is a shortcut for one economist to explain their ideas, might be completely the wrong path for another, leading people away from what they believe is the correct destination.

‘The shortcut might be leading you down an extremely dangerous hole,’ she says. ‘The quote I’ve come to love is from a mathematician called George Box: all models are wrong, but some are useful.’

The doughnut became one of seven new diagrams that Raworth offers in Doughnut Economics as a shortcut to a new economic destination. As she remembers the writing of the book, she admits that just like digging that tunnel through a mountain, creating these shortcuts was hard work.

But it was work that is urgent given the direction that the planet and humanity are going in.

‘To rewrite economics so it’s a fit tool for the twenty-first century,’ says Raworth, ‘we need to use all the shortcuts we can because we have little time!’

6

The Differential Shortcut

If you rolled balls down each of these slopes, which one would get the ball to the end fastest? Which is the shortcut, A, B or C?

6.1 Which is the fastest line of descent?

As astronaut Lieutenant Colonel John Glenn orbited Earth for the third time he began to prepare his spacecraft for its re-entry into the planet’s atmosphere. It was 20 February 1962 and Glenn had just become the first American to orbit the Earth. But before the mission could be counted a success, he still needed to get home safely. The trajectory he chose was going to be crucial. Get the angle of descent wrong and the spacecraft would burn up on entry. Land the craft too far out to sea and the navy would not get there in time to stop the capsule plummeting to the seabed.

Glenn was putting his life in the hands of the calculators that had crunched the numbers. In 1962 these calculators were not machines. They were a group of women since immortalised in the 2016 Hollywood film Hidden Figures. In the film Glenn sits on the launch pad before initiating blast-off and asks mission control, ‘Get the girl to check the numbers.’ The girl in question was Katherine Johnson, one of the team of calculators NASA was using. In the movie it takes her 25 seconds to run the maths and confirm that everything is on course.

In reality Johnson’s calculations were made weeks before blast-off and probably took two to three days. Even that’s still an impressively fast time to navigate such a complex range of possible paths and scenarios. But Johnson had a shortcut up her sleeve that allowed NASA and every agency that has ever sent objects into space to know where their spacecraft would end up: calculus, probably the most powerful tool for finding shortcuts ever invented by mathematicians. From landing a probe on a comet to sending a craft on a fly-past of the planets, calculus is the signpost that will point the spaceship in the right direction so that it will reach its destination.

And it’s not just the space industry that has capitalised on the power of this mathematical shortcut. Many companies have, in the effort to maximise output while minimising cost and to find the most efficient way to make their product. As have aerospace manufacturers aiming to create a wing that causes the least amount of drag so that fuel isn’t wasted. And tankers needing to find the fastest route through turbulent waters. Brokers trying to spot the moment a stock hits its maximum value before it crashes. Architects who want to design buildings that maximise space given the constraints of the surrounding environment. Engineers constructing bridges that need to minimise the use of materials without compromising structural stability.

All of these need calculus to achieve their goals. If you’ve got a complicated equation describing the economy or energy consumption, or whatever you’re interested in, calculus is a way of analysing that equation and finding the points where the output is greatest or least.

It’s also a tool that gave the scientists of the seventeenth century the ability to understand a world in flux. Apples were falling. Planets were orbiting. Fluids were flowing. Gases were swirling. Scientists wanted a way to take a snapshot of all these dynamic scenarios. And calculus provided a way to freeze-frame all this motion. It is striking that it mirrored the interests of artists working at the same time. Baroque painters depicted soldiers falling from horses, architects designed buildings with sweeping dynamic curves, sculptors captured in stone the moment Daphne metamorphoses into a tree in Apollo’s arms.

The scientific revolution that happened during the second half of the seventeenth century owes its development to two of the great mathematicians of the age: Isaac Newton and Gottfried Wilhelm Leibniz. The development of calculus by these two great men provided the most staggering shortcut to navigating our dynamic universe. Richard Feynman once described it as ‘the language God talks’.

So if you haven’t learned calculus yet, now is the time. It will require some equations but I promise you, they are worth it.

A universe in flux

Even before John Glenn had completed his orbits of the Earth, calculus helped get him up there. As he sat on the launch pad he knew the spaceship would need to achieve a particular speed to be able to clear the gravitational pull of the Earth, called the escape velocity. But knowing what the speed of a spaceship is at any point as it is propelled into space is not an easy task. Things are constantly changing: the mass of the craft is shrinking as it burns off fuel, the pull of gravity is decreasing as it travels further and further away from Earth. As the push of the jets and the pull of gravity compete, it seems an impossible puzzle to unpick. But the real strength of calculus is that it can accommodate a hugely complex range of changing variables to give a snapshot of what is happening at any particular moment in time.

And it all began with that apple falling from the tree in Newton’s garden in Woolsthorpe Manor in Lincolnshire. He had retreated back to his family home from his college in Cambridge after plague had hit. Lockdown during a pandemic has certainly been a productive time for some. Shakespeare is said to have completed King Lear while the Globe was closed during a lockdown. As Newton sat in his garden, he wanted to make sense of the challenge of calculating the speed of his apple at any point on its journey from the tree to the ground. Speed is distance travelled divided by the time it takes to travel that distance. That’s fine if the speed is constant. But the trouble was that because of the pull of gravity the speed was constantly changing. Any measurements that Newton did would only give him the average speed over the period of time he was measuring.

To get a better calculation of the speed he could take smaller and smaller time intervals. But to get the exact speed at any moment really means taking an infinitely small time interval. Ultimately you want to divide distance by zero time. But how do you divide by 0? Newton’s calculus made sense of this.

Galileo had already discovered the formula for finding out how far the apple had fallen after any length of time. After t seconds the apple falls a distance of 5t^2 metres. The 5 here is a measure of the particular pull of gravity on Earth. An apple tree on the Moon will have a smaller number in the equation because its gravity is smaller and the apple falls more slowly. Glenn’s spacecraft would have to keep track of this number changing as he got further from the Earth.

Let’s take the apple and throw it directly up in the air. I’m going to launch it from my hand at a speed of 25 metres per second. Baseball pitchers can hit speeds of over 40 metres per second so this isn’t unreasonable. The formula for how high the ball is from my hand after launch becomes 25t − 5t^2.

I can use this formula to calculate the time it takes to reach my hand again, which is when the height above my hand, 25t – 5t^2, becomes 0 once more. Put t = 5 into the equation and I get 0. So the total time for the apple to travel up and back down is 5 seconds.

But what Newton wanted to be able to understand was how fast the apple is travelling at each point during its trajectory. Yet this speed is constantly changing as the apple slows down and speeds up again.

Let’s try and calculate the speed after 3 seconds, using our distance travelled divided by the time it takes to travel that distance formula. So the distance that the apple travels from 3 to 4 seconds is

[25 × 4 − 5 × 4^2] − [25 × 3 − 5 × 3^2] = 20 − 30 = −10 metres

The minus indicates that it is travelling in the opposite direction from the one that I threw it in. It’s already heading down. So the average speed over this period is 10 metres per second. But that’s just the average speed over this one-second interval. It’s not the actual speed of the apple at 3 seconds. What if I try to take a smaller time interval? If I keep making the interval smaller, what I find is that the speed gets closer and closer to 5 metres per second. But Newton was after the instantaneous speed, which is captured when the time interval becomes zero. His analysis produced a way to make sense of why the instantaneous speed at 3 seconds should be 5 metres per second.

6.2 The graph of the height of the apple with respect to time. The average speed of the apple between two points in time is the gradient of the line through the graph at these points.

It is possible to interpret this speed in a graph of the distance travelled over time. The average speed between 3 and 4 seconds is the gradient of a line drawn between the two points on the graph at 3 and 4 seconds. As I make the time interval smaller, my line gets closer to the line just touching the curve at time t = 3. What Newton’s calculus calculates is the gradient of the line that touches at this point, a line called the tangent to the curve. In general at time t, calculus tells us that the speed and the gradient is given by the formula

25 − 10t

Here is an explanation why: suppose we are trying to calculate the speed at time t. Let’s see how far the apple travels in a small interval following t, say from time t to time t + d.

[25(t + d) − 5(t + d)2] − [25t − 5t^2]

= 25t + 25d − 5t^2 − 10td − 5d^2 − 25t + 5t^2

= 25d − 10td − 5d^2

Now let’s divide by the time interval d:

25d − 10td − 5d^2/d = 25 − 10t − 5d

Let d get very small and the speed becomes

25 − 10t

This is called the derivative of the equation 25t − 5t^2. This clever algorithm takes the equation for distance travelled over time and produces a new equation that gives us the speed at any point in time. The power of this toolkit is that it doesn’t just apply to apples and spaceships. It gives a way of analysing anything in flux.

If you’re a manufacturer it is important to know the cost of creating your product so that you can set a price that will generate a profit. The cost of making the first product is going to be very high because of the cost of setting up the factory and employing workers, and so on. But as you make more and more products the marginal cost for each extra one will change. To start with, the marginal cost will go down because it becomes more and more efficient to make your product. But if you push production too far, costs can go up again. Increased production eventually leads to things like overtime, use of less efficient, older plants, and competition for scarce raw materials. As a result, the cost of additional units increases.

It’s a bit like throwing the ball in the air and at first the ball travels far but each subsequent second the ball slows and covers less ground. Calculus can help a manufacturer to understand how the costs of goods are changing as they vary the output, and to find the sweet spot of how many goods to produce to reduce the marginal cost to its smallest.

Newton’s shortcut to navigating a world in flux marked the beginning of modern science. I would rank Newton alongside Gauss as one of the all-time great shortcutters. I’ve even gone as far as making a pilgrimage to Woolsthorpe Manor where Newton is reputed to have sat under the apple tree that sparked the creation of this inspired shortcut. I was amazed to see that the tree is still there! The person who showed me round allowed me to take two apples from the tree and I managed to grow an apple tree in our garden from one of the pips. I spend many hours sitting under that tree hoping to find the shortcut that will guide me to the other side of the problem I’m currently working on.

Like me, Gauss was a huge fan of Newton’s work. ‘There have been only three epoch-making mathematicians,’ he wrote. ‘Archimedes, Newton, and Eisenstein.’ That last one is not a misprint. It is a young Prussian number theorist called Gotthold Eisenstein who had impressed Gauss by solving a couple of problems that he couldn’t crack.

Gauss was always rather sceptical about the story of the apple being key to Newton’s discoveries. ‘The history of the apple is too absurd,’ he wrote. ‘Whether the apple fell or not, how can anyone believe that such a discovery could in that way be accelerated or retarded? Undoubtedly, the occurrence was something of this sort. There comes to Newton a stupid, importunate man, who asks him how he hit upon his great discovery. When Newton had convinced himself what a noodle he had to do with, and wanted to get rid of the man, he told him that an apple fell on his nose; and this made the matter quite clear to the man, and he went away satisfied.’

It’s true that Newton had little time for publicising his ideas. For him calculus was not so much a device for optimising solutions but a personal tool that helped him to reach the scientific conclusions that he documented in the Principia Mathematica, the great treatise that he published in 1687 describing his ideas on gravity and the laws of motion. He explained that his calculus was key to the scientific discoveries contained inside: ‘By the help of this new Analysis Mr Newton found out most of the propositions in the Principia.’

He liked to refer to himself rather grandly in the third person. But no account of the ‘new analysis’ was published. Instead he privately circulated the ideas among friends but they were not ideas that he felt any urge to publish for others to appreciate. This decision not to formally publish his ideas was to have ugly consequences. Because some years after Newton’s discovery another mathematician also came up with the mathematics of calculus: Gottfried Leibniz. And it was his approach that highlighted the optimising power of this tool.

Maxing out

While Newton needed calculus to understand the fluctuating physical world that surrounded him, Gottfried Leibniz came at the ideas from a more mathematical, philosophical direction. He was fascinated by logic and language and was driven by the idea of capturing a whole range of different things in a state of flux. Leibniz had big ambitions. He believed in an incredibly rationalist approach to the world. If everything could be reduced to mathematical language, where everything was unambiguously expressed, there was hope to put an end to human strife: ‘The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, to see who is right.’

Although his dream of a universal problem-solving language was not realised, Leibniz was successful in creating his own language that could solve the problem of capturing things in flux. At the heart of his new theory is an algorithm a bit like a computer program or a mechanical set of rules that could be implemented to solve a vast array of unsolved problems. Leibniz was very pleased with his invention: ‘For what I love most about my calculus is that it gives us the same advantages over the Ancients in the geometry of Archimedes, that Viète and Descartes have given us in the geometry of Euclid or Apollonius in freeing us from having to work with the imagination.’

Just as Descartes’s idea of coordinates had converted geometry into numbers, Leibniz’s calculus had provided a new language to master and pin down explicitly the world of change.

Although Newton and Leibniz are credited with the great breakthroughs on making calculus into the powerful subject that is taught today, it was Pierre de Fermat, better known for his Last Theorem, who recognised that calculus can find shortcuts to the optimal solution to a problem.

Fermat was interested in trying to find a way to solve the following sort of challenge. A king has promised his trusty adviser a piece of land by the sea for his good services. The king has given the adviser 10 kilometres of fencing to mark out a rectangular plot of land bordered by the sea. The adviser obviously wants to maximise the area of the land. How should he arrange the fences?

Essentially he has one variable to play with: the length of the side of the rectangle that is perpendicular to the sea, which I will call X. As that grows the length along the beach that he can capture gets smaller. What is the balance between the two lengths that will make the area of land it surrounds largest? One’s first intuition might be to choose a square shape. Making things as symmetrical as possible is often a good strategy for finding the shortcut to a solution. A bubble, for example, chooses the symmetrical sphere as the shape that uses the least surface area to enclose the air inside. But is the symmetry of the square the right answer for our trusty adviser?

There is a very simple formula for the area of land depending on X, the length of the side that is varying. Since the beach length is 10 − 2X, the area A must be

X × (10 − 2X) = 10X − 2X^2

What is the value of X that makes this biggest? One strategy might be just to keep trying out values until we get a feel for the X that seems to be making the area largest. That’s the long way round this problem. Fermat realised that there was an easier way.

He found the shortcut was to turn the equation for the area into a picture. Draw a graph of the equation 10X − 2X^2. The shortcut will eventually spare you drawing the picture but to find shortcuts sometimes you need to take a detour first. The shape of the graph is a curve that climbs from zero area if X = 0 through to a peak that then decreases until you get zero area again if X = 5. The key is working out where that peak is. That’s where the area is greatest. What is the value of X that produces the peak?

6.3 The graph of the area of land with respect to the distance along one side. The area is maximal when the horizontal line intersects the graph at one point not two.

Draw a horizontal line across the graph. In general it will cut the graph’s curve at two points – except for that one spot right at the top when the horizontal line will just sit on top, touching at one point. This is the point we are after, the top of the graph where the area is biggest. Fermat found a strategy that identifies this point without having to draw the graph. It revealed that setting X = 2.5 would optimise the area of the land. The region was not a square but a rectangle, whose long side was twice the short side. If you are feeling brave enough to do a bit of algebra, here are the details of Fermat’s idea.

Suppose I set X = a then the horizontal line through this point will hit the graph on the other side at a point X = b where the height above this point is the same as the height above X = a. So this is the point at which

10a − 2a^2 = 10b − 2b^2.

I can use some algebraic tricks to simplify this. Put the square terms on one side:

2a^2 − 2b^2 = 10a − 10b

But I can factorise the side with the squares:

2(a − b)(a + b) = 10(a − b)

To factorise an equation means to recognise that an algebraic expression can be written as two simpler expressions multiplied together. In this case the difference of two squares is actually just the product of (a − b) and (a + b). But now I can see that both sides of this new equation are multiplied by (a − b) so I can cancel these terms on both sides to get

(a + b) = 5.

But Fermat is interested in the moment when a and b are actually equal because that is the top of the curve. That is the point where b = a. Putting this into the equation we get

2a = 5.

The point at which the graph tops out is when a is 2.5. This is the length of the side of the rectangle that will maximise the area of land. We get a rectangle which is 2.5 by 5.

There is an interesting moment in the calculation above when I divided by a − b. That was all OK unless a = b, in which case I just divided by 0 which isn’t allowed. But hold on. Didn’t Fermat want to find the place where a = b? So does that invalidate the whole shebang?

This is the point of calculus. It makes sense of how to divide by zero.

There were calculations here, but where was calculus? Calculus gives you the gradient of the line tangent to each point on the curve. Fermat has identified the maximum area as the point when the tangent is horizontal. This is the point when the gradient, or the derivative, is zero. This is the strategy for using calculus to find optimal solutions to the outputs of equations: find the point at which the derivative of the equation is zero.

The curve describing the area of land looks remarkably like the one that Newton drew to keep track of the height of his apple. The equation for the area of the land 10X − 2X^2 and the equation for the distance of the apple from my hand 25t − 5t^2 are essentially the same equation. The second equation is simply the first one multiplied by 2.5. This is one of the great shortcuts of mathematics. The same equation can cover a multitude of different scenarios. In the case of the apple, this maximum point of its height in the air is the moment that the speed hits zero and the apple begins to move in the opposite direction.

But this sort of equation might represent many other things as well: energy consumption, quantity of building materials, time to one’s destination. Having a tool that can find you the best way to maximise or minimise these different quantities has been transformative. If the formula is the profit of a company that will depend on various ingredients that your company might vary, who wouldn’t want a tool that could tell you what value to set these variable inputs to in order to maximise the profit output? Calculus is the shortcut to maximum profitability.

Mathematical scaffolding

Although calculus was principally created to analyse the world as it changed over time, it was also good at analysing change outside of time. In particular calculus has become a very powerful tool for looking at the different ways you might design a building and to find the version that optimises energy efficiency, acoustic quality, or building costs while still making a structure that stands the test of time.

One such building, completed in 1710, still stands proud not too far from where I live in London: St Paul’s Cathedral. I have a soft spot for this building partly because it was designed by a mathematician who trained at the same college in Oxford that I attended as an undergraduate. Before becoming one of England’s leading architects, Christopher Wren had cut his teeth learning mathematics at Wadham College. As a student he equipped himself with a whole range of techniques that would allow him to find shortcuts to creating some of the great buildings across the country.

One of his first great achievements was the Sheldonian Theatre in Oxford, the building where university students receive their degrees. The beauty of this building is the fact that it has a huge roof with no supporting columns. This was not apparently so parents could see their loved ones getting their scrolls but because the space was to be used principally for dancing. The way Wren achieved this extraordinary expanse of roof with no visible supports was with a lattice structure of beams that moved the load bearing to those at the edge that were sitting on the perimeter wall. But in order to find the arrangement that would work, Wren had been faced with solving 25 simultaneous linear equations. Despite having trained as a mathematician, he was somewhat defeated by the problem and ended up having to get help from the Savilian Professor of Geometry, John Wallis. Asking for help is often an important shortcut!

But Wren’s mathematics would truly come to the fore in the building of the dome of St Paul’s. As you approach the cathedral, the dome that you see is spherical in shape. A sphere has a beauty and perfection about it that is particularly appealing when seen from a distance. The shape also tapped into the idea of the church representing the shape of the cosmos. But the sphere has a crucial flaw when it comes to buildings. It can’t stand by itself. Indeed it is too shallow to support itself, meaning the dome would come tumbling down into the heart of the cathedral if left unsupported. Which is why St Paul’s doesn’t have just one dome – it has three.

The one you see when you are inside the cathedral is not the inside of the external dome. It is in fact a second dome whose shape has been made from a new curve called a caternary, later explicitly identified by Leibniz among others using calculus, that is able to stand freely without support. This is the shape that a chain makes when it is hung between two ends. Just as a ball left to roll on a mountain finds the point of lowest energy at which to rest, the hanging chain minimises the amount of potential energy it possesses. Nature is very good at finding these low-energy states. But for an architect like Wren the key point is that when this low-energy solution is inverted it becomes a shape that can support its own weight.

So what is the shape of this curve? Leibniz experimented by varying the shape and produced an equation for the potential energy contained in each. He then used calculus to identify the curve that had the smallest energy. This would be the shape that the chain would hang in. Once identified, the shape could be used by subsequent generations of architects to make freestanding domes without having to scale up physical chains hanging in the space they were designing. Wren particularly liked a dome in the shape of the catenary because when you look up it creates a forced perspective, making the dome seem higher than it actually is. The use of mathematics in this way to create an optical illusion was a big theme of the architecture of the Baroque period.

There was still the problem of how to make sure the external dome didn’t collapse into the cathedral and destroy this beautiful inner dome. This is why there is a third dome hidden between the two domes that you can see. On a visit to St Paul’s recently I had the chance to go inside the two domes and to see the third that is doing all the work of supporting the external spherical dome. This hidden dome also uses the catenary curve: to determine the shape of the arch that Wren would need to support the cupola sitting atop the outer dome. If you hang a weight on a chain it pulls the chain down. You can then use calculus to give a mathematical description of this new shape with minimum energy. But the clever thing is that if you invert the new shape, this arch can support a weight sitting on top of the arch equivalent to the weight that you hung on the chain. That is how Wren worked out the shape of the inner dome that is supporting the top of the spherical dome that you see from outside.

The most extraordinary use of these weighted chains to build domes can be found if you go down into the basement of the Sagrada Familia Basilica in Barcelona. Antoni Gaudí used the principle for the design of the roof of his unfinished chapel. He attached huge numbers of bags of sand to a web of strings which hung in these catenary curves to represent the load of the structure that needed to be supported. Turn the shape that these strings were making upside down and you had the shape of a roof that could be built without falling down. By adding and moving the bags around, Gaudi was able to create the shape of the chapel roof that he wanted, while still being confident that it wouldn’t fall in when he tried to build it. But to make a mathematical description of all those curves that could be given to the manufacturer, you are going to need the shortcut of calculus. Today’s architects have replaced chains and bags of sand manipulated by hand with calculus and equations manipulated by computers in order to create the curvaceous buildings that grace our city skylines.

Yet it isn’t just cathedrals and skyscrapers that calculus helps to build. Another of Leibniz’s successes in finding curves with optimal properties was the discovery of the best curves for making rollercoasters!

Rollercoasters

I love rollercoaster rides. It’s not just the thrill of it. If you’re a nerdy mathematician like me, it’s the buzz of all the geometry and calculus that has gone into constructing a ride that pushes things to the limit while keeping the train attached to the track. There is one rollercoaster in Europe that gets my mathematical blood racing more than any other: the Grand National in Blackpool. When you race round this track you are actually experiencing not just the power of calculus but also one of the most exciting shapes in the mathematician’s cabinet of curiosities: the Möbius strip.

As the name of the ride suggests, the Grand National is a race between two trains. When you get in your carriage at the top of the ride there appear to be two parallel tracks. Riders on the two trains are within touching distance of one another as the tracks race through the twists and turns, passing through features named after some of the notorious jumps of the famous horse race. But as the trains make the final dash for the winning post something rather strange happens. They arrive at the opposite stations to the ones that they embarked from. Very curious. The tracks never meet and cross each other. How on Earth did the designers create this feat?

The effect is achieved at the infamous Becher’s Brook jump where one track races over the top of the other and from that point the tracks have swapped sides so that by the end of the ride the trains arrive at the opposite station.

And it is this simple twist at Becher’s Brook that is the key to the Möbius strip, the beautiful mathematical shape that underpins the design of this track. To construct your own Möbius strip, take a long strip of paper about 2 centimetres in width. Now make a loop but before you join the ends of the paper, make a twist of 180 degrees in one of them. If you imagine a piece of paper running between the two tracks of the Grand National ride, then at Becher’s Brook the paper is twisted 180 degrees as the tracks run under and over each other before they are joined to the track at the beginning of the ride.

The Möbius strip has some very curious properties. The shape has only one edge. Put your finger on it and follow it round. You will be able to reach any other point on the edge. This means that the rollercoaster at Blackpool is actually just one continuous track rather than two parallel tracks. But what rollercoasters like the one at Blackpool really want is speed!

If you want the fastest rollercoaster, it turns out that calculus will help you to design the quickest path to your destination. In fact this is the challenge I set at the beginning of the chapter. Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time?

It was a problem that was first posed not by the creator of a theme park but by Swiss mathematician Johann Bernoulli in 1696. He chose it as a challenge to the two great minds of the day, his friend Leibniz and his adversary in London, Newton:

I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.

The challenge was to design a ramp that would get the ball from the top point A to the bottom point B in the fastest time possible. You might think that a straight ramp would be quickest. Or possibly an inverted curved parabola like the path that a ball follows when it is thrown in the air. In fact, it’s neither of these. The quickest path turns out to be a shape called a cycloid – the path traced by a point on the rim of a moving bicycle wheel.

6.4 The cycloid: the curve traced out by a point on a circle as it rolls along a straight line.

If I invert this curve, it is the fastest way to get from A to B. The curve descends below the level of the destination, gathering more speed that is then used to get it up the final climb to the end before any of the other curves.

Because calculus can find the minimum and maximum of a variable in a particular set of constraints, it doesn’t matter that there are infinitely many curves from A to B. The equations will always allow us to find the fastest.

Newton and Leibniz ended up in a terrible fight over who discovered this amazing shortcut for finding the optimal solutions to problems. After years of acrimony and accusation, in 1712 the Royal Society in London was asked to adjudicate between the rival claims: was Newton’s method of fluxions, as it was known, discovered first and did Leibniz’s plagiarise those ideas with his invention of the differential method. In 1714 the Royal Society officially credited Newton with the discovery of calculus and although recognising Leibniz as the first to publish nevertheless accused Leibniz of plagiarism. However, the Royal Society’s report was probably not the most impartial: it was in fact written by their president, one Sir Isaac Newton.

Leibniz was incredibly hurt: he admired Newton and never really recovered. The irony is that it is Leibniz’s account of calculus that eventually triumphed, not Newton’s.

Although the underlying ideas of Leibniz’s shared much in common with Newton’s development of calculus, there was a big difference. Leibniz was coming at his calculus from a more linguistic, mathematical direction. He was not concerned with whether it was an apple falling and capturing its speed over time, but was considering a much more general setting. If something’s behaviour depended on several factors, his calculus was designed to study how that behaviour varied when you changed the things it depended on.

Newton was a physicist at heart and was probably handicapped by his goal of describing the physical world. The language and notation that Leibniz introduced was far more flexible and able to cope with different settings. And it is Leibniz’s notation that stood the test of time and is taught in schools and universities.

If the truth be told, both Leibniz and Newton had only begun the whole process of developing calculus. Both their treatises and analysis left a lot to be desired. It would be up to the next generation to put calculus on sound logical foundations. But there is no denying that the advances of the next generation were made possible by the breakthroughs that both Newton and Leibniz had made. As Newton once famously said: ‘If I have seen further it is only by standing on the shoulders of giants.’

Do dogs do calculus?

But perhaps Newton and Leibniz were both beaten to the discovery of calculus by another adversary. There is evidence that the animal kingdom has known how to home in on optimal solutions since long before humans came up with the shortcut of calculus.

Let’s return to our trusty adviser who is now relaxing on the beachfront after having claimed the maximum bit of land possible using his calculus. He suddenly sees a swimmer out at sea in distress. He shouts to the lifeguard on the beach to rescue the stranded swimmer.

6.5 What is the quickest path for the lifeguard to reach the drowning swimmer?

Assuming the lifeguard can run twice as fast as she can swim, where should she enter the water to achieve the quickest rescue?

If the lifeguard was trying to minimise the distance travelled, she would just draw a straight line between the start and finish. But since the lifeguard is slower at sea than on land she actually wants to choose a path that cuts down on the time spent at sea. But if she went for the point that minimised the time at sea, there is a problem. It means a longer path across the beach, which could end up increasing the total time. The optimal path looks like it will send the lifeguard to the right of the centre but not quite as far as the point where the line from the swimmer to the land is perpendicular. So where is the best place to enter the water to find the true shortcut to the drowning swimmer?

This was another of the problems that Fermat thought about. It is an optimisation problem again. Fermat encountered it not as a lifeguard finding the fastest path but the challenge of the path that a light beam would take.

You might have experienced the rather strange illusion that happens in a swimming pool when a stick inserted into the water appears to bend suddenly as it enters. It is not the stick but the light travelling from the stick to your eye that is bending. As I described in Chapter 4, light loves taking shortcuts. It tries to find the fastest way to travel from the stick to your eye. But light travels more slowly in water than in air. So just like our lifeguard, it tries to spend as little time in the water as possible while balancing that against not making the time in the air too long. The same explanation is key to the strange experience of seeing a mirage in the desert. The light from a patch of sky exploits a shortcut through the warm air closer to the ground before sweeping back up to your eye, making the sky appear to sit in the desert looking like water.

Just as the adviser did with his fence, the lifeguard needs to cook up an equation for the time it takes to reach the swimmer based on entering the sea X metres from the start point. Then by applying the tool of calculus she can find the value of X which makes this time smallest. But what if you haven’t got access to pen and paper? What if you haven’t invented algebra and calculus? What if you are just relying on intuition and feeling? What if you were a dog! How good is a dog at judging the right place to enter the water?

Tim Pennings, a professor of maths at Hope University in Michigan and dog owner, decided to do some experiments to see whether his dog was any good at cracking this calculus problem. Like many dogs, his Welsh corgi, Elvis, was a sucker for chasing a ball. So Pennings decided that instead of the challenge of rescuing a drowning swimmer, he would experiment with chucking a ball into Lake Michigan during his walks with Elvis and see which path Elvis would choose to retrieve the ball.

Of course it might be possible that Elvis’s main objective would be to minimise the amount of energy expended in retrieving the ball, in which case the smart solution would be to take the time in the water to its minimum and run to the point where his path in the lake would be perpendicular to the shore. But Pennings could see by the glint in Elvis’s eye and the elevated excitement as the ball left his hand that retrieving it as fast as possible was going to be the dog’s goal. The stage was set for his experiment into Elvis’s intuitive grasp of calculus.

On a day when the waves on Lake Michigan were low and the ball wouldn’t move about too much once it landed in the water, he set off with Elvis. With the help of a friend, Pennings launched the ball into the water then ran after the dog, slamming a screwdriver into the point at which Elvis entered the water, and then used a tape measure to see how far Elvis swam before retrieving the ball.

There were a number of false starts when Elvis just ran directly into the water in a clearly sub-optimal route. Pennings decided to delete these data points from his analysis. As he said: ‘Even an A student can have a bad day.’ But by the end of the day he’d managed to gather 35 data points for Elvis’s solution to the challenge. So how did the dog do? Remarkably well! In most cases Elvis was close enough to the optimal point of entry. The variables that obviously existed in the experiment could easily have accounted for Elvis’s approximation.

Does that mean Elvis knows the shortcut of calculus? Of course not. But it is striking how the animal’s brain has evolved to find these shortcuts without the power of formal mathematical language. Nature favours those who can optimise solutions so the brains of animals that could intuitively solve these challenges survived ahead of those who got it wrong. But there are limits to what an intuitive brain can estimate. That’s why sitting on the launch pad at Cape Canaveral, John Glenn wanted the numbers to be run through this advanced tool we’ve developed called calculus to work out the best path home rather than trusting to his intuition.

Sometimes animals use teamwork to solve the problem Elvis the dog was challenged with. There is evidence that an ant colony faced with a similar problem to the lifeguard scenario can do as well as Elvis in finding the optimal path. This time the ball was replaced by food: a cockroach. In the experiment conducted with fire ants by a team of researchers from Germany, France and China, the ants found the best paths across two different domains to retrieve food for the colony. Here they had the chance for many ants to strike out and experiment with different routes. The ants lay pheromone trails for others to follow. As more ants home in on the optimal solution, that path’s trail gets stronger.

The ants actually do something similar to how we believe light finds the optimal path. How does one photon of light know to find the best way? Quantum physics asserts that the photon tries out all the paths simultaneously, collapsing into the optimal one once observed. The ants use a similar strategy of trying everything out using lots of ants before finding the best way.

Nature is very good at finding optimal solutions. Light finds the quickest path to its destination. Gravity in modern physics is interpreted as matter falling through the geometry of space-time finding the quickest journey through. Hanging chains solved the problem of creating a stable dome for Wren. Bubbles take advantage of the sphere’s minimal energy. In more recent times, soap films were used by Frei Otto to design the 1972 Munich Olympic stadium. The strange undulating canopy that covers the stadium is structurally stable thanks to Otto’s analysis of how soap bubbles form on a metal frame.

This strange property of nature to find low-energy optimal solutions was captured mathematically in the first half of the eighteenth century by Pierre-Louis Maupertuis in his principle of least action. As Maupertuis explained, the mathematics translates into the dogma: ‘Nature is thrifty in all its actions.’ Quite why nature is so parsimonious is still something of a mystery. But sometimes dogs and ants or soap film aren’t on hand to help find the answer we’re after. Instead we can turn to the incredible tool created by Newton and Leibniz. Calculus has been and will continue to be the most amazing shortcut to the optimal solutions to the challenges we face.

As the ultimate shortcutter Gauss himself observed about calculus: ‘Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius.’

Shortcut to the shortcut

Although calculus is one of our greatest shortcuts, it does take some technical expertise to be able to employ this tool. Even if the idea of doing a crash course in calculus is beyond most people, it is at least worth knowing that this technique exists to find optimal solutions. Many shortcuts require a technical guide to help us navigate potentially tricky terrain. So if you’ve got parameters that can be varied and you want to know the optimal setting for these variables, contacting an expert in calculus might be your best shortcut. As Newton recognised, standing on the shoulders of giants has always been a clever shortcut. And sometimes you might find that the technical guide isn’t your local mathematician but nature. It is always worth looking to see if nature has already homed in on an optimal solution to your problem. A soap film might reveal the low-energy solution to an engineering problem. The path of light might be pointing you in the direction of the shortcut. Or perhaps following a colony of ants will shortcut trying out too many options.

Pitstop: Art

One of the key lessons of mathematics is the power of an algorithm to shortcut hard work. Instead of treating each problem on a case-by-case basis, an algorithm crystallises what unifies all the problems and then presents a recipe that anyone can apply, regardless of their particular setting. Calculus is one such algorithm. It doesn’t matter whether your equation is describing profit margins, the speed of a spacecraft or energy consumption, calculus is an algorithm you can set on each scenario to find the optimal solution.

I was rather surprised to discover that algorithms might also help make art. I learned this in a recent conversation with Hans Ulrich Obrist, curator at the Serpentine Gallery in London. I was intrigued because I’ve always been terrified by a blank canvas, and I wondered whether having a shortcut might help me turn my creative ideas into something real.

Obrist’s idea grew out of the challenges of globalisation in the art market. At the start of his career, the art world was still oriented toward the west. An exhibition would travel to Cologne or New York and might take a detour to London or Zurich. But with art galleries opening across the globe, Obrist was keen to crack the challenge of how you get a new exhibition into spaces in South America or Asia. It becomes logistically challenging to take a big exhibition to all the places that started wanting to host them. In collaboration with artists Christian Boltanski and Bertrand Lavier, Obrist came up with a way to overcome this: do it. The idea was to create a set of instructions or recipes for a piece of art that someone else can use to make it where they are, whether they are in China, Mexico or Australia.

For Obrist do it was the shortcut to the challenge of globalisation. You don’t bother trying to transport material works in big crates. Just create instructions that can be realised anywhere and in simultaneous timeframes. A generative exhibition. An artistic algorithm. The instruction becomes the shortcut. These do it instructions are similar to musical scores, which, like an opera or symphony, go through countless realisations as they are carried out and interpreted by others.

The idea of instructional art isn’t new. It has its origins in the work of Marcel Duchamp, who in 1919 sent instructions from Argentina for his sister Suzanne and Jean Crotti to make his gift for their marriage. To create the oddly named wedding present, Unhappy Ready-Made, the couple were told to hang a geometry text on their balcony so that the wind could ‘go through the book and choose its own problems’. Art by instruction exploded in the late 1960s, driven by the works of John Cage and Yoko Ono. But it was Obrist who realised that instructions could be more than an interesting conceptual idea and a genuine shortcut around the logistical problems of a global art world.

One of the exciting by-products of do it is that it has empowered those who might otherwise be frightened of trying to create art. We talked during the 2020 lockdown due to the coronavirus in Europe and Obrist was excited by a new role that the do it instructions had played during this difficult global period.

‘The shortcut became a sponge,’ he said. ‘Every place it went it could learn and take on board new instructions. So it became this growing archive. We started seeing a Chinese version. A Middle Eastern version. And over the last few months, I’ve been getting all these messages, first from China and then from Italy and then Spain. Little by little as all the shutdowns started happening, people started to take their do it books from the shelves and realise some of these artists’ instructions at home.’

I wondered if Hans Ulrich could give me an example of one of the do it instructions. Taking out his do it compendium, a glorious large orange volume, he turned to Austrian artist Franz West’s do it:

WEST, Franz

Home do it (1989)

Take a broomstick and tightly bandage both the handle and the bristles with cotton gauze so that the bristles stand on end.

Take 35 decagrams of plaster and mix with the appropriate amount of water. Distribute the plaster over the entire bandaged surface. Take another strip of gauze and bandage the plastered work again. Apply another layer of plaster to totally cover the work.

Repeat this procedure once again and let the ‘Passstueck’ dry completely.

The result of this procedure is that the object can be used as a ‘Passstueck’, either alone, in front of a mirror, or in front of guests. Deal with it however you feel suitable.

Encourage your guests to act out their intuitive thoughts for possible uses of the object.

The Passstueck, or Adaptives, was a project that West began in the 1970s taking small objects and covering them with a coat of plaster so that they morph into something alien, yet vaguely recognisable. His do it was a shortcut for others to create their own examples. As Obrist told me, ‘It’s not only doing things with your broom with Franz West’s instructions. It’s also about doing something with someone else.’ For example, Louise Bourgeois’s do it instructions are ‘When you are walking, stop and smile at a stranger.’

As I have experienced in my own work, shortcuts often appear only after a long journey. The same is true for Obrist: ‘In art we often need detours and in exhibitions we need detours. But detours are in some ways the opposite of shortcuts. I talked once to David Hockney and he said he needs to write a novel or he needs to do a film or a scientific treatise on perspective or he goes on the iPad and makes iPad drawings. It always leads him back to painting but it’s almost as if he needs these detours.

‘The project seemed so straightforward when we made a little booklet with 12 instructions which people could interpret as a shortcut. But it turned out to be my most complex project ever with many side routes and detours. It became this sort of learning system. I thought that was really fascinating because I thought that do it was an extreme shortcut because it was this idea that basically you take a route more direct than the one ordinarily taken. With the instructions you go directly from the artist to the one that realises them, there is no one in between. You can just do it. One can do it more quickly. It would lead to more immediate results. Yet this project has turned out to be my longest project. So in that sense, it’s a strange paradox that the shortcut was the biggest detour.’

For Obrist these instructions are a bit like a good virus. A virus spreads so effectively because at its heart is a set of instructions on how to replicate itself using the cellular material of the host. Interestingly one of the shortcuts that it uses is the concept of symmetry. A virus is often put together like a symmetrical die, which has the advantage that the same instruction can be used at different regions of the shape, meaning that you don’t need bespoke instructions for separate regions.

But symmetry also turns out to be a shortcut that another artist has exploited in creating his work. Conrad Shawcross is a sculptor who loves exploring the interface between art and science. His work has been recognised worldwide and in 2013 he was elected an academician at the prestigious Royal Academy of Arts. Shawcross’s studio is a short cycle ride from my home in east London, so I was intrigued to meet up to find out if there are any shortcuts that he has used to becoming an internationally recognised artist. He told me that he considers shortcuts a way of making ambitious achievements manageable.

‘You have to be very efficient and very smart about your processes in order to achieve something that otherwise would be impossible. It’s about creating templates or jigs or repeat parts that can be put together to create complexity.’

Shawcross has often been inspired by rule-based artists. He is an admirer of American artist Carl Andre’s work, which takes the brick as the element that will be repeated, or Claude Monet who would return to the same lilypads at the same time of day to paint the incremental changes. For Shawcross, the seed for many of his early explorations was an important mathematical shape called the tetrahedron, a triangular-based pyramid.

Part of the appeal of the tetrahedron is the ancient Greek belief that this was indeed one of the building blocks of the universe itself. The Greeks thought that matter was made from earth, wind, fire and water and that each element had its own symmetrical shape. The tetrahedron was the shape of fire. Shawcross’s first piece to explore the shape as a building block of art was a structure he was asked to build in 2006 at Sudeley Castle. He made 2000 oak tetrahedrons and then spent two weeks trying to assemble them into a structure. The process was unruly and precarious. ‘They form these non-tessellating, fiery tendrils that can never join back onto themselves. It was driving me rather than me driving it. On the one hand it was a bit frustrating, but it was also an awakening, a failure which taught me a lot and was the beginning of many themes for me.’

Shawcross needed to find a way to make something that was both beautiful and structurally sound. He eventually found the insight he needed from a mathematician, who pointed out that, if you take three tetrahedrons, there’s only one way to assemble them.

Here was a perfect example of the power of symmetry to provide a shortcut. If you try to find a different way to fuse three tetrahedrons together, you will discover that by a rotation you can always transform the new proposal into the first configuration. Rather than having 2000 building blocks, Shawcross realised he could actually make a larger building block out of each three fused tetrahedrons.

‘It immediately cut my problem down by a third,’ he says. ‘Suddenly the task became much more surmountable.’ With this shortcut at hand, Shawcross just needed to find a way to piece together 667 units made up of three tetrahedrons, a task that was much more achievable in the time he had to make the piece.

But when I talked to Shawcross in his studio, it turned out that some shortcuts are simply beyond the pale as a sculptor and artist. His extraordinary piece called ADA, a moving sculpture that maps out complex geometries in space programmed by a series of gears, made an appearance as part of a dance piece at the Royal Opera House in London. As always Shawcross was working to a tight deadline and it was touch and go whether the installation would be ready for the evening performance.

As they were painting ADA someone suggested that there was no need to paint the back of the sculpture as it wouldn’t be seen by the audience. A clever shortcut, you might think. But Shawcross just couldn’t bring himself to cheat the audience in this way. With all his pieces, even if there are sides to them that will never be seen, it is important that they be given the same treatment as the parts in view. The audience may not be able to see the back of the piece but for a sculptor like Shawcross, this was a shortcut too far.

Here are some more do it artistic algorithms, your shortcut to creating art at home.

AL MARIA, Sophia

(2012)

Locate a television with a generous selection of satellite offerings.

Utilize the Fibonacci sequence of numbers to select channels in order.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 and so on.

Alternatively use a Fibonacci calculator.

Take a photo with a digital device of each channel in passing.

When you have exhausted your satellite channel options as prescribed by the golden ratio – collate the data in the reverse order you have collected it and compile into a mosaic.

The resulting image is a simplistic representation of one edge of the multifaceted media matrix.

Marvel the stunning mediocrity of our manmade wonder.

EMIN, Tracey

What Would Tracey Do? (2007)

Take a table. On the table place 27 bottles – all of different sizes and colours. Take a reel of red cotton and wrap it around the bottles, like a strange web that joins them all together. You can, if you wish, take the reel of cotton underneath the table.

KNOWLES, Alison

Homage to Each Red Thing (1996)

Divide the exhibition space floor into squares of any size.

Put one red thing into each square. For example:

• a piece of fruit

• a doll with a red hat

• a shoe

Completely cover the floor in this way.

ONO, Yoko

Wish Piece (1996)

Make a wish.

Write it down on a piece of paper.

Fold it and tie it around a branch of a Wish Tree. Ask your friends to do the same.

Keep wishing.

Until the branches are covered with wishes.

7

The Data Shortcut

You are invited on to a game show. There are 21 boxes and inside each box is a cash prize. You are allowed to open one box at a time. You can keep the money in the last box you opened. But once a new box has been opened, you can’t go back and claim the previous box’s money. The trouble is you have no idea what the size of the prizes might be. There could be a box with a million pounds in. Or they could all just contain prizes less than a pound. Your challenge is: how many boxes should you open to give yourself the best chance of getting the biggest prize that is in all the boxes?

Every day we are generating more and more data as we meander round the expanding digital world that we are helping to populate. Humankind currently produces in two days the same amount of data it took us from the dawn of civilisation until 2003 to generate. That is a vast digital landscape to explore. Hidden inside that data is treasure for any company that can discern patterns that might help to predict your next digital move. Finding your way in this data jungle is not easy but mathematicians have discovered a clever suite of shortcuts that can uncover this treasure without having to survey the whole territory.

As soon as the scientific revolution ignited in the seventeenth century we began to be overwhelmed by the data we were generating. John Graunt, one of the first demographers, complained in 1663 about ‘the overwhelming amounts of information’ he was inundated with as a result of his studies of the bubonic plague ravaging Europe at the time. You need those numbers in order to deal with a pandemic. This is why the World Health Organization’s director-general Tedros Adhanom Ghebreyesus told a news conference in Geneva that the key to surviving the 2020 coronavirus outbreak was ‘testing, testing, testing’. Without the data, governments would have no idea what resources to deploy, or to where.

Yet without ways to find the signal in the noise, data is useless. The US census board complained in 1880 that the data it had collected was so extensive that it would take over ten years to analyse the numbers, by which time they would be inundated by yet more data from the 1890 census. Tools were needed to shortcut the messages inside these vast swathes of numbers we were generating and collecting.

My hero Carl Friedrich Gauss had always been a fan of data. He’d revelled in a book full of numbers he’d been given as a fifteen-year-old for his birthday, including tables of logarithms and a list of prime numbers at the back. ‘You have no idea how much poetry there is in a table of logarithms,’ he wrote. He spent hours trying to tease out some pattern hidden inside the random-looking primes, eventually realising that there was a connection with the logarithms at the front of the book, a revelation that would lead to the prime number theorem predicting the chance that a number taken at random is prime.

He’d successfully forecast the passage of Ceres through the night sky from the observations gathered by astronomers before the asteroid disappeared behind the Sun. He’d signed up to analyse census data from the Hanoverian government, declaring: ‘I hope to get the editing of the census, the birth and death lists in local districts, not as a job, but for my pleasure and satisfaction.’ He even spent time analysing Göttingen University’s pension scheme for the widows of professors, concluding that contrary to what others had feared, the fund was in great shape and could afford to pay the widows more salary.

The success at recovering Ceres from the noise of the night sky was due to a strategy he developed called the method of least squares. If you have some noisy data and you want to plot the most likely line or curve through the data, Gauss showed that you want to choose the curve such that if you calculate how far each bit of data is from the curve, square this distance and then add up all the answers, it is as small as possible.

7.1 Gauss’s method of least squares.

In the paper he published in 1809 where he outlined the method, he also explained how data likes to distribute itself into a shape we now call the Gaussian distribution. Essentially if you were to plot many different data sets – people’s heights, blood pressure, test results, errors in astronomical or surveying measurements – you would see the same spread: a majority of cases clustering in the middle with a few outliers at the edges. The curve is often called a bell curve because it has a bell shape.

The statistical tools that Gauss and others cooked up are now the go-to shortcuts for anyone navigating our modern data-rich world.

8 out of 10 cats

I was always intrigued as a kid by a cat food advert that would regularly appear on TV. It claimed that 8 out of 10 cats preferred Whiskas, the brand being advertised. I found it curious because I couldn’t remember anyone coming and asking our cat what cat food she preferred. How many cats, I wondered, had they asked to make such a bold claim?

You might think that to be able to legitimately make such a statement would involve a huge amount of work. After all there are estimated to be 7 million cat owners in the UK. Clearly the manufacturers of Whiskas didn’t knock on 7 million doors to check. It turns out that the mathematics of statistics provides an amazing shortcut to finding out the nation’s favourite cat food. In exchange for a little bit of uncertainty, the number of cats you need to ask turns out to be remarkably small. Suppose that I am happy to tolerate a 5 per cent error in the proportion of cats that claim to like Whiskas. I could miss out asking 5 per cent of the cats in exchange for this added error. That’s fine but 5 per cent of 7 million is only 350,000 cats. That leaves a lot of cats I still need to ask.

The thing is that I’ve got to be really unlucky to have left out 350,000 cats and for all of them to not like Whiskas. Most of the time that 350,000 will be split in a fairly similar way to the whole population. Here is the clever shortcut. What if I say that I am happy to take a sample size such that 19 times out of 20 the proportion of Whiskas-loving cats that I get in the survey is 5 per cent away from the proportion that I would get if I had surveyed the whole population of cats? How big would that sample size need to be? Amazingly, you only need to ask 246 cats to gain this level of certainty that you are truly representing the preferences of all the UK’s 7 million cats. That is an astonishingly small number of cats. This is the power of mathematical statistics, to be able to provide you with the confidence to make such a claim based on asking so few cats. Once I’d done a course on mathematical statistics, I understood why our cat had never been asked what cat food it liked.

Even the ancient Greeks recognised the power of inferring a lot from a little. When an alliance of city-states planned an attack on the city of Plataea in 479 BC, they needed to know the length of the ladders they would require in order to scale its walls. Soldiers were sent out to measure the size of a sample of the bricks that had been used to build the wall. By taking an average size and multiplying it by the number of bricks that could be seen to make up the wall, they got a good estimate of how high the walls were.

But it wasn’t until the seventeenth century that more sophisticated approaches began to emerge. In 1662 John Graunt used data about the number of funerals held in London to give the first estimate of the city’s population. Based on his data gathered from parish records, he’d estimated that 3 people died each year per 11 families and that the average family size was 8. Given that the number of funerals recorded annually was 13,000, this led him to estimate the population of London to be 384,000. In 1802 the French mathematician Pierre-Simon Laplace went even further and used a sampling of registered baptisms in 30 parishes to reach an estimate for the population of the whole of France. His data analysis indicated that there was one baptism for every 28.35 people living in each parish. Given a record of how many baptisms in total had been recorded in France that year, he was able to come to an estimate for the population of 28.3 million people.

Even knowing how many cats there are in the UK requires the kind of statistical shortcut of going from the small to the big. In the case of the UK’s feline population, we can apply a similar strategy to that of the soldiers in Greece: survey a small sample and scale up. If you know the proportion of cats per person in the smaller sample, you can simply multiply by the total population of the country to get an estimate. But what if you wanted to estimate the total number of badgers in the wild in the UK? As no badgers are owned by people we can’t simply use the number of humans like we did for cats.

Instead ecologists use a clever shortcut called capture-recapture, a strategy that is at the heart of how Laplace made his estimate. Suppose they are trying to estimate the badger population in the county of Gloucestershire. Ecologists will begin by setting up a number of traps to capture badgers over a particular period. But how do they know what proportion of badgers they’ve caught? They don’t. Yet here’s the ingenious trick. Tag all the badgers you’ve caught and release them back into the wild, allowing the tagged badgers time to reintegrate into the total population. Then set up cameras throughout the county to record badgers. Now you get two different numbers: the total number of badgers sighted and the number of those that are tagged. This gives ecologists the proportion of tagged badgers among the badgers sighted. They can now scale this up. Given that they know the total number of tagged badgers in the county and they know now what proportion this represents of the total badger population, they can get an estimate for the total number of badgers in the county.

For example, suppose 100 badgers were captured and tagged in the first capture and in the subsequent video sightings sample 1 in 10 were tagged. We can estimate the total badger population as being 1000, in order to get the same proportion as our cameras recorded. In Laplace’s case the babies born (whose number is known) in the total population (whose number is unknown) represents the tagged sample and then counting the number of babies in the 30 parishes (both numbers known) represents the recapture part of the experiment.

This tactic has been used to make estimates of everything from the number of people enslaved in the UK today to the number of tanks that the Germans were manufacturing during the Second World War.

The trouble sometimes with shortcuts is that they aren’t always pathways to knowledge. Occasionally they can lead you astray, giving you the illusion that you’ve reached an answer when in reality the destination that the shortcut has led you to is miles from where you want to be. This is one of the dangers of statistical shortcuts. They can be cutting corners rather than genuine shortcuts.

Although you can get away with asking 246 cats to get some insight into the preferences of the population of 7 million cats, you certainly wouldn’t hope to gain much understanding from a sample of 10 cats. And yet the scientific literature has numerous examples of apparent discoveries based on such ridiculously small samples. This crops up quite often in many psychophysical and neurophysiological studies reported in major journals because it’s just too difficult to include too many people in such research. But can you really infer anything from research done on two rhesus monkeys or four rats?

Unfortunately headline discoveries that ‘8 out of 10 Xs prefer Y’ are often trumpeted without any inkling of what sample size was used, leaving you with little appreciation of how likely the discovery is to be true.

The gold standard when it comes to legitimately reporting a significant discovery is given by the parameters that I set for ascertaining a good sample size for the cat food survey. There I was happy with a sample size that 19 times out of 20 would correctly represent the food preferences of the cat population.

When it comes to scientific discoveries and their potential significance, for example a new drug to address a medical condition, they can be regarded as significant if there is less than a 1 in 20 chance that the discovery would have happened without taking the drug. Let’s say that you invented a spell for making a coin land heads. Most people would be rather sceptical, so what would you need to do to convince them? Suppose that after you cast your spell you get 15 heads in the toss of 20 coins. Would that be an indication that you might be on to something? In fact if you calculate the chances of a fair coin randomly landing heads 15 times in 20 tosses (without using your spell), there is less than a 1 in 20 chance of this happening. So the fact that with your spell you got 15 heads means that you would be justified in thinking that your spell might be working.

Since the 1920s this 1 in 20 random possibility has been the threshold that you need to pass for your discovery to be considered ‘statistically significant’, which would be acceptable for publication. It’s called having a P value of less than 0.05. One in 20 represents a 5 per cent chance that the thing might happen randomly.

The trouble is that you only need twenty research groups and one is very likely to get this random result. Nineteen groups would have moved on to other ideas but the twentieth group would have got extremely excited and known they’d passed the threshold for publication of a significant result. You can see why, with such a threshold, so many cranky hypotheses might have made it into the literature. This is why there has been a call to try to reproduce many of the results that have been published thanks to passing this test of statistical significance.

Conversely if something has a P value of 0.06 (or a 6 per cent chance of randomly happening), this is regarded as too weak to be statistically significant and will often be rejected. And yet that can be equally dangerous as a reason to reject a hypothesis. Yet negative results don’t make good news stories. So nineteen research groups don’t publish their discovery of no connection.

Great care needs to be taken with these thresholds. If you are trying to ascertain whether a coin is fair or not, this threshold might be OK. But imagine that you are trying to pick up whether a doctor’s failure rate is due to malpractice. You wouldn’t want to be bringing in 1 in 20 doctors for investigation. And yet at what point should you start to be concerned?

For example, in September 1998, Dr Harold Shipman, a well-respected family doctor, was arrested for injecting at least 215 of his patients with lethal doses of opiates. A team of statisticians led by David Spiegelhalter subsequently suggested that using a test originally introduced in the Second World War to maintain quality control of military supplies, they could have detected something strange in Shipman’s data much earlier and potentially saved 175 lives.

Thresholds for significance need to be handled carefully. In March 2019, 850 scientists wrote a letter to Nature striking back at what they regarded as the obsession of the scientific community with the P value as the benchmark for scientific discovery. ‘We are not calling for a ban on P values,’ the letter said. ‘Nor are we saying they cannot be used as a decision criterion in certain specialized applications (such as determining whether a manufacturing process meets some quality-control standard). And we are also not advocating for an anything-goes situation, in which weak evidence suddenly becomes credible … we are calling for a stop to the use of P values in the conventional, dichotomous way – to decide whether a result refutes or supports a scientific hypothesis.’

Wisdom of the crowd

One smart shortcut that statistician Sir Francis Galton came up with was to consult lots of ordinary people, getting them to do all the hard work, and then use a shrewd bit of maths to finish the job. Galton is rightly criticised today for his immoral racist theories of eugenics but his theory of the wisdom of the crowd is still considered a valuable tool in analysing big data. He actually stumbled on his discovery while he was trying to prove that the opposite was true. Indeed he had such little faith in the collective wisdom of the average members of society that he was very critical of allowing the public a say in politics: ‘The stupidity and wrong-headedness of many men and women being so great as to be scarcely credible.’

Hoping to prove his point, Galton decided to do an experiment, using the county fair at his home town of Plymouth. It had a competition to guess the weight of an ox after it had been slaughtered and dressed. The challenge attracted 800 people who for sixpence submitted their estimates. Although a few entrants might have been farmers, most were visitors with little knowledge to tap into. ‘The average competitor was probably as well fitted for making a just estimate of the dressed weight of the ox, as an average voter is of judging the merits of most political issues on which he votes,’ wrote Galton dismissively.

But when he took the guesses away with him and statistically analysed the data he got something of a shock. Although many were way off the mark, some grossly underestimating the weight, others greatly overestimating it, he discovered that if you took the average of all the values it was amazingly close to the true figure. (Galton actually started his analysis by taking the value that was exactly at the halfway point of all the guesses, called the median, which also turned out to be very accurate.) The crowd’s average guess of the weight of the ox came to 1197 pounds. The true value was 1198 pounds. Just one pound away.

Galton was stunned. ‘The result seems more credible to the trustworthiness of a democratic judgement than might have been expected,’ he wrote. He had let the crowd do all the hard work of making guesses and then used maths to shortcut his way to a solution: truly the wisdom of the crowd.

I recently received a thank-you note from a member of the public who after hearing me speak about the idea used exactly this strategy at his local fair. The challenge was estimating the number of jelly beans in a jar. He’d waited until the fair’s final moments before loading all of his fellow fair-goers’ guesses into an Excel spreadsheet, taking the average and making his guess. It turned out that his estimate using the wisdom of the crowd was closest, just 5 beans away from the actual number of 4532. He included a few jelly beans in his letter to me as my percentage cut of the prize in reward for informing him of this cunning shortcut.

Another example of the wisdom of the crowd crops up in the famous game show Who Wants to Be a Millionaire?. Most of the time you’re on your own answering questions in your attempt to get 15 correct and bag the million-pound prize. However, there are a couple of life lines that you can cling to if you haven’t got a clue. One allows you to phone a friend while a second option is to ask the audience. A team of academics from Switzerland gathered data on the German version of the show revealing that in their sample the audience had been asked 1337 times and got it wrong on just 147 of those occasions. That is a remarkable hit rate of 89 per cent. Compare that to the statistic for phoning a friend, which failed to get the right answer 46 per cent of the time.

If you are going to use the audience it is important not to give your views on the possible answer. As a species we are terribly susceptible to being led astray. Take the case of the contestant who would get to a quarter of a million pounds if she correctly answered the following question:

The Norwegian explorer Roald Amundsen reached the South Pole on 14 December of which year?

A: 1891 B: 1901 C: 1911 D: 1921

She was pretty convinced that Robert Scott who Amundsen had beaten to the pole was Victorian so she was sure that answers C and D were incorrect. But she really didn’t know between the first two. So she asked the audience. Look at the results she got back.

A: 28 per cent B: 48 per cent C: 24 per cent D: 0 per cent

Of course one’s instinct is to go for B. But look at answer C. Why have so many people gone for that given that the contestant was pretty sure that it was wrong? The answer is that the contestant was wrong. Indeed she probably led many people astray by airing her thoughts, ending up with people voting for B when left to their own devices they would have voted C, the correct answer.

However, the tactic of trusting the audience might depend on which country you are playing in. Apparently Russian audiences are notorious for leading contestants astray, deliberately choosing the wrong answer. Of course you could always try the shortcut that Major Charles Ingram is accused of using to win the £1 million: cheating. He apparently had someone in the audience who would cough whenever the host read out the correct answer. It turned out that if you knew your maths you’d have been OK without a coughing assistant. The last question on which the million-pound prize depended was to identify the name of the number consisting of 1 followed by 100 zeros. Is it A) a googol, B) a megatron, C) a gigabit or D) a nanomole. If you need any help, I’d be coughing on answer A.

If the crowd is so wise, who needs experts? Well, it all depends on the task at hand. Despite the Conservative politician Michael Gove’s declaration during the Brexit debacle that ‘we’ve had enough of experts’, I wouldn’t want to get on a plane flown collectively by the passengers. And it doesn’t matter if you amassed every amateur chess player in the world to play collectively against Magnus Carlsen, I still know who I’d put my money on. On which questions might the crowd provide a shortcut to the answer and on which will they lead you astray? One of the key pointers is to make sure your crowd are all answering independently. Remember how the woman in Who Wants to Be a Millionaire? influenced the audience by her belief that Scott of the Antarctic was Victorian.

The psychologist Solomon Asch illustrated how the crowd can be especially persuasive in influencing people to go against their instincts. In an experiment conducted in the 1950s, Asch asked a group of seven people to identify which one of the three lines in Figure 7.2 had the same length as the line to the left.

7.2 Asch’s experiment: which line has the same length as the line on the left?

The twist was that the first six people to answer were plants. They were all told to nominate B as the answer. Time and again when the seventh person came to respond, they would not trust their eyes and answer C. Instead the desire to conform with the choices of the group overrode what they were seeing and they would respond with the same choice as the first six participants.

In an age of social media, this desire for conformity is potentially having a devastating effect on our ability to choose independently of others. Social media means that it is quite hard to keep the crowd independent.

But there is some evidence to suggest that total independence isn’t necessarily great for making a wise crowd either. In a fascinating study conducted by a team based in Argentina, it was discovered that if you allow some conferring among members of the crowd before aggregating the results, the answer beats a totally independent crowd.

In a live event in Buenos Aires the research team began by asking the 5180 audience members to each answer eight questions without talking to their neighbour. For example, what is the height of the Eiffel Tower? Or, how many goals were scored in the 2010 World Cup? The answers were collected and the average answer was calculated. But then the researchers asked the audience to get into teams of five and to discuss the questions before giving their revised answers. When these were collected and aggregated, the results were much more accurate.

The point is that a few people will have some expert knowledge that might help steer those who frankly haven’t a clue. And so the crowd can benefit from some expertise. If you haven’t a clue about football, estimating the number of goals in the World Cup is going to be a total guess. But if in your team of five you get someone who knows a bit about football and explains that on average you might get 2 to 3 goals per game and also that there are 64 matches played in the World Cup, you’ve got good material on which to base a guess. That would give you say 2.5 × 64 = 160 goals. The actual answer was 145. But the point is that when you go back to make your new guess based on the discussion you’ve had, you could well take into account what sounded like quite convincing expert knowledge that one of your group members offered.

Of course there are people who believe they are experts and in fact can lead people astray, so we don’t want the crowd influenced by one confident leader. Yet it seems like this combination of a crowd of small teams is more effective than a crowd of individuals.

Another quality that can make a big difference is making sure the crowd has a diverse range of opinions. The audience that took part in the Buenos Aires event might have come from a particular social class interested in attending such events, and as a result you might have missed out on a more diverse spectrum of society. This has been illustrated in some interesting cases where the public were asked to help in making budgetary decisions rather than leaving them up to the politicians. The idea of participatory budgeting was first studied in Porto Alegre in Brazil in 1989. When the Iceland economy failed after the financial crash of 2008, its government decided to invite members of the public to help set budgets. Yet the initiative wasn’t generally regarded as a success. By inviting people to apply to take part, it appears that only people who were interested in politics came forward. The group that formed had an inbuilt bias and did not represent the diversity that the system hopes to exploit.

So when the same experiment was conducted in British Columbia, people were instead selected randomly and sent letters with an expectation that they attend, rather like doing jury service. By randomly choosing people rather than letting them self-select, the crowd had a much more diverse range of opinions and was far more successful at implementing the ideals of participatory budgeting.

Who wants to be a scientist?

The idea of using a crowd as a shortcut to scientific discovery is the key to the surge of citizen science projects that we have witnessed over the last few years. One of the first and most successful has been run out of my university in Oxford. It is called Galaxy Zoo. The university’s astronomy department has used it to help classify the different types of galaxies that our universe contains. There are a fantastic range of telescopes taking wonderful pictures of the galaxies, but there just weren’t enough research students to look through all the images. When the project kicked off, computer vision was still in its infancy and was unable to distinguish between a spiral galaxy and a spherical galaxy.

But for a human, distinguishing between the two was straightforward. Indeed the team in Oxford realised that you didn’t need a PhD in astrophysics to help out. You just needed a large number of eyes looking through the data. To take part in the project, members of the public would be given a quick online tutorial to explain what they were looking for and shown the difference between a spiral galaxy and a spherical galaxy. Then they were let loose on the mass of unclassified images that had been captured by the telescopes around the world.

By using the crowd, the university’s astronomy department was able to shortcut the huge task of classifying all this data. It’s a bit like the moment Tom Sawyer gets his friends to whitewash the fence he had been told to paint as a punishment. He turns work into play and suddenly all his friends are lining up to help paint.

But the Galaxy Zoo crowd went one better. They discovered a completely new sort of galaxy hiding in the data. Some of the images didn’t fit into either of the categories that they were asked to label the data with. Professional astronomers had encountered these images and had just dismissed them as anomalies. But the Galaxy Zoo crowd started to run into more and more of the images, which looked like green peas sitting in the blackness of space. A thread emerged on the Galaxy Zoo blog called ‘Give Peas a Chance’, asking not to dismiss these green blobs. The humorous wordplay on the John Lennon song led to the galaxies becoming known as Green Pea Galaxies.

The discovery made by the citizen scientists eventually led to a paper ‘Galaxy Zoo Green Peas: Discovery of a Class of Compact Extremely Star-Forming Galaxies’ being published in the Monthly Notices of the Royal Astronomical Society.

Using the crowd as a shortcut to scientific discovery is not new. In 1715 astronomer Edmond Halley enlisted the help of 200 volunteers to work out how fast the shadow of the Moon swept across the country during the eclipse of 3 May that year. Stationed at various points across the country, members of the public were asked to record the time and duration of the total eclipse. In Oxford, unfortunately, the skies were overcast and so volunteers were unable to contribute any data. The team stationed in Cambridge were more lucky with the weather. Only they got distracted and missed it! As Reverend Cotes who was in charge of the Cambridge team wrote to Halley, ‘We had the misfortune to be oppressed by too much company.’ They were serving cups of tea to the visiting crowd and by the time they were ready to do their observations the eclipse had finished.

Halley did manage to collect enough data to be able to estimate that the shadow swept over the Earth at an impressive 2800 kilometres per hour. He published the results in the journals of the Royal Society, where he was a fellow.

Emboldened by Halley’s success, Benjamin Robins, another fellow of the Royal Society, enlisted the help of citizens in an experiment to discover how high fireworks reached into the sky. He had the perfect occasion to do it on the night of 27 April 1749 when King George II held a celebratory firework display to mark the end of the Austrian War of Succession, accompanied by music composed especially for the occasion by the king’s favourite composer, George Frideric Handel.

Taking out an advert in the Gentleman’s Magazine, Robins asked people to record the height of the fireworks from their location.

For if such as are curious and are from 15 to 50 miles distant from London, would carefully look out in all proper situations on the night when these fireworks are play’d off, we should then know the greatest distance to which rockets can possibly be seen; which if both the situation of the observer, and the evening be favourable, will not, I conceive, be less than 40 miles. And if ingenious gentlemen who are within 1, 2 or 3 miles of the fireworks, would observe, as nicely as they can, the angle that the generality of the rockets shall make to the horizon, at their greatest height, this will determine the perpendicular ascent of those rockets to sufficient exactness.

This was not some idle research project. Given the importance of rockets to the military, knowing the range of a firework could be very useful in developing weapons. Unfortunately the instructions that Robins provided in the Gentleman’s Magazine were so obscure that everybody was put off taking part except for one gentleman 180 miles away from London in Carmarthen in Wales. Patiently waiting on a hilltop, he claimed to have seen two flashes at an elevation of 15 degrees above the horizon. Given the curvature of the Earth and the Brecon Beacons in the way, it is highly unlikely that he actually saw any of the 6000 fireworks that were set off. Having heard how many were used in the display and what little impact it had out in Wales, the volunteer thought the whole thing a huge waste of public money.

The power of the crowd to help in scientific investigation has today been much more successful than Robins’ failed attempt. From counting penguins in video footage from Antarctica to folding proteins to discover the key to degenerative diseases, enlisting the crowd has been a very cunning shortcut to new insights.

And the power of the crowd as a shortcut to knowledge hasn’t been missed by the corporate world either. In fact the success of Facebook and Google has depended on the crowd freely giving away valuable data in exchange for their services.

Machine learning

Galaxy Zoo kicked off in 2007 when computer vision was still very poor. However, the last few years have seen a step change in the ability of computers to detect what is in an image. This is due to a new way that code is being written called machine learning where code changes and mutates through its interaction with data. The power of allowing code to learn from the bottom up rather than trying to work out the code in a top-down manner has provided an amazing shortcut to writing powerful algorithms. The code itself may not be very efficient and slick but with today’s computing power that isn’t such a problem as it used to be.

One of the great successes of machine learning has been computer vision. The key to this revolution has been the power of statistical analysis of data to provide a shortcut to seeing. The computer isn’t infallible but that’s fine. Provided it gets the right answer most of the time that is good enough. And this is the point about our original 8 out of 10 cats shortcut. To get a success rate of 99 per cent in distinguishing a cat from a dog requires being exposed to data, but how much? We don’t want to have to give the computer all the pictures of cats and dogs online, of which there are many!

The general rule for training an algorithm to distinguish different categories of images is that you need 1000 images to represent each one. To create a cat recognition algorithm you need 1000 images of cats for the code to learn from. For standard machine-learning algorithms more data doesn’t really improve the success rate. The algorithm seems to plateau. But for more sophisticated deep-learning models more data does show a logarithmic improvement.

Knowing how much data you can get away with is essential when it comes, for example, to wanting to know what variables might be impacting sales. Perhaps you think the day of the week has an impact, or the weather, or whether the news is positive. The way to try to understand what is influencing your sales is to gather data. Take the variables that you think might be affecting the sales and then record sales for different values of all the variables.

To know how much data you can get away with to make an informed inference, we can look to regression analysis and the rule of 1 in 10. If there are 5 variables that you are tracking, then 10 × 5 = 50 pieces of data would be the ballpark number to glean what the impact of variations of these parameters might have on sales.

But we have to be careful with this kind of shortcut because it can lead us astray. Just as it is important to have diversity in a crowd if you are hoping to glean some wisdom, equally you need to make sure that data is diverse. When Amazon hoped to develop artificial intelligence to help sift through applications for jobs, the company used the profiles of current employees as its role model. A sensible decision you might think, given that Amazon was happy with the quality of its employees to date. But when the AI started rejecting any CV that wasn’t from a white twenty-year-old male the company realised that the algorithm was unfairly discriminating against a whole swathe of individuals applying for jobs.

The Algorithmic Justice League started by Joy Buolamwini is calling out such algorithmic shortcuts that fail to get us to new destinations and just bring us back to old prejudices.

It is also important not to track too many variables at any one time, because the more variables you track the more likely you are to find patterns inside. The dangers of tracking too many variables were illustrated when an fMRI (functional magnetic resonance imaging) scanner was used in an experiment to examine 8064 regions of the brain to see which might be involved when the participant was shown various images of human expressions. Sure enough 16 regions showed a statistically significant response. The trouble was that the subject being scanned was a large dead Atlantic salmon. The researchers had been using inanimate objects like the salmon in order to correct for false positives. But it illustrates the dangers of simply measuring too many things and hoping to pick out a pattern. The team received that year’s Ig Nobel Prize awarded for achievements that ‘first make people laugh, and then make them think’.

As Craig Bennet, who was one of the researchers on the team, explained: ‘If you have a 1 per cent chance of hitting a bull’s-eye when playing darts and you throw one dart, then you have a 1 per cent chance of hitting the target. If you have 30,000 darts then, well, let’s just say that you are probably going to hit the target a few times. The more chances you have to find a result, the more likely you are to find one, even by chance.’

How much data before you make up your mind

The game show I described at the beginning of the chapter is actually a good model for many of the challenges that we face in life. Your first boyfriend or girlfriend might be amazing but should you marry them or is there a nagging feeling that maybe you could do better? There are many more fish in the sea and perhaps someone else out there is ‘the one’. But if you dump your current partner, there’s generally no way back. So at what point should you just cut your losses and settle for what you’ve got?

Flat hunting is another classic example. How many times do you see a fantastic flat on your first search but then feel that you need to see more before you commit yourself, only to risk losing the first great flat?

The key to optimising your chances of getting the best prize possible is mathematics’ second most popular number: e = 2.71828 … Like π, mathematics’ number-one number, e has an infinite decimal expansion that never repeats itself and is a number that keeps cropping up in all sorts of different settings. It crops up in Euler’s beautiful equation that unites five of the most important numbers in mathematics that I introduced in Chapter 2. It is also intimately related to the way the interest accumulates in your bank account.

But e turns out to be the shortcut to get the best chance of choosing the winning box in our hypothetical game show. The mathematics proves that if you have N boxes, you need to gather data from N/e of the boxes to get some idea of what the prize money looks like. 1/e = 0.37 … This represents 37 per cent of the boxes. Once you have opened these boxes, the strategy is to choose the next box that beats all the boxes that you have opened already. This doesn’t guarantee you the best prize but 1 in 3 times you will end up with the maximum amount possible. If you based your decision on seeing fewer or more boxes, these chances decrease. The figure of 37 per cent is the optimal amount of data to gather before taking the plunge, whether it’s boxes in a game show, flats, restaurants or even your partner for life. Though perhaps it’s best not to let your partner know you were so calculating when it came to love.

Shortcut to the shortcut

Making a decision on which direction to take your ideas for a new project is often improved by doing a survey of people’s preferences. As has often been touted, data is the new oil but it is still important to know how much of it you need to power your ideas. Too much data and you can drown. Too little and your project won’t get off the ground. The shortcut of statistics reveals that you can often get a long way on a surprisingly small sample size. Finding clever shortcuts to collecting data is also important. As Mark Twain illustrated, it takes one person a long time to paint a fence but many hands can quickly finish the job. Tapping in to the wisdom of the crowd is a way to glean insights, whether it’s setting up a Twitter poll, devising an online game that yields data or exploiting Google analytics to understand engagement with your website.

Pitstop: Therapy

When I first told Shani, my wife, that I was writing a book about shortcuts, she was horrified. She is a psychologist and believes that often there is no replacement for the deep, protracted work one needs to undergo in therapy in order to rewire the brain. And yet Shani did admit that even therapy has found shortcuts to tackle the huge mental health problems that society faces.

Traditionally the idea of going to a therapist calls up the image of years spent lying on a couch talking about your childhood. But for some conditions there are very powerful techniques that can shortcut these years in therapy. Shani suggested that I talk to Dr Fiona Kennedy who, having practised as a psychologist for many years, now trains others in the range of intensive therapies that have emerged to tackle mental health issues. These interventions can help patients with phobias, anxiety, depression and post-traumatic stress without the need for years of treatment.

For Kennedy one of the reasons these therapies have been successful is that they take a more scientific approach. ‘If you were going to a surgeon to have a heart operation and you had two candidates to do the surgery. And the first said “this is my history of heart operations. These are the techniques I’ve used and these are the success rates I’ve had” and the other one said “well, I don’t really collect any data, but I am a very creative person that people find totally inspiring. I’ve done many operations and I’ve enjoyed them very much”, who would you get to do your surgery?’ Although evidence-based scientific thinking hit the world of psychotherapy only recently, it has been key to the successful introduction of these methods into health services worldwide.

Probably the most well-known psychological shortcut is CBT or cognitive behavioural therapy. Developed by psychiatrist Aaron Beck in the late 1960s and early 1970s, CBT focuses on how your thoughts, beliefs and attitudes affect your feelings and behaviour, and teaches you coping skills for dealing with different problems.

Kennedy recalls how as a student she took part in an experiment in which rats and students were asked to perform various tasks: ‘The rats beat the students hands down. We were all thinking too hard about what was going on.’ The experiment illustrated how cognition can interfere with a journey to a successful outcome. For Beck and others the key was to find ways to change the cognition.

Kennedy has quite a mathematical description of what is going on: ‘It’s all about networks. You’ve got a very complex set of relational networks that determines who you are, but also how you respond to the world. So changing that network becomes important.’

In Beck’s initial model for CBT our behaviours were viewed in a very algorithmic way. A trigger acts as an input which then gets processed to produce thoughts, feelings and behaviours which then might trigger an action or output. Beck proposed CBT as a way to break down this algorithm into smaller pieces to identify the bug in the program, the faulty cognition. The behavioural part of the therapy consists of exercises that the therapist gives the client to prove to them that certain parts of the algorithm are erroneous. For example, a fear of spiders might be addressed by gradual short exposure to a spider to reveal that the patient’s fears of the consequences are unfounded.

The striking thing is how quickly in certain cases awareness of the faulty cognition can lead to positive changes in behaviour. Better thinking leads to better well-being. The fact that this can be achieved in 8 one-hour sessions has led to the explosion of CBT and other therapies as a shortcut to getting people back to work. The highly structured nature of the therapy means that it can often be provided in different formats including groups, self-help books and even as an app on your phone.

The shortcut is believed to be so effective that it became the backbone of the British initiative Improving Access to Psychological Therapies (IAPT), which began in 2008 and has transformed the treatment of adult anxiety disorders and depression in England. Economics professor Lord Richard Layard persuaded the Labour government at the time that the amount of money that would be saved by getting people back to work would mean that the scheme would end up paying for itself. In 2009, £300 million was provided by the government over three years to train an army of more than 3000 therapists. IAPT is today widely recognised as the most ambitious programme of talking therapies in the world. In 2019 more than one million people accessed IAPT services for help to overcome their depression and anxiety.

Sometimes circumstances don’t allow for anything but a very short intervention, yet Kennedy points me towards data to support the effectiveness of a CBT model that relies on just three sessions. First proposed by Michael Barkham, a professor of clinical psychology, it is called two-plus-one. Clients are seen for two one-hour sessions a week apart followed by a third session three months later. A growing body of research is showing that even this very short shortcut can be effective. For example, data published in the Lancet in 2020 revealed how such an intensive two-plus-one model had resulted in meaningful reductions in psychological distress among South Sudanese female refugees in Uganda. As the researchers stressed in their paper, innovative solutions are required to provide mental health support at scale in such low-resource humanitarian contexts.

The other aspect of Kennedy’s approach that resonated with me was the use of diagrams as tools for exploring new perspectives. One such diagram is the cognitive triangle. This is a diagram that helps the therapist and patient to understand the integrated nature of thoughts, feelings and behaviours. Sometimes a square is drawn, with feelings divided into two: emotions and bodily sensations. The idea is that without intervention in the flow round the shape, thoughts will trigger feelings that lead to unhelpful behaviours that the patient would like to address, like a fear of going outside or a phobia of spiders. But by understanding and being aware of the cycle it is possible to intervene earlier to change behaviour. The diagram is like a map of the patient’s mental terrain. By lifting themselves outside of the network of thought, the patient is able to see that there are choices of different paths that they might take.

Kennedy describes another diagram that she gives to therapists rather than patients to think about during sessions. ‘Imagine that you’re the therapist and I’m the client, and we’re both sitting on opposite ends of a seesaw balanced on a rope and the rope is stretched across the Grand Canyon. It’s very important for us both to maintain this balance. I come along to therapy one day and I’m in a great mood because I’ve done my homework and I made these changes. So I’ve moved towards you on the seesaw, and then you naturally, being a very enthusiastic and caring therapist, will move towards me on the seesaw.

‘But the next week I come along and I don’t think I can do this any more. I’ve had a terrible week. Nothing’s worked. I just feel like giving up. I’ve moved away from you on the seesaw. Your instinct would be to move towards me. In which case we’ll crash into the Grand Canyon. The harder you try, the more I’m going to resist you. And so what you need to do is to move away from me.’

It’s a fascinating image because Kennedy has changed the therapy into an equation that, like the seesaw, needs to be kept balanced.

For Kennedy and others the evidence of the effective nature of these shortcuts is in the data, much of it collected by David Clark, professor of psychology at Oxford University. Tens of thousands of therapists send in data from their clients every week, and Clark has been collecting data for a decade. He places all this data in the public domain in order to promote transparency about mental health outcomes.

But sometimes cognition is not enough. Sometimes there is no shortcut that can replace the deep, protracted work one needs to undergo in therapy in order to rewire the brain. Kennedy acknowledges that formulaic therapies have their drawbacks.

‘CBT is all based on logic. But actually there is other stuff to therapy as well. About self-acceptance and attachment and being part of a family and part of the group and part of the world that comes with a good enough upbringing that if you want to fix, you cannot do in eight sessions.’

As a result, CBT has sometimes been seen as a sticking plaster that is being put on a gaping wound. It might stop the bleeding in the short term but if you don’t address the cause of the wound, it will just reopen again after some time. How can you rewire the brain in 8 one-hour sessions? Some therapists fear that CBT is sometimes cutting corners rather than being a genuine shortcut.

I think partners of therapists are always curious about what exactly is happening behind the closed door of a therapy session. It was one of the reasons I pulled In Therapy by psychoanalyst Susie Orbach off Shani’s bookshelf. It turns out that this was one of the motivations for Orbach to write the book, which is dedicated to Orbach’s own partner Jeanette Winterson ‘who has always wanted to know what goes on in the consulting room’.

Orbach became famous for treating Princess Diana’s eating disorder. As she explains in the book, therapy isn’t just about training the mind and body to do something new like play the cello or speak Russian. You’ve got to start by doing the much harder task of unlearning something.

Therapy can take so long because you have to address the basic ways your mind makes sense of the world. As Orbach puts it, ‘In therapy you don’t just learn a new language to add to your repertoire, you relinquish unhelpful parts of the mother tongue and weave them together with the knowledge of a new grammar.’

When I contacted Orbach to explore this idea more she stressed this point. But she also acknowledged that there are still shortcuts that she uses in her sessions with patients. The interesting thing when talking to her is the role that patterns can play. The therapist spots patterns of behaviour that correspond to previous case studies to help in plotting a course of action for the new patient in the room. But this needs to be balanced with a recognition that each case is individual and unique.

‘The kind of therapy I do is that you draw the lesson from an in-depth study of one person,’ says Orbach. ‘That’s Freud’s legacy. It’s the case study. It doesn’t mean they fit but it means maybe 50 per cent of it fits. So that’s a shortcut in a way if it’s embedded in you, in your thinking, in your cognitive and your emotional repertoire as a therapist.’

This is one of the fascinating tensions that exist in psychology. On the one hand it borders on being a science because there are things like case studies, and patients come in with particular ailments. A medical doctor looks to match symptoms to previous case studies to be able to navigate the patient through the ailment based on those prior histories. Patterns of behaviour can similarly provide the therapist with a shortcut to understanding a patient in the same way that patterns help me as a mathematician to apply a previous methodology to solve a seemingly new case. And yet the unique nature of each individual’s psychology means that you never get replication. Each case requires personalised treatment. This is the art rather than the science of being a therapist.

‘Therapy is a bespoke craft with each therapeutic pair or group creating novel circumstances to respond to,’ says Orbach. ‘One truth can open to another which may shade what is first understood. The intricate constructions of the human mind shift during the course of therapy. Being a participant observer to the changing of internal structure and of the expansion of feelings is very satisfying. Seeing how defences are used, and how they can be worked around and in time dissolve, has a beauty which is perhaps akin to the mathematician or physicist’s experience of finding an equation elegant.’

Orbach suggests to me that the way she approaches each new patient is not so dissimilar to the way that I as a mathematician will approach each individual new problem.

‘If I do an assessment of a potential patient I’ve got a physical sense, maybe even a sort of geometric picture in my mind of internal object relations, defence structures, emotional this that and the other. I’ve got so many things going on, but I don’t even know that they are going on until I have to write it out. So that constitutes a shortcut but then again that has come from the fact that I’ve been at it for forty bloody years.’

As ever that recurring theme that shortcuts are hard-earned, requiring many years of work. I wondered what Orbach’s thoughts were on the way CBT is being used in therapy as a shortcut. She is sceptical of this almost algorithmic approach to therapy.

‘I don’t believe in manualised therapy. Does that mean it’s useless? No. Something is better than nothing. But are you supposed to be better within eight weeks or eight sessions? The trouble with a lot of the Access to Therapy is that they’re often not therapists doing it and it’s a highly skilled job.’ Indeed some of the CBT treatment is even being delivered by AI therapists. Orbach doesn’t believe you can reduce therapy to a formula you follow. ‘Human subjectivity is not trivial,’ she says. ‘It is infinitely complex and beautiful.’

CBT might have the power to construct frameworks for patients to be able to see certain patterns of thought and understand where they come from. With awareness they can then act to short circuit these negative automatic thoughts. But they miss an essential quality of therapy for Orbach, which is that these patterns often work at the level of ideas, but not emotions. And this is key to why she believes they can’t truly shortcut therapy. Emotions play a critical role in high-level cognition and consciousness. You can’t change the latter without addressing things at an emotional level. Emotions create cognitive structures that develop over decades. For example, Orbach says, ‘You have a defence structure so you might understand, yes I’m reproducing this particular behaviour because it’s embedded in me and that is how I understand for example “love means hate” or “love means hitting” or whatever. I understood that but the emotional component of that is unbelievably complex. And so of course, it’s an aid, but basically it’s …’ – at this point she lets out a huge release of air – ‘… it’s not easy.’

8

The Probable Shortcut

Which should you put your money on?

1. Throwing six dice and getting at least one 6.

2. Throwing twelve dice and getting at least two 6s.

3. Throwing eighteen dice and getting at least three 6s.

Modern life consists of a whole sequence of decisions based on assessing a range of probable outcomes. Risk analysis is an integral part of how we negotiate our way through the day. There is a 28 per cent chance of rain today. Should I take an umbrella? The newspapers declare a 20 per cent increase in the chance of bowel cancer if I eat bacon. Should I stop with those bacon sandwiches? Is my car insurance too high given the risk of an accident? Is there any point in me buying a lottery ticket? What are the chances that my next throw of the dice when I’m playing a board game will send me down a ladder?

Many professions are faced with calculating odds to make crucial decisions. What is the chance of a stock rising or falling? Is the accused guilty of the crime given the DNA evidence presented? Should patients be concerned by false positives in medical tests? Where should a footballer aim when taking a penalty? Negotiating an uncertain world is a challenging task. But finding a path through the fog isn’t impossible. Mathematics has developed a powerful shortcut to help us navigate the uncertainties of everything from games to health, from gambling to financial investment: the mathematics of probability.

Throwing dice is one of the best ways to explore the power of this shortcut. This chapter’s opening challenge was one that had exercised the famous seventeenth-century diarist Samuel Pepys. Pepys had a fascination with games of chance but he was always rather cautious about risking his hard-earned cash on the roll of a dice. In his diary entry of 1 January 1668 he writes how returning home from the theatre he stumbled upon ‘the dirty ’prentices and idle people playing’, recalling how a servant had taken him there as a child to see men throwing dice in an attempt to win their fortune. Pepys records seeing ‘how differently one man took his losing from another, one cursing and swearing, and another only muttering and grumbling to himself, a third without any apparent discontent at all’. His friend, Mr Brisband, offered to give him ten pieces to try his own luck saying that ‘no man was ever known to lose the first time, the devil being too cunning to discourage the gamester’. But Pepys refused, fleeing back to his chambers.

When Pepys had witnessed the game as a child there were no shortcuts that would have given him an edge. But in the intervening years between his youth and adulthood all that had changed because across the Channel two mathematicians, Pierre de Fermat and Blaise Pascal, had come up with a new way of thinking that would give the gambler a potential shortcut to making money, or at least losing less of it. It may be that Pepys had not yet heard of the breakthroughs Fermat and Pascal had made in wrestling the dice from the hands of the devil and into those of the mathematicians. Today in casinos across the world, from Vegas to Macau, the mathematics of probability that they initiated is the key to keeping those places in business at the expense of the ‘idle people playing’.

What are the chances of that?

Fermat and Pascal had been inspired to come up with a shortcut after they’d heard about a similar challenge to the one that Pepys was contemplating. A mutual acquaintance of the two, Chevalier de Méré, wanted to know which was the better bet out of the following:

A. Getting a six after throwing a dice 4 times.

B. Getting double six if you throw two dice 24 times.

The Chevalier was not in fact a knight but a scholar by the name of Antoine Gombaud who began using the title to represent his views in the dialogues he liked to write. But it stuck and his friends started calling him Chevalier. He’d tried to solve the conundrum of the dice by taking the long route and doing lots of experiments, throwing dice over and over. But the results had been inconclusive.

Instead he decided to take the problem to the salon organised by a Jesuit monk named Marin Mersenne from his monastic cell. Mersenne proved something of a hub for intellectual activity in Paris at the time, receiving interesting problems and sending them out to his other correspondents who he thought might have some clever insights. He certainly chose well when it came to Chevalier de Méré’s challenge. Fermat and Pascal’s response was to establish the shortcut of this chapter: the theory of probability.

It’s not surprising that the long route hadn’t really helped the Chevalier to decide which option was the best bet. Once Fermat and Pascal applied their new shortcut of probability to the dice, it turned out that option A occurs 52 per cent of the time while option B occurs 49 per cent of the time. Play the game 100 times and the errors that creep into random games of dice will easily obscure this difference. Only after about 1000 games might the true pattern of play emerge. That’s why this shortcut is so powerful. It saves you having to do a lot of hard work repeating experiments which might still give you the wrong feel for the problem.

The interesting thing about the shortcut that Fermat and Pascal came up with is that it only really helps you to gain an edge in the long term. It wasn’t a shortcut that helped the gambler win in any individual game. That was still in the lap of the gods. But in the long run it made a huge difference. Which is why it was great news for the casinos and less good news for the idle punter hoping to make a quick buck on a single throw.

Back in London, Pepys wrote about the fascination of seeing the players on his walk home trying to throw a seven: ‘To hear their cursing and damning to no purpose, as one man being to throw a seven if he could, and, failing to do it after a great many throws, cried he would be damned if ever he flung seven more while he lived, his despair of throwing it being so great, while others did it as their luck served almost every throw.’

Was the man particularly unlucky for not ever getting a seven? The strategy that Fermat and Pascal came up with for calculating the chances of getting a particular score with two dice was to analyse all the different ways that the dice could land and then to look at the proportion of those possibilities that give the score seven. There are 6 ways the first dice can land, which combined with the 6 different ways the second dice lands gives a total of 36 different combinations. Of these, 6 give you the score seven: 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, 6 + 1. Given that each of the different combinations is equally likely to occur, they argued that 6 out of 36 times the dice will land on a score of seven. It is actually the most likely score that you’ll get with two dice. But there is still a 5 in 6 chance of not getting a seven. With that in mind, how unlucky really was the gentleman Pepys saw who felt such despair at failing so many times to roll a seven?

What are the chances that he failed to get seven after rolling the dice 4 times? To go through all the different scenarios looks rather daunting since there are 36^4 = 1,679,619 different outcomes. But Fermat and Pascal came to the rescue because there is a shortcut. To work out the chances of not getting a seven in 4 rolls you simply have to multiply the chances on each roll: 5/6 × 5/6 × 5/6 × 5/6 = 0.48. This means the chances of not getting a seven 4 times in a row is still only about evens.

Conversely this means that in 4 rolls of 2 dice there is an evens chance of seeing a seven. The same analysis proves that getting a six after rolling a dice 4 times is also evens. So the fact that the gentleman Pepys had observed hadn’t got a seven after 4 throws isn’t that surprising. It’s the same as not getting a head after one toss of a coin.

The fact that seven is the most likely throw of the dice can be used to your advantage when playing many games of dice like backgammon or Monopoly. For example, Jail is the most visited square on the Monopoly board. Combining this with an analysis of the likely score of two dice means that after being on the Jail square, many players will visit the orange region of properties with greater frequency than anywhere else. So if you can grab the orange properties and stack them with hotels, you’ll give yourself a crucial edge in playing the game.

A cunning shortcut: consider the opposite

Hidden in the calculation that Fermat and Pascal made is another ingenious shortcut often used by mathematicians. What if I’d started with the challenge of trying to calculate the probability of getting a seven in 4 rolls of the dice? It clearly isn’t achieved by multiplying the chance of getting a seven 4 times. This gives me the rare case of getting 4 sevens in a row. Instead I have to go through all the possible combinations in which seven might occur. I need to calculate the chance of getting seven on the first throw and no sevens after that or no sevens on the first 2 throws and then two sevens on the last 2 throws. Again a mess of work. But here is the powerful shortcut. There is only one case I’m not interested in: failing to get any sevens. But this probability was easy to calculate. So instead of going at the problem head on, look at the negative.

I find this a very effective shortcut whatever problem I am working on. If tackling something head on is too complicated, try looking at the opposite side. For example, understanding consciousness is a tough scientific problem but analysing when something is not conscious can sometimes give you new insights into the more direct challenge. This is why analysing patients in deep sleep or a coma can help scientists to understand what it is that makes an awake brain conscious.

This shortcut via the negative challenge is key to working out the following problem: each weekend in the UK there are ten football matches in the Premier League. What proportion of them have two people on the pitch with the same birthday?

At first glance that would seem to be a pretty rare occurrence. Maybe one out of ten? I think one’s intuition on this is influenced by thinking that this question is the same as asking: if I was playing football this weekend what are the chances that someone on the pitch with me has the same birthday as mine? There would be about a 5 per cent chance of that happening.

Yet you are only thinking about pairing yourself up with each player on the pitch. What about all the other possible pairings? It doesn’t have to be you that shares the birthday. This starts to get more complicated and one begins to see that there are lots of different ways that people can pair up.

But by using the shortcut of looking at the negative problem, there is a much more efficient way to solve this challenge. What are the chances that no one on the pitch has the same birthday? If I can calculate this, then by subtracting it from 1 I can get the probability that there are two people who share a birthday.

The match is about to start. The teams run out, for our sakes entering the pitch in turn. I run on first. Then the next player comes on. What is the chance that he has a different birthday to me: 364/365. He just has to avoid my birthday: 26 August.

Now the next player runs on. They must have a different birthday to me and the second player on the pitch. There are still 363 days to choose from, so the chances they don’t have either of our birthdays is 363/365. With three of us on the pitch the chances that neither of us have the same birthday is 364/365 × 363/365.

Now I just keep going, bringing on all 22 players … and the referee. Each time another person runs onto the pitch the number of possible birthdays that I have to avoid goes up. By the time the referee comes on, he or she has to avoid the 22 birthdays already on the pitch so the chances of that are (365 − 22)/365 = 343/365. The probability after all 23 people are on the pitch that none have the same birthday requires us calculating:

364/365 × 363/365 × 362/365 × … × 344/365 × 343/365 = 0.4927

I’ve calculated the opposite of what we want. Now I just need to flip this. The chances that there are two people with the same birthday is 1 − 0.4927 = 0.5073. Incredibly it’s more than likely that there is a birthday coincidence. Which means that on average, 5 out of the 10 matches in the Premier League each weekend will see two people on the pitch with the same birthday.

Interestingly it is probably higher than evens because there is evidence that footballers are more likely to have a birthday in the months of September or October. Why? At school the impact of being born early in the school year means that you are likely to be physically more developed than someone like me born in August and hence will be stronger and faster and more likely to be picked for the school football team and to get experience playing the game. I vividly remember wondering why I never won races at school. Then one summer at a fair in our town I took part in a race run according to age group. It being the summer I hadn’t yet had my birthday while all the people in my school year had already had theirs, which meant I was up against the kids from the year below at school. As I left my opponents behind, I was totally shocked to find myself crossing the finishing line first for the first time ever.

But weedy young du Sautoy still had to make do with sitting in the library and becoming a maths whizz!

The shortcut to the casino

Mathematicians are in high demand in Las Vegas because casinos are constantly looking for new shortcuts that can rig their games in the house’s favour. Take the craps table, the game that evolved from the one Pepys observed. Betting on craps is quite a complicated business because of the dynamics of the game but at any one time you can make a bet that the next roll of the dice hits a seven. I explained that on average this happens 1 in 6 times. And yet if you place a dollar on this bet and win, the casino pays you only $4 on top of your $1 stake. To make this a fair game they’d need to pay out $5. This bet is one of the worst that you can make at the craps table as it gives the house a 16.67 per cent edge. This is the profit (on average) that the casino rakes in each time a punter makes a bet.

If you are absolutely insistent on betting on seven, there is a better way to decrease the house’s edge, which is to spread your bet three ways. Instead of placing one bet on getting a 7, you place three bets. One bet goes on the dice landing on 1 and 6. The second bet on 2 and 5, and the third on 3 and 4. This is called a hop bet. Although making these three bets is effectively the same as betting on a combined total of 7, the payout on each of these is better odds than betting straight on 7. The house now makes only 11.11% profit (on average) every time you bet.

Every game in Vegas has been carefully analysed to make sure that in the long run the house gets the edge, but as a punter you can also use the tools that Pascal and Fermat developed to find the places where you have the best chance of losing money slower than anywhere else.

For example, in craps there is a bet where the house actually pays out according to the odds of winning. It is about the only place in the casino where the game hasn’t been biased in the house’s favour. Craps works by the player throwing the dice and setting a target score. The target score needs to be one of 4, 5, 6, 8, 9 or 10. If the dice land on 2, 3, 7, 11 or 12, the game is over. The scores 7 and 11 win the game for the player, while 2, 3 or 12 are losing scores and are referred to as ‘crapping out’. If a target has been set, the aim is to get this score again with the dice before a 7 shows up.

The fair bet that you can make is to bet on this target appearing before the 7 appears. Suppose that a target of 4 has been set. If you bet $1 on seeing a 4 again before the 7, the casino will pay you $2 on top of your $1 stake if this happens, giving you back $3. This is exactly the odds that this will happen. There are three ways to get a 4 and six ways to get a 7, so you will only win this bet 1 out of 3 times. It is a bet where the casino pays without first taking a cut by stacking the odds in its favour. Not a shortcut to making money but at least the probability shortcut shows that you aren’t giving money away. Betting on this scenario means that in the long run you should come out even.

Here’s a little challenge for you. Let’s move to the roulette wheel. You have $20. Your goal is to try to double your money. If you put money on red and it comes up, you get double your money back. Which strategy is more likely to work? Strategy A: put all your money in one go on red. Strategy B: put down $1 on red on each spin of the wheel.

At first sight it might not seem to matter but there is a slight twist to the roulette wheel. There are 36 numbers, half red, half black, but there is also a 37th number: the zero, which is green. If the ball lands here you lose your money, whether you bet on red or black. This is the place where the house beats everyone. It looks pretty innocent, but the casino has calculated that this is its shortcut to profitability. At least in the long run!

It means that there isn’t an evens chance of you winning your bet if you put your money on red. You have a slightly smaller than evens chance: 18/37. Suppose that you make 37 bets of $1 on red and by a fluke over the 37 spins the roulette wheel turns up each number once. So on 18 of them you win $1, but on 19 you lose the $1, so at the end you’ve got only $36. It means that essentially every time you play, you pay the house $1/$37 = $0.027 for each $1 you bet. A house edge of 2.7 per cent. The more you play the more you pay.

In Strategy A the chance of you doubling your money in one go by putting down your $20 is 18/37 or 48 per cent, just less than evens. But if you play Strategy B, because you are paying for each $1 bet you make, it means that this strategy is gradually getting you further and further from your goal of doubling your money. In fact there is only a 25 per cent chance that this strategy nets you double your money in the long run.

Although Strategy A is the best bet, it does mean you have rather a short time in the casino. Strategy B might be a more fun evening, but you pay for your entertainment.

You might have heard that the place to go in the casino to get an edge over the house is the blackjack table. In the 1960s mathematician Edward Thorp worked out that by studying the hands of the dealers and the other players, you can get an edge. The method is called card counting. In blackjack you are trying to get cards that add up to 21 or less that beats the dealer’s hand. If you go over 21 you are bust and lose. The key fact that makes counting cards work is that if the dealer’s cards add up to 16 or lower, the dealer will always take another card.

A pack of cards has 16 cards that have a value of 10 points (ten, jack, queen, king). If you know that a lot of these cards are still in the pack, it means the dealer has a higher chance of going bust if they have to take a card and so it makes sense for you to bet more on your cards. Counting cards is a simple method of keeping track of how many high-value cards have already been played and how many are left in the pack. Casinos in general don’t use just one pack of cards but six to eight to minimise the effect of counting but it can still give you an edge. The film 21 dramatised the true story of a visit to Vegas by a team of MIT mathematicians who implemented Thorp’s shortcut. The nerdy mathematicians come off looking so sexy and cool that the film has probably done more for university admissions in mathematics than the combined efforts of all the maths departments up and down the country.

At first sight this looks a great shortcut to getting rich. The only trouble is that when I did an analysis of how long it would take you in reality to make much money from applying this strategy, it turned out that the time you would need to put in meant you were earning less than the minimum wage. It seems that lady luck played a part in the MIT team’s success.

Entry fee

How much would you be prepared to pay to play the following game? I roll a dice and pay you in dollars the number that appears on the dice. There is a one in six chance that I roll a six and you win $6. Each of the other options also occur with a one in six chance. Over six rolls you might make 1 + 2 + 3 + 4 + 5 + 6 = $21. So the average payout per roll is 21/6 = $3.50. If anyone offers to let you play for less than this, it would be worth playing because you’ll be the winner in the long term. Every time you play a game for money it is sensible to assess what the average payout is likely to be to determine whether the game is worth playing.

Although it was the correspondence between Fermat and Pascal that led to the discovery that you could apply mathematics to games of chance, the mathematics of chance really crystallised with the publication by Swiss mathematician Jakob Bernoulli of Ars Conjectandi (or ‘the Art of Conjecturing’). Jakob was part of the Bernoulli dynasty that had backed Leibniz in the controversy over calculus. It is here that you find the formula for the fair price you should pay for any game.

Suppose there are N possible outcomes. You win W(1) dollars if outcome 1 occurs. This happens with probability P(1). Similarly outcome 2 occurs with probability P(2) in which case you win W(2) dollars. On average this game earns you W(1) × P(1) + … + W(N) × P(N) dollars each time you play. So if anyone offers you less than this to play, you’re going to be the winner in the long run. For example, in the dice game there are 6 outcomes, the probabilities P(1), …, P(6) are all 1/6 and the winnings W(1), …, W(6) run from $1 to $6.

The formula seemed sound until Jakob Bernoulli’s cousin Nicolaus, in an almost oedipal act, came up with the following game. I toss a coin. If it lands heads I pay you $2 and the game ends. If it lands tails I toss again. If the second toss is heads I pay you $4. If it is tails I toss again. Each time I toss, the payout doubles. So if I toss 6 tails followed by a head I’ll pay you 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^7 = $128. How much would you be prepared to pay to play Nicolaus’s game? $4? $20? $100?

There’s a 50 per cent chance that you’ll only win $2. After all the probability that it lands heads first toss is ½. So P(1) = 1/2 and W(1) = 2. But you’re hoping for a long run of tails followed by a head to get as big a prize as possible. The probability that you get a tail followed by a head will be ½ × ½ = 1/4. But this time you win $4. So the second outcome has P(2) = 1/4 but W(2) = 4. As you keep going the probabilities get smaller but the payout bigger. For example, 6 tails followed by a head has a probability of (1/2)7 = 1/128 but wins you 2^7 = $128.

If you stopped the game after 7 tosses, you would only lose if there were seven tails in a row. Using Jakob’s formula the average payout would be W(1) × P(1) + … + W(7) × P(7) = (1/2 × 2) + (1/4 × 4) + … + (1/128 × 128) = 1 + 1 + … + 1 = $7. It’s worth playing the game therefore if anyone offers you a price of less than $7 to play.

But here is the sting. Nicolaus is prepared to play the game indefinitely until a head appears. You’re a winner every time. So how much will you pay to play the game? There are infinitely many options now. The formula says that the average payout will be 1 + 1 + 1 + … namely infinity dollars! If anyone offers to play this game with you, it’s worth playing whatever the cost to play. If the entrance fee is over $2 then 50 per cent of the time when a head appears on the first throw you lose out. And yet in the long run, if you keep playing the game over and over, the maths says you come out on top.

But why is it that most of us wouldn’t play the game for anything more than about $10 max? It’s called the St Petersburg Paradox, after Nicolaus’s cousin Daniel who, while working at the Academy in St Petersburg, came up with the first explanation of why no rational person would pay any price to play the game. The answer is what any billionaire will tell you. The first million you earn is worth so much more than the second million. You shouldn’t put in the formula the exact amount you win but what that prize is worth to you. In this way the price to play this game will vary according to how you value the outcomes. Daniel’s resolution goes far beyond just the curiosity of a mathematical game: it is essentially the foundation of modern economics.

To illustrate again how this shortcut to becoming a billionaire isn’t quite the shortcut it appears to be: if you could play a game a second, how long would it take to play 2^60 games? This is the number of games you might expect to play to break even in the Petersburg game if the entry price was $60. The answer is over 36 billion years. The universe is at most 14 billion years old. Another explanation for why most people wouldn’t pay an arbitrary price to play the game.

Goats and cars

Back in the 1990s a problem from an American game show called Let’s Make a Deal had people all over the world, including professional mathematicians, up in arms about the best strategy. The game show had a final round that went a bit like this.

I’ve got 3 doors. Behind two of the doors there is a goat and behind the other is a brand-new sports car. In the following analysis I am assuming the contestant wants to win the car rather than a goat. The contestant gets to pick a door, say door A. Fairly straightforward so far – there’s a one in three chance that the car is behind the door, right? But here comes the twist. The host, who knows where the goats are, opens one of the other doors and reveals a goat. Then he offers you the choice of sticking with your original door or swapping. What would you do?

Most people’s intuition is that there are now two doors so there is a fifty-fifty chance that the door they chose originally has the car behind it. If you switched at this point it wouldn’t change the odds of winning and you’d kick yourself if in fact you’d actually chosen the right door at the outset. So most people stick.

But you actually double your odds of winning by switching. It sounds strange, but here’s why. To calculate the probability of you winning I have to go through all the different scenarios where you switch and then count how many give you a win.

Scenario A: the car is behind door A, the door you chose. You swap and then get a goat.

Scenario B: the car is behind door B. The game show host opens door C to reveal the goat. You swap to door B and win a car.

Scenario C: the car is behind door C. The game show host opens door B to reveal the goat. You swap to door C and win a car.

Each of these scenarios is equally likely. Yet 2 out of 3 of them you win the car. If you stick, that strategy gives you only a 1 in 3 chance of winning. By swapping you actually double your chances of winning!

Don’t worry if you don’t quite get or believe that. This same explanation was published in a magazine and more than 10,000 people, including hundreds of mathematicians, wrote in to complain that it was wrong. Even Paul Erdös, one of the greatest mathematicians of the twentieth century, got it wrong before he thought about it properly.

But if you’re not convinced, how about this? Imagine that instead of 3 doors, there are a million doors. The game show host knows which one hides the prize. You’ve picked your door at random. There is a million-to-one chance you chose the right door. Now the game show host opens all but one of the other doors revealing 999,998 goats. There are two doors left. The one you chose and the door that the game show host hasn’t opened. Wouldn’t you switch now?

The point is that you are being given information by the game show host when the other doors are being opened. He knows where the goats are. If I alter the set-up things can change. Suppose you are up against another contestant. You choose a door. They get to choose one of the remaining doors. Their door is opened to reveal a goat. What do you do now? Strangely, although it looks like you’ve got the same information – two doors, one with a car one with a goat – this time it really is fifty-fifty that you get the car if you stick. The difference is that there is another scenario which has to be taken into account this time: if your door contains a goat the second contestant could have chosen the door with the car. That couldn’t happen in the previous scenario because the game show host always reveals a goat (they know where the goats are). Imagine the million doors scenario. Your opponent opens 999,998 doors, all with goats. They have been extraordinarily unlucky not to get the car but you learn nothing about either of the remaining doors by this bad luck. It’s now down to a fifty-fifty between the last two doors.

The Reverend Bayes

Probability seems to make a lot of sense when you are considering future events. If I am about to throw two dice then 1 out of 6 possible scenarios sees the dice land with the sum of 7. The probability is the same for you or for me because we are putting a value to something in the future.

But what if you throw the dice, they land, and you cover the result from me. So this throw has happened. It is in the past. It is definitely either 7 or not. There isn’t anything in between. The trouble is that I don’t know the answer. We still, some believe controversially, can assign a probability to this event. The probability for you is different because you have information. My probability is quantifying my lack of knowledge of the situation. Suddenly probabilities are depending on the amount of information we each have. This is quantifying epistemic uncertainty, things one could know in principle but do not in practice.

As I get more information about the event my probabilities are going to change. But coming up with a mathematics to keep track of what value I should be assigning the event given new data has resulted in different schools of thought.

For instance, you throw a white billiard ball on a table randomly, secretly mark its position, then remove the ball. If I was to draw a line to guess where the ball might have landed, I have no information so I might as well draw the line in the middle. But what if I now throw five red balls, and you tell me between which red balls your original white ball landed. Suppose there are three on one side and two on the other. This is going to push my guess towards the side of the table with the two balls. But how far should I push the line with this new knowledge?

Some schools of thought say that you should draw the line two-fifths of the way along the table. But a controversial figure in the theory of probability, Thomas Bayes, suggested that actually it should be drawn at the three-sevenths position because there is some extra information that you are missing in your analysis, namely the fact that before you knew anything, there was a fifty-fifty chance of a random ball being either on the left-hand side or right-hand side. Bayes throws these two extra balls into the mix in determining where to draw the line.

Bayes was a nonconformist minister preaching in Tunbridge Wells but was also something of an amateur mathematician. He died in 1761 but among the documents he left behind was a manuscript explaining these ideas of assigning probabilities to facts, given only partial information. This manuscript was later published by the Royal Society under the title: An Essay towards Solving a Problem in the Doctrine of Chances. The ideas in this paper have become hugely influential in the modern practice of assigning probability values to facts we have limited information about.

In court cases, lawyers will try to put probabilities on the chances that someone was guilty of a crime. They are either guilty or not. This assignment of a probability is in some sense quite strange. It is merely meant to be a measure of our epistemological uncertainty. But according to Bayes, the probabilities change as we feed in new information that we’ve gathered. Often juries and judges don’t understand the subtleties of Bayes’ ideas to the extent that judges have tried to dismiss such mathematical tools as being inadmissible in court.

Assigning probabilities to events as a shortcut to understanding our uncertainty often gets misused. Unfortunately the general public don’t have a good intuition about chance. That’s why we have to resort to maths as our shortcut so we don’t get lost on the way. Take the following example.

We are told the person who committed a crime came from London. The person in the dock is from London. But that is pretty flimsy evidence. There is a 1 in 10 million chance at the moment that we have the criminal.

The jury are now told that DNA at the scene matches the suspect’s and that there is a one in a million chance that the DNA found at a crime scene would produce such a match. One in a million sounds like certainty. Most people would convict on that evidence alone. Bayes helps explain how we should update the probability of the suspect being guilty. If London has a population of 10 million people, it means there are 10 people in London whose DNA would match the DNA at the crime scene. So that’s just a 1 in 10 chance that the person in the dock is guilty. What looked like a certain conviction is no longer so clear cut. This case is quite easy to understand but many uses of Bayes’ theorem in court cases are much more complex, involving many different types of evidence that require computer software to be able to analyse the probabilities of guilt. Sadly judges frequently don’t understand the maths and throw the expert evidence out of court, resulting in some horrendous miscarriages of justice.

Medicine is another area in which probabilities are applied and once again if you don’t understand how the shortcut is being used, you can be led far from the destination you were hoping for. If you go for a scan for breast or prostate cancer and you are told that the scan is 90 per cent accurate in detecting cancer, most people will be freaked out if they get a positive result. But should they be? It is important to have the extra data that only 1 in 100 patients is likely to have cancer. So out of 100 people tested, one will probably have cancer and the test generally gives a positive result. It’s the false positives that cause the problem. Of the 99 who are tested, with a 90 per cent success rate the test actually gets it wrong for 10 out of these 99 healthy patients. So if you test positive there is only a 1 in 11 chance that actually you are the one with cancer!

It is important to understand these numbers because the media love abusing them to make a scary story. What about that news story with which I opened the chapter, that bacon increases your chance of bowel cancer by 20 per cent. That sounds frightening. Should I give up those bacon sandwiches I love? If you look at the proportion of people who have bowel cancer it’s 5 in 100. If you eat bacon that goes up to 6 out of 100. That’s a less frightening way to couch this probability.

Pepys

What about Pepys’ challenge of throwing dice to get a 6 that opened this chapter? What are the chances of getting at least one 6 in six throws? Again the shortcut is to consider the opposite. The chance of not getting a 6 six times is (5/6)6 = 33.49 per cent. So the chances of getting at least one 6 is quite high at 66.51 per cent.

What about throwing twelve dice and getting at least two 6s? There are really too many different scenarios to consider again here, so let’s use the trick of working out the opposite: the chances of getting (a) no 6s and (b) exactly one 6. Case (a) is the same principle as before: (5/6)12 = 11.216 per cent. What about getting exactly one 6. There are twelve different scenarios here, according to which throw the 6 occurs on. The chance that the first throw is a 6 and the rest are not is (1/6) × (5/6)11. This is actually the same for all the other scenarios so the total probability is 12 × (1/6) × (5/6)11 = 26.918 per cent. So the chance of getting two or more 6s in twelve throws is

100 − 11.216 − 26.918 = 61.866 per cent

So option (a) is the better one to gamble on. If you do a similar case analysis on option (c) which is a bit messier than option (b), the odds get even worse at 59.73 per cent.

Pepys had written to Isaac Newton about the problem in three letters he sent at the end of 1693. Pepys’ intuition was that (c) was the most likely option but, applying the shortcuts of Fermat and Pascal, Newton replied that the mathematics implied the opposite was true. Given that Pepys was about to stake £10, or the equivalent of £1000 in today’s money, it was lucky that Newton’s advice saved him from a shortcut to penury.

Shortcut to the shortcut

At every step in our journey through life we encounter junctions with many different pathways leading into the distance. Each choice involves uncertainty about which will get you to your destination. Trusting our intuition to make the decision often ends up with us making the suboptimal choice. Turning the uncertainty into numbers has proved a potent way of analysing the paths and finding the shortcut to your goal. The mathematical theory of probability hasn’t eliminated risk but allows us to manage it more effectively. The strategy enables us to analyse all the possible scenarios that the future holds and then to see what proportion of them lead to success or to failure. This gives you a much better map of the future on which to base your decisions about which path to choose.

Pitstop: Finance

Everyone is after the shortcut to great riches. Purchasing a lottery ticket. Placing a bet on a horse. Founding the next Facebook. Writing the next Harry Potter. Investing in the next Microsoft. While maths can’t guarantee your path to riches, it does still offer some of the best ways of maximising your chances.

You might think that Isaac Newton, with all his mathematical techniques for optimising solutions, would have been a successful investor but after losing a great amount of money in a market collapse he declared, ‘I can calculate the movements of stars but not the madness of men.’

Yet since Newton’s day mathematicians have understood that there are clever shortcuts to making money in the markets. That is why the funds that are consistently doing well in good times and bad are invariably staffed by people with mathematics PhDs. A great shortcut to finding the best fund for your savings would be to count up the number of mathematics PhDs on the fund’s books. But how does knowing your mathematics help? Isn’t it all driven by human whims and moods? Wouldn’t a psychology PhD be more useful?

At the beginning of the twentieth century French mathematician Louis Bachelier proposed that investing in stocks is actually no different from betting on the toss of a coin. His was the first model that emerged of how prices vary over time. With complete knowledge of the markets, Bachelier believed that stock prices would move up and down randomly. This behaviour is known as the drunkard’s walk because on a graph it looks like the path of a drunken person staggering down the street. Sure, overall prices might be affected by the outbreak of a pandemic. But having taken that knowledge into account, a stock might from that moment on head up or down randomly. Knowing this doesn’t really give you an advantage. But it does if you recognise that in fact this model is wrong. Mathematicians in the 1960s realised that the randomness of a coin toss wasn’t quite correct because that would imply there was a chance that a stock might become negative in price. So a new model emerged that was still random but accepted that a stock has limits on its lowest value while potentially still able to climb as high in value as it wanted.

One way you can beat the markets is if you can glean some hidden information in the prices. This can give you an edge. For example, a bookie assigning odds for a three-horse race will ensure that you can’t win money by simply betting on all three horses. But what if you know for some reason that one of those horses can’t win? Then it is possible to spread the bet across the other two to guarantee a win.

Essentially this is the origin of the idea that Ed Thorp suggested in his 1967 book Beat the Markets. As I mentioned in the previous chapter, Thorp had already cracked getting an edge at blackjack by counting cards. He’d even used a device to analyse the spin of a roulette wheel to make clever bets, before he was chucked out of the casino for cheating. But his new idea would lead to the creation of the concept of the hedge fund. The key was finding a way to invest in two financial horses such that you profited regardless of whichever one was successful.

Thorp had discovered that certain financial products called warrants were overpriced, a bit like a bet in the casino is overpriced to give the house an edge. Unfortunately in the casino you can’t bet that you’ll lose, so it’s impossible for a punter to take advantage of this knowledge. But Thorp realised that there was a way to exploit the overpricing of warrants using something called shorting the market. You could borrow the expensive warrants owned by someone else with the promise of giving them back the warrants at a later date plus some interest. You could now sell the warrants and when the time came to give them back, you buy back the warrants before returning them to the borrower. The point is that the overpricing will in general mean that the price of the warrant would now be lower than the price you had sold it at, making you a profit.

The only problem is that sometimes this won’t be the case. The warrant could possibly go up over time, just as the bet at the casino can still be successful even if the house has an edge. And if the warrant does amazingly, you could be liable for a big loss. But here was the clever hedge. A warrant is an option to buy a stock. If the warrant does well that’s because the underlying stock has done well. So in tandem with selling the warrant you borrowed, you also bought some stock such that if you happened to be unlucky that the warrant did well, although the bet on the warrant failed, you’d get money from the stock doing well. It’s not a guaranteed money earner but Thorp understood that most of the time, whether prices go up or down, you profit.

The key was pricing this balancing act to your advantage, just like the punter who spread bets on two horses knowing the third couldn’t win. It’s all about exploiting knowledge to your advantage. The casino does this, but the clever thing is that hedge funds had also spotted this shortcut to making money from the markets.

But mathematics isn’t the only shortcut available to the investor. Helen Rodriguez, a friend of mine who is a very successful financial analyst, prepared for her career by studying history not mathematics. And as it turns out, the skills of the historian offer a shortcut Helen routinely exploits to give her an edge in understanding when a company is under- or overvalued.

Helen specialises in high-yield bonds, also known as junk bonds, which are commonly used in buying and financing companies. If you buy a bond you are lending a company money with the promise of a fixed rate of interest in return plus your money back at maturity. Junk bonds have a higher risk of default, and for that reason they also have higher returns.

‘Here’s your first shortcut: we use a rating scale of the credit quality of companies defined by the company’s willingness and ability to pay,’ says Helen. ‘It starts at triple A, for companies that pose little risk right down to C for companies where there is little chance of recovering interest or principal. Bonds are high yield if the rating is below triple B minus. If a company has a low rating the interest has got to be higher to make it worth investing in. Hence the name high-yield bond.’

You often hear in the news about a country’s or a bank’s credit rating being downgraded by someone like Moody’s, which is one of the corporations that issues these evaluations. The ratings take the multi-dimensional corporate world and attempt to project that in all its complexity and mess on to a one-dimensional line, with C at one end and triple A at the other.

Helen uses her historical toolkit to work backwards, looking at the story of a particular company that’s been given a credit rating and trying to understand if the bond is under- or overvalued. Attempting to squeeze out new information like this might allow her to get an edge. It is quite an art to identify an aspect of a company’s story that others have overlooked that can give new insight to the value of a bond. And that skill of seeing the bigger picture is one that historians often excel at.

‘I was covering these 2500 German beauty shops, and the bonds were still stuck above par and I was thinking, What a waste of time,’ she says. ‘Then they had a bad quarter which they blamed on terrorism in Germany. Then they had another bad quarter and they still blamed terrorism, at which point I thought, This is a bit odd. But the bonds were still stuck above par. So I started doing a bit of reading and discovered that Asian companies were coming into Europe and exploiting the internet to sell the same cosmetics but maybe six months out of fashion at half the price. It’s called the grey market. There were a couple of disruptor companies that were doing this and absolutely killing the German beauty market. So we sell at 103 and, within a year, the bonds are in the 40s. People hadn’t woken up to this grey market.’

Essentially Helen used a similar trick to Ed Thorp. She borrowed the bonds and sold them at 103 but was able at a later date to give back the bonds to the original owner from whom she’d borrowed them by buying them at 40, making a huge profit. She’d managed to use her hunch about the imminent collapse of the bond to her advantage. Often it’s about seeing through a company’s bravado about its worth.

‘Companies will often not be as open as they should be about the fact that they have a problem,’ says Helen. ‘It’s something as stupid as it’s run by fifty-five-year-old men that don’t understand teenage girls. It’s often arrogance or vanity or just not understanding the Way of the World. We’ve seen it massively in retail, the whole disintermediation and disruption that’s happened because of the internet. It’s shocking how late some of the managers of companies have seen it.’

It seems to me that in some ways it’s quite hard to come up with shortcuts because in reality you have to get to know your company quite intimately to gain that kind of insight. There’s a lot of storytelling involved. Helen compares it to watching a soap opera. ‘I’ve been following this one company, a Spanish gaming company. The restructuring took a year and a half and literally every single day, I’d open the Argentinian newspapers and have to read them because Cristina Kirchner, the former Argentinian president, was using the gaming sector as a political football. That was the story that was driving the bond!’

It’s the skills Helen learned training to be a historian that she believes give her the shortcut to tell the story of each company she is assessing. As she is watching the soap opera of each company play out, she needs to guess what’s going to happen in the next episode before it airs. As Helen sees it, she needs to be able to synthesise a huge amount of information into something useful. This is what historians are good at. ‘It’s like a puzzle you’re trying to understand. It’s just like doing history. With ten different sources I need to come up with my narrative of what I think happened. That’s why someone else might take the same sources and come up with a different narrative. You need this for there to be a market. You need the guy who thinks this is a great thing and you need another guy who thinks this is the end of the world. Then there’s a trade.’

One of her other shortcuts is top of my list of mathematical shortcuts: the power of spotting a pattern. ‘You can also find patterns in what happens to companies and what goes wrong, because they’ve all got the same problem, but maybe their segmentation of what they sell is a little bit different. I’m trying to spot the patterns in what’s going to happen before everybody else and making that recommendation.’

Having worked for many years for Deutsche Bank and Merrill Lynch among others, making investments, Helen now works for a company providing independent corporate bond research for investors, like the analysis she did on the Spanish gaming company.

So if you were reading this pitstop in the hope that I have some cunning shortcuts to how to invest your savings, my advice would be to combine the skills of the mathematicians together with exploiting the deep knowledge that someone like Helen has gleaned through her training as a historian to be able to guess the next episode in the soap opera that we call the markets. As Newton articulated, sometimes the best shortcut is to stand on the shoulders of giants.

9

The Network Shortcut

Draw the following figure without taking your pen off the page and without running over a line twice:

9.1 Drawing challenge.

Our journey through the modern world is increasingly mapped by networks. Systems of roads, railways and flight paths allow us to get from one side of the planet to the other. A variety of apps offer the most efficient path through this intricate web. Companies like Facebook and Twitter have stretched our social interactions far beyond the inhabitants of our local village. The ultimate network that humanity spends hours navigating daily is that alternative world: the internet. Google rose to prominence with a shortcutting algorithm called PageRank, which helps users navigate round this web of nearly 2 billion sites. Although we regard the internet as a relatively new phenomenon, the first inklings of this network were actually hatched in the nineteenth century by my favourite shortcutter.

Carl Friedrich Gauss was a big fan of physics as well as mathematics and collaborated with Wilhelm Weber, one of the leading physicists in Göttingen, on numerous projects. Gauss even came up with a way to shortcut having to walk from the observatory in Göttingen to Weber’s lab. Instead of meeting in person, he rigged up a telegraph line between the two. It stretched across the roofs of the town for a distance of three kilometres. Gauss and Weber had understood the potential of electromagnetism to communicate over a distance. They cooked up a code in which each letter was represented by a sequence of positive and negative electrical pulses. This was in 1833, several years before Samuel Morse came up with a similar idea.

Gauss thought the idea something of a curiosity but Weber saw the import of such a technology: ‘When the globe is covered with a net of railroads and telegraph wires this net will render services comparable to those of the nervous system in the human body, partly as a means of transport, partly as a means for the propagation of ideas and sensations with the speed of lightning.’ The rapid spread of the telegraph makes Gauss and Weber the grandfathers of the internet. Their collaboration has been immortalised in a statue of the pair in the city of Göttingen.

Today this network, as Weber predicted, extends well beyond the few kilometres of wire that the two scientists trailed across the roofs of Göttingen. Indeed it is so complex that finding shortcuts across networks has become one of the central subjects of modern mathematics. These networks can be made not just of wires, but also bridges, as I explored on a recent trip to Russia.

Read Euler. Read Euler. He is the master of us all

When I flew to Kaliningrad a few years ago I made sure I got a window seat for the short flight from St Petersburg. I was on a pilgrimage to a city that is home to one of the stories that every mathematician is brought up on, of one of the cleverest shortcuts dreamed up in the history of mathematics.

As we came in to land in Kaliningrad, a small exclave of the Russian Federation cut off from mainland Russia by Lithuania and Poland, I could see the River Pregel running through the city. The river has two branches that join at Kaliningrad from where it flows west to emerge at the Baltic. There is an island in the centre of the city around which the two branches run. It is the bridges connecting the banks of the rivers and this island that are at the heart of the mathematical story for which Kaliningrad is famous.

The tale dates back to the eighteenth century when the town had a different name: Königsberg, birthplace of Immanuel Kant and the famous mathematician David Hilbert. At the time part of Prussia, there were seven bridges spanning the Pregel. It had become a Sunday afternoon pastime among residents of the city to see whether they could find a way to cross all the bridges once and once only. But however hard they tried, they always found that there was one bridge that they couldn’t get to. Was it really impossible or was there some way the residents hadn’t tried that meant you could cross all seven bridges?

For the residents of Königsberg there didn’t seem to be any way to avoid the hard work of trying every possible route round the bridges until they had exhausted all the possibilities. There was always a sneaky feeling that they might have missed a clever way round that meant the challenge was possible.

9.2 The seven bridges across the River Pregel in eighteenth-century Königsberg.

It took the arrival of one of my mathematical heroes, Leonhard Euler, to resolve the puzzle once and for all: it was impossible to cross all the bridges once and once only. To figure this out, Euler discovered a shortcut that avoided having to try every route round the bridges.

I introduced the Swiss mathematician Euler in Chapter 2 when I revealed the extraordinary formula he came up with linking five of the most important numbers in mathematics. ‘Read Euler. Read Euler. He is the master of us all,’ wrote Pierre-Simon Laplace, one of France’s foremost mathematicians, about Euler’s importance to the subject. Most mathematicians would agree, ranking him alongside Gauss as one of the greats. Even Gauss was a fan: ‘The study of Euler’s works will remain the best school for the different fields of mathematics and nothing else can replace it.’

Euler’s contributions range far and wide, including the shortcut to solving the challenge of the bridges of Königsberg, which he first learned about when he was a professor at the Imperial Russian Academy of Sciences in St Petersburg. Not a native of St Petersburg, he had travelled there from his home town of Basel, where he’d been unable to find a job as a mathematician. Evidently, all the positions in mathematics had been filled. Strange that such a small city should be so oversubscribed with mathematicians. Even stranger: all of them came from the same family: the Bernoulli family.

Basel couldn’t even accommodate all the Bernoullis. Daniel Bernoulli had already decamped to St Petersburg and it was his invitation that secured a position for Euler at the academy. Before Euler set off, Daniel sent him a letter listing all the Swiss creature comforts they were missing in St Petersburg: ‘Please bring fifteen pounds of coffee, one pound of the best green tea, six bottles of brandy, twelve dozen fine tobacco pipes and a few dozen packs of playing cards.’

Weighed down with all these provisions, and travelling by boat, on foot and by post wagon, Euler took seven weeks to reach St Petersburg from Basel, arriving in May 1727 to take up his position.

The bridges of Königsberg

At first, the problem of the bridges of Königsberg was little more for Euler than a bit of light relief from all the complicated calculations he had been engaged in. In 1736 he wrote a letter to Giovanni Marinoni, court astronomer in Vienna, describing what he thought of the problem: ‘This question is so banal, but seemed to me worthy of attention in that neither geometry, nor algebra, nor even the art of counting was sufficient to solve it. In view of this, it occurred to me to wonder whether it belonged to the geometry of position, which Leibniz had once so much longed for. And so, after some deliberation, I obtained a simple, yet completely established, rule with whose help one can immediately decide for all examples of this kind whether such a round trip is possible.’

The important conceptual leap that Euler had made was that the physical dimensions of the town were irrelevant. What was important was how the bridges were connected together. The same principle applies to the map of the London Underground: that is not a physically accurate map but one that retains information about how the stations are connected. If one analyses the map of Königsberg, then just as locations in London become points on the Underground map, the four regions of land connected by the bridges could each be condensed to a point, with the bridges represented by lines connecting the points. The problem of whether there was a journey round the bridges was then equivalent to being able to draw the resulting picture without taking your pen off the paper and not running over any line twice.

9.3 The network diagram of the bridges of Königsberg.

So why was this impossible? Although Euler probably never drew this graphical picture of Königsberg explicitly, his analysis shows that if a journey was possible, each point visited in the middle of a journey must have one line into it and one line out. If you visited that point again, it would be on a new bridge into it, and a new bridge out of it. And so there must be an even number of lines attached to each point. The only exceptions to this were at the beginning and end of a journey. Where you set off would have one line coming from it. And the end of the journey would also have one line running into it. For a journey on any graph to be possible, no more than two points – the beginning and end points – can have odd numbers of lines emerging from them.

But if you look at the plan of the seven bridges of Königsberg, each point has an odd number of bridges connected to it. With so many points with an odd number of bridges sprouting from them, it meant that a path round the city crossing each of the bridges only once was impossible.

This is one of my favourite examples of a shortcut. Instead of trying lots of different ways to trace out a route round the map, this simple analysis of the number of points with an odd number of bridges emerging reveals immediately that such a route isn’t possible.

The beauty of Euler’s analysis is that it doesn’t just apply to Königsberg. In any network you draw of points joined by lines, if the number of edges emerging from a point is always even then Euler proved that there will always be a path running over all the lines once and once only. Also if there are precisely two places with an odd number of edges, such a path is possible where these points will be the beginning and end of your journey. It doesn’t matter how complex the map, this simple analysis of the tally of odd-number nodes provides a shortcut to knowing whether the network is navigable or not.

Königsberg had only seven bridges, but recently mathematicians in Bristol applied Euler’s shortcut to the 45 bridges that span the complex system of waterways that run through their city. And while Königsberg has one island, Bristol has three: Spike Island, St Philips and Redcliffe.

It is far from clear at first that a walk round all 45 bridges would be possible, but using Euler’s shortcut you can see that the tally of odd-number nodes in the map charting the way the land was connected by bridges was small enough that a path had to be possible. A route for Bristol’s walkable bridges was first devised in 2013 by Dr Thilo Gross, a former reader in engineering mathematics at Bristol University. ‘Having found the solution I naturally had to walk it,’ he said. ‘The first bridge-walk took 11 hours and was about 53 kilometres long.’

Euler’s shortcut actually helped me during a set of psychometric tests that I was required to take for a job application when I was younger. The test included a series of networks and the task was to draw the networks shown without taking your pen off the paper and without running over a line twice. The implication of the task was that it was possible and it looked like they were testing your ability to complete each task. But the task was really testing the honesty of the applicants because one of the three networks was impossible to draw. Like the map of the bridges of Königsberg, there were more than two nodes with an odd number of lines emerging from the point.

Obviously my clever-clogs essay next to the challenge about why the task was impossible thanks to Euler’s shortcut didn’t go down too well. I didn’t get the job.

Human heuristics

Euler’s great insight was to home in on the essential quality of the Königsberg map that was important for solving the problem. It didn’t matter how far you had to travel or what the bridges looked like. The art here was to throw away all the extraneous information and retain the quality of the map that was critical for navigating the journey. This idea of throwing away unimportant information is key to many shortcuts. It’s the idea behind humans’ use of heuristics: the way we ignore or approximate information, either consciously or unconsciously, in order to simplify the task at hand and ease the cognitive load. As humans we often have to make decisions based on limited time or mental resources, so we have to find efficient ways to pick out aspects of the problem that contribute to a solution and don’t use up precious mental space unnecessarily.

In their groundbreaking work, psychologists Amos Tversky and Daniel Kahneman identified three key strategies that we use as mental shortcuts to making decisions. We use the idea of patterns between different events, which they called representativeness. This is certainly a quality that I exploit in mathematics to shortcut rethinking a problem. The second strategy is called anchoring and adjustment. This is a process that starts with an initial piece of information that we understand or know, the anchor, which other situations are then judged against. The final strategy is the availability heuristic, which uses our local knowledge to judge a more general situation.

Clearly the last two are very prone to produce bias because in general we don’t have very good anchors or terribly representative local knowledge. In his hugely influential book on the limitations of human heuristics Thinking, Fast and Slow, Kahneman gives examples of how just mentioning a number before you ask a question can skew people’s estimates. For example, mentioning the years 1215 and 1992 contaminated people’s estimates for the first year that Albert Einstein visited the US (1921), dragging them lower or higher than those participants who were asked the question without an anchor, even though the anchoring dates were clearly unrelated to the question being asked.

The mathematical shortcuts that we have come up with over the centuries are an attempt to override the evolutionary shortcuts that can fail as our questions become more complex. These heuristics might have helped us to navigate our patch of the savannah, where things were less likely to vary too much, but they are not helpful in trying to understand universal truths.

The key to a good heuristic is to understand, like Euler did in Königsberg, that the nature of the bridges, the distances involved, the geography of the city, were not relevant to the problem. Only the way the land masses were connected was relevant to solving the challenge.

Once I had arrived in Kaliningrad, I was intrigued to see on my visit to the modern city how many of the seven bridges were still standing. Being an important port on the Baltic, it had been a strategic location for the German fleet during the Second World War and had suffered devastating bombing by the Allies. Much of the historic city had been levelled to the ground, including the famous university situated on the island at its heart where Kant and Hilbert had learned their academic trade. So how had the bridges fared?

Three of the pre-war bridges were still there. Two had vanished completely. The two remaining bridges had been bombed during the war but had subsequently been rebuilt to carry a huge dual carriageway through the city. Yet two new bridges had appeared: a railway bridge, which I discovered pedestrians can cross over, joining the two banks of the Pregel out to the west of the city, and a footbridge called the Kaiserbridge. There are once again seven bridges but now in a slightly different arrangement from the eighteenth-century bridges that Euler analysed. The beauty of his shortcut, of course, is that it applies whatever the number and arrangement of the bridges. So my immediate thought was to see whether a journey around today’s bridges might be possible.

9.4 The seven bridges of twenty-first-century Kaliningrad.

Remember Euler’s mathematical analysis showed that if there were exactly two places with an odd number of bridges emerging from them, a path would always be possible: you start at one of the odd-numbered points and end at the other. Checking the plan of today’s bridges of Kaliningrad reveals that such a journey is possible. Starting on the island in the middle of the city, I set off excitedly on my pilgrimage round the seven modern bridges of Kaliningrad.

The story of the bridges of Königsberg is also the beginning of a very important branch of mathematics that is highly relevant to our digitally connected world: network theory. And the development of shortcuts around complex networks like the internet has made some mathematicians a lot of money.

Internet shortcuts

There are over 1.7 billion websites on the internet. Yet despite this extraordinary number, the Google search engine still manages to quickly find the information you want to retrieve. You might think that this is the result of huge computing power, and that is certainly part of the equation. But it’s the way Google searches that has made it such an indispensable tool.

In the past, search engines would look for those websites that mentioned your search term the most times. If you were looking for biographical details about Gauss’s life, searching ‘Gauss biography’ would have thrown up the sites that included these two words most often.

But if I wanted to spread some false biographical details about Gauss, by loading the meta-data of my site with many copies of the words ‘Gauss’ and ‘biography’ I could ensure that my false news site hit the top of the list. Just using a word search did not provide a powerful way to find the sites you were after.

Two Stanford graduate students, Larry Page and Sergey Brin, working out of a garage in Menlo Park, came up with a much more robust solution to find the best way to rank which biography of Gauss to put at the top of a search request. They decided to exploit an ingenious tactic: to use the internet itself to tell it which pages were the most important. The idea was that the relevance of a website could be judged by the number of other websites that linked to it. A legitimate page detailing the biography of Gauss would likely be linked to by other websites interested in the topic.

But if the importance of a website is simply judged by the number of links from other websites, there would be an easy way for me to hack my false website to the top of the list. By making thousands of spoof websites and linking them to my ‘Gauss Biography’ page, I could seemingly make my page look the most important.

Page and Brin had a strategy to stall such a hack. A website would only climb high in the rankings if the websites that linked to it were also highly regarded. But hold on. This sounds rather circular. I need to know which of these sites linking to my Gauss biography sites are highly valued. But they get their value from high-value sites linking to them. I seem to have gotten into an infinite regress.

The way to resolve this was to regard all websites as having equal status at the outset. I start by giving each website a score of 10 stars. But now I proceed to redistribute the stars. If a website links to five other websites, I give each of those websites two of its stars. If it links to just two websites, each of the linked websites receives five stars. Although the original website has given away all its stars, it will hopefully be linked to by other websites that will give it some of their stars.

By continuing to redistribute stars from one website to another, I start to see the dominant websites collecting more and more stars. Simply being linked to by my thousand spoof websites will be revealed for the hoax it is. After a single round, my thousand sites are starless and can no longer help maintain the value of my false site. Very quickly my site gets drained of its stars and plummets down the list of sites that the algorithm is valuing. There is a little more work one has to do to implement the idea but this is the essence of the way Google ranks websites.

However, it takes time and computing power to analyse how the stars flow around the network. But then Brin and Page realised that there was a shortcut to work out the ranking. As undergraduates they’d been taught what at first sight seems a rather esoteric bit of arcane mathematics called ‘the eigenvalue of a matrix’.

What this mathematical tool does is to identify in different dynamic settings certain parts of the system that remain stable. It was first used by Euler in the context of a rotating ball. If you take a globe with the countries of the Earth painted on the surface, then no matter how you might twist and turn the globe in your hand, it is possible to take its final position and to fix two antipodal points such that a rotation around the axis through these points returns the globe to its starting position. Essentially it means that every possible rearrangement of the globe can be realised by a simple rotation through some axis.

The eigenvalue of a matrix provides both a proof that such an axis of rotation always exists and a method for finding the two stable points through which the axis runs. It is remarkable how this technique allows us to identify points of stability in so many different dynamic settings. For instance, eigenvalues of a matrix are central to identifying the stable energy levels in a quantum system. They are also key to picking out the resonant frequencies of musical instruments.

Brin and Page realised that they were also the secret to identifying how the stars would stabilise once they had been distributed around the network. Just as the eigenvalues find the stable energy levels in an atom or the stable points on the rotating sphere, they also help to determine how to assign stars so that the numbers aren’t subsequently altered by the redistribution round the network. So instead of running an iterative process waiting for things to reach an equilibrium, the eigenvalue of a matrix was a smart shortcut to calculate the page rank of any website on the internet.

Although my attempts to elevate my fake Gauss biography site were thwarted, it is still important for companies to understand how Brin and Page’s shortcut works. There are things a company can do to make sure that the Google shortcut charts a path through its website. Small perturbations to the Google algorithm can see the shortcut alter course slightly, resulting in your website dropping down the rankings. You need to know what changes to make to get your site back on the pathway.

Social shortcuts

Sometimes the challenge is how to get from one point in the network to another point via the shortest path possible. Are there cunning shortcuts across? Take the network of social connections across the population of the planet. If I choose two random people, how short a chain of friendship connections can I find to get from one to the other? The surprise is that it takes very few.

This question was first posed in a short story entitled ‘Chain-Links’, written by Hungarian author Frigyes Karinthy in 1929. In it, the central character speculates that this network has amazing shortcuts across its chain of links:

A fascinating game grew out of this discussion. One of us suggested performing the following experiment to prove that the population of the Earth is closer together now than they have ever been before. We should select any person from the 1.5 billion inhabitants of the Earth – anyone, anywhere at all. He bet us that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the selected individual using nothing except the network of personal acquaintances.

It took just over 30 years for this fictional game to be put to the test. In a famous experiment conducted by American psychologist Stanley Milgram in the 1960s, a stockbroker friend of Milgram’s living in Boston was chosen as the target. Milgram decided to nominate two cities in America that he felt were furthest both geographically and socially from the target in Boston: Omaha in Nebraska and Wichita in Kansas. Letters were then sent to random people living in these two towns with instructions to forward the letter on to the named stockbroker. The catch was that no address was provided. If the recipient did not know the named target, they were asked to forward it to a friend in their network of acquaintances who they believed might be better placed to forward the letter.

Of 296 letters that got sent out, 232 never reached the target in Boston. But of those that did, on average it took six forwardings of the letter to get from the original recipient to the target. There were indeed five individuals between the start and end of the chain.

This experiment led to the famous phenomenon called six degrees of separation. The phrase was popularised by John Guare in his play of the same name. As one of the characters declares towards the end of the play: ‘I read somewhere that everybody on this planet is separated by only six other people. Six degrees of separation. Between us and everybody else on this planet. The president of the United States. A gondolier in Venice. Fill in the names. It’s not just big names. It’s anyone. A native in a rainforest. A Tierra del Fuegan. An Eskimo. I am bound to everyone on this planet by a trail of six people.’

In our digital age we are more interconnected than ever and it is a network we can explore much more easily than by forwarding letters via the US postal system. In 2007 a data set of messages composed of 30 billion conversations among 240 million people revealed that the average path length between users was indeed 6. A paper published in 2011 found that across Twitter you can connect any two tweeters with a chain that is on average just 3.43 users.

Why do social networks have these shortcuts? It’s certainly not true of all networks. Just arrange 100 nodes on a circle and only join those that are next to each other. It requires 50 handshakes to get from one side of this network to the other. A network in which you can move between any two points with only a small number of connections is called a small world.

It turns out that an extraordinarily large number of networks are examples of small worlds. Not just our social and internet connections. The neural connections in everything from the C. elegans nematode worm with its 302 nodes through to the human brain with its 86 billion neurons seem to be examples of small-world networks. This allows one neuron in the system to quickly communicate with any other neuron via a small number of synapses. Electrical power grids are small worlds, as are airport networks and food webs. What is it about these networks that makes them small worlds?

Two mathematicians, Duncan Watts and Steve Strogatz, discovered the secret, which they published in a paper in Nature in 1998. If you take a set of nodes and create local links between those that are close to each other then typically you get a picture like our circle, with long journeys that are necessary to connect randomly chosen nodes across the network. But Watts and Strogatz discovered that it required just a few global links across the network for shortcuts to appear. It’s like everyone in Boston knows each other but then someone in Boston happens to have an aunt living in Kansas, which will provide the way of connecting these local neighbourhoods more globally. In the C. elegans worm you can see the same architecture. The neurons are arranged in a circle but across the circle you see links connecting distant neurons. It seems that the human brain also has a similar architecture. Lots of local connections with a few long synapses linking disparate bits of the brain.

9.5 Example of small world network.

Airport networks work in a similar fashion, with a few airports that act as hubs connecting the world with long-range flights. Then within a region there are lots of short-haul flights that take you from the hub to local destinations.

Using their mathematical model Watts and Strogatz were able to show that in a network with N nodes where each node has K acquaintances linked in this local-global manner, the average path between two randomly chosen points in the network is given by the formula

log N/log K

where log is the logarithm function cooked up by John Napier to shortcut calculations. Put N equal to 6 billion and connect them up with 30 acquaintances each and the number of degrees of separation comes out at … 6.6.

If you are building a network, be it social, physical or virtual, often you will want there to be shortcuts across the web of connections. But now we know how to cook up such systems. To create a network with this small world’s characteristic of amazing shortcuts from one end to the other, it’s adding this randomly selected batch of global connections that seems to do the job.

Gauss’s brain

When Gauss died in 1855 he left his brain to science for investigation. His friend and colleague Rudolf Wagner, a physiologist at the University of Göttingen, took on the task of dissecting the brain to see if there was anything special about it that might have made Gauss so adept at finding mathematical shortcuts. It was part of a larger project undertaken at the university to understand whether there was any special architectural difference between elite brains and those of the general public. Instead of crude measures like volume or weight, Wagner proposed that Gauss’s brain was more highly convoluted in the cortex than a normal brain.

His work was complemented by a set of copper engravings and lithograms that one of Wagner’s team made. More recently, with the aid of modern high-resolution fMRI a team in Göttingen has indeed confirmed a rather rare connectivity between two regions of the left hemisphere of Gauss’s brain. However the team had to contend with a strange mix-up that happened in the collection. It turns out that for years what had been considered Gauss’s brain was in fact that of another of Göttingen’s elite, Conrad Heinrich Fuchs, who died the same year as Gauss. It seems that the specimens had been mixed up after Wagner made his analysis and the diagrams were drawn. It was only when the team came to compare the fMRI scans with the original drawings that the mix-up was identified.

9.6 Gauss’s brain.

Göttingen’s nineteenth-century project to understand the different architecture of the brains of elite thinkers continues to this day. More recently, at the University of Louisville in Kentucky, the anatomy department has been studying the brains of deceased scientists, or ‘supernormals’ as they are called by the lab. Professor Manuel Casanova, who led the investigation, detected structural differences in the brains of scientific specialists.

It seems that an abundance of short, local connections give rise to brains that specialise in focused modes of thinking. These are individuals who are tapping into the power of single regions in the brain. In contrast, brains with long connections that link diverse regions of the brain help to create new ideas and out-of-box thinking.

It’s interesting that this seems to correspond with a dichotomy that has emerged between styles of thinking. ‘The fox knows many things but the hedgehog knows one big thing,’ wrote the ancient Greek poet Archilochus. It was the springboard for the foxy philosopher Isaiah Berlin’s essay that seeks to divide thinkers into two categories. Foxes tap into a broad range of interests, a horizontal thought process. The hedgehogs think deeply, a thought process that runs vertically, perpendicular to the foxes’. Foxes are interested in everything. The hedgehog is single-minded in its obsessions.

If the abundance of short-range connections characterises the hedgehog and long-range connections the fox, wouldn’t a brain that could combine lots of these short connections with long links give rise to someone who could combine the skills of the fox and the hedgehog? That would be ideal but the fact is that the wiring within the brain requires both space and metabolic activity. And as long as you have the constraint of the geometry of the skull, it is impossible to fuse the two.

But there is an alternative. Collaboration. Gauss collaborated with Weber to produce the first telegraph wire that would give rise to the modern internet. It is by sharing our specialisms and creating these long-range connections between brains that can specialise that we have the potential to get something new and exciting to emerge. The low-hanging fruit can be found in the hinterland between subjects. By learning the language spoken by someone outside your discipline and applying it to the problems in your own field, easy gains can be made. This is why, whatever your field of work, learning the ideas from another discipline might enable both collectively to find a shortcut to the other side.

Perhaps the perfect fusion of fox and hedgehog is the collaboration of human and machine. Although my book is meant as a celebration of the distinctly human trait to sniff out the shortcut, maybe I shouldn’t be so dismissive of what the machine can offer. While the machine can use brute force to calculate faster and further, ultimately it is in combination with human cunning to find clever shortcuts that together both can reach destinations that are beyond the grasp of individual human or machine.

Solution to the puzzle

The puzzle at the start of the chapter is the challenge that I was offered in my psychometric job application. Thanks to Euler’s shortcut, I know that it is impossible to draw because there are more than two nodes that have an odd number of lines emerging from them. However, if you use a trick, there is a way to draw the shape. Take a piece of paper and fold up the bottom quarter. Draw a square, starting from the top-left corner and making sure that the bottom line of the square goes over the paper you folded up, keeping your pen on the paper as you complete the square. Fold the page back down to leave three lines of the square with your pen on the top-left corner. If you now analyse the diagram that is left to draw it passes Euler’s test.

9.7 The trick to drawing the figure: fold the paper.

Shortcut to the shortcut

Networks are everywhere. The structure of a company. The electrical wiring of a computer. The interdependence of different stock options. Transport networks. The cellular interactions in our bodies. The relationship between characters in a novel. Our social networks. Whenever you’ve got a set of objects and connections between them, you’ve got a network. It is worth analysing whatever structure you are trying to understand to see if there is a network hidden inside. Because once you have identified a network, mathematics is ready with shortcuts to help you navigate its architecture. Tools to identify the most important nodes in the network. Strategies to transform networks into small worlds with fast paths from one end to the other. Topological maps that throw away extraneous information and help you to see what’s really going on.

Pitstop: Neuroscience

Often the best ideas seem to come from nowhere. It’s as if not thinking helps the brain to find shortcuts to answers. The philosopher Michael Polanyi believed that this tacit thought process in which the brain taps into subconscious unarticulated arguments was key to the power of human thought. He summed up his thesis with the phrase: ‘We can know more than we can tell.’

This is certainly my experience of creating mathematics. That sense of ‘seeing’ the answer even though I’m not quite sure why I feel it’s right. It’s how I manage to come up with conjectures about what I think the lie of the mathematical land is like. I sense the existence of a far distant peak without quite knowing how to navigate a path to get there.

Many mathematicians have talked about flashes of insight, the way the brain seems to launch an idea into our consciousness. It works away subconsciously and once it hits a solution it knows to throw that up into the conscious arena. I have had these flashes of insight, which are then followed by the frequently painful task of piecing together the logic of how my subconscious came to this conclusion.

The mathematician Henri Poincaré described a famous occasion when he’d been working away on a problem unable to make any headway. It was only when he was away from his desk, letting his mind idle, that, as he stepped onto a bus in Paris, he had a sudden realisation of how to solve the challenge: ‘At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations that I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry.’

Alan Turing had a similar experience when he was working on his idea of Turing machines. After working hard in his rooms, Turing used to enjoy running along the banks of the River Cam in Cambridge as a way of relaxing. It was while lying on his back in the meadow near Grantchester that he understood how the mathematics of irrational numbers could be used to show why his Turing machines had limitations in what they could compute.

To learn more about solving a problem by not thinking about it, I decided to contact a neuroscientist, Ognjen Amidzic, who has been exploring brain function in people engaged in their particular fields of expertise.

Amidzic hadn’t intended to become a neuroscientist. His dream had been to become a chess grandmaster. He spent thousands of hours practising, and even moved from his home in the former Yugoslavia to Russia, so he could train with the world’s best teachers. But he eventually plateaued. He couldn’t increase his rank beyond the level of expert.

Amidzic decided to investigate whether there was something about the way his brain was wired that was handicapping him. So he trained instead to become a neuroscientist and started to research whether he could identify a different brain activity between amateurs and grandmasters.

To demonstrate his findings he got me to play a game of chess against one of Britain’s grandmasters, Stuart Conquest, and had us both wired up via a magnetoencephalogram to demonstrate the difference in brain activity. I certainly don’t rank anywhere near grandmaster or expert but I can think logically and analyse a chess position to see what the best next move might be.

I lost the game quite quickly. But that wasn’t the result that interested me. It was the results of the magnetoencephalogram that were striking. It turned out that we were using very different bits of our brain to play. I seemed to be using up more brain activity but achieving less success.

Amidzic’s research has revealed that an amateur chess player like myself is using the medial temporal lobe, which is at the centre of the brain. This is consistent with the interpretation that the amateur’s mental acuity is focused on analysing unusual new moves during the game. This could be equated to a spoken, conscious analysis of the consequences of each potential move and could probably be expressed out loud by the amateur player to provide a commentary of their thought process.

In contrast the grandmaster was using the frontal and parietal cortices and bypassing completely the medial temporal lobe. This is an area of the brain that is more often associated with intuition. It is where we access long-term memory and is involved with a more subconscious thought process. A grandmaster might sense that a move is a good one, even if they can’t articulate why. The brain isn’t working hard to produce a logical rationale for the feeling, like the amateur would, and so isn’t wasting energy in the medial temporal lobe. The grandmaster was shortcutting conscious thought to arrive at a solution.

It was as if my brain was running round like a mad gazelle while the grandmaster’s sat there like a lion hiding in the grass wasting no excess energy before making the killing move.

Controversially, Amidzic believes that people’s brain activity doesn’t change much with practice. He thinks that by scanning an amateur chess player you can tell already whether they have the brain to become a grandmaster because even at the outset of their career they are already accessing the frontal and parietal cortices as they play: ‘Everyone wants to think that you can achieve, that you can be what you want to be and if they are not able to achieve it in life then you have someone responsible for this, their mother or the government or father’s support … lack of money or whatsoever so they have some explanations.’

But Amidzic believes that it’s not down to hours or to access to great teaching and education but fundamentally to genetics. ‘You are born a grandmaster or you are born an average chess player or you are born a great mathematician or musician or soccer player, whatever,’ he says. ‘People are born, not created. I just don’t believe and I don’t see any evidence whatsoever that you can make, create a genius.’

Amidzic recalls scanning a child whose father was desperate for him to become a grandmaster. Amidzic could tell that the brain was stuck analysing things in the medial temporal lobe. He believed that this boy was never going to push his rating beyond expert and advised the father to think about another pursuit. Apparently the father ignored the advice and Amidzic was subsequently proved right in his assessment.

For Amidzic, the key appears to be finding that activity for which the brain seems to have a good intuition. In his own case he believes that in the end it was neuroscience not chess that he was predisposed to excel in: ‘Life is funny. I’m more famous for this than I would have been as a chess player.’

After analysing my brain activity while I was playing chess it transpired that I too would probably never make the cut as a grandmaster. My brain wasn’t finding the shortcuts to good moves but was getting bogged down, taking the long road through my medial temporal lobe. In contrast Amidzic suggested that if we scanned my brain while I was doing maths, then indeed it would be this intuitive part of my brain that I would be accessing.

It’s not clear from his research whether this is truly all down to genetics or whether you can train a brain. But his research does seem to have identified that when it is performing at its peak, the brain takes advantage of shortcuts to avoid too much thought cluttering the journey to a solution.

10

The Impossible Shortcut

At Glastonbury Festival I often perform a gig in the Astrolabe theatre. Afterwards I try to visit all the other stages. Can you find me the shortest path that starts and finishes at the Astrolabe and visits all the other stages on the map once and once only?

10.1 Map of Glastonbury Festival.

Not every problem has a shortcut. We have seen how any challenge that demands a physical change to the body, like learning a musical instrument, rewiring the brain through therapy, training to be an athlete, necessitates time and effort to achieve. But it turns out that there might be another range of challenges that have no shortcuts. Mathematicians now believe that there is a whole slew of problems that cannot be solved without doing the hard slog of checking all the possible solutions.

Are you a teacher trying to plan next year’s timetable? A haulier planning the best route for your fleet of lorries to deliver goods? A supermarket stacker trying to find an efficient way to pack boxes on the shelves? A football supporter keen to know if your team can still come top of the league? A sudoku fan in search of a good strategy to solve those fiendish puzzles? All after shortcuts. But alas these might be challenges where better thinking won’t help to find a solution. Even Gauss would have to stick in the hard work of checking all the possible scenarios to find the solution. Perhaps the most striking thing is that mathematics, the art of the shortcut, is out to prove that for certain problems there are no shortcuts.

The classic problem that mathematicians believe has no shortcut to solve it is called the Travelling Salesman Problem. This is the challenge of finding the shortest path round a network of cities. The name seems to have its origins in a handbook for travelling salesmen published in 1832 in which the problem is formulated, together with some example tours through Germany and Switzerland. To date, mathematicians have come up with nothing cleverer than trying all the different possible paths to guarantee finding the shortest.

The trouble is that as I add more cities the number of possible routes escalates, and testing every possible route becomes totally impracticable even when implemented on a computer. Surely there must be a quicker way to pick out the solution? Couldn’t an Euler or a Gauss or a Newton find some clever strategy to sniff out the shortest route? What about, for example, just always choosing the nearest city to the one that you are currently visiting? This is called the nearest neighbour algorithm. Often this can produce a pretty good route that is only 25 per cent longer than the optimal path. But it is quite easy to rig up networks in which the algorithm ends up producing the longest path through the cities instead of the shortest.

Some algorithms have been developed that do guarantee a route that is always at most 50 per cent longer than the optimal path whatever network the algorithm is given. But what I am after is that ingenious shortcut that will sniff out the best route without an exhaustive search, and this problem has so vexed mathematicians that many have begun to suspect that no such shortcut exists. Indeed this very challenge of proving that there is no such shortcut is one of the seven Millennium Prize Problems that were chosen at the beginning of the twenty-first century as the greatest unsolved mathematical problems. The mathematician who can prove there is no shortcut to solving the Travelling Salesman Problem will earn themselves a $1 million prize as a reward.

What is a shortcut?

To win the $1 million prize it is important to actually define what would constitute a shortcut mathematically in this context. The difference between the long way and the shortcut mathematically translates into an algorithm that takes an exponential amount of time to reach a solution versus an algorithm that needs only polynomial time. What exactly do I mean by this?

The task at the heart of this challenge is to come up with a method that will work not just on one puzzle but an algorithm that can work whatever version or size of the puzzle it encounters. The question is how long the algorithm takes depending on the size of the puzzle that I give it. For instance, suppose I have a set of 9 tiles with different patterns on each. I want to arrange the 9 tiles in a 3 × 3 grid such that the patterns match those on adjacent squares.

10.2 Nine tiles arranged so patterns along the sides match the patterns on adjacent squares.

How many different ways are there to arrange the tiles? I’ve got 9 choices for the tile I put in the top-left corner of the grid. This tile can be arranged in 4 different orientations. That’s a total of 9 × 4 = 36 different choices. The next position along has a choice of 8 remaining tiles, each of which has 4 different orientations again. As I go through the grid I find that the total number of ways of arranging all these tiles is:

9! × 4^9

where 9! is shorthand for 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, called 9 factorial. If a computer can perform 100 million checks in a second this would take just over 15 minutes. Not too bad. But see how quickly the time escalates if I increase the number of tiles. What if I consider 16 tiles in a 4 × 4 grid. Using the same analysis, the number of combinations to check is:

16! × 4^16

This ramps up the amount of time needed to check them all to 28.5 million years. Move to a simple 5 × 5 grid and I’m well outside the lifetime of the universe, a mere 13.8 billion years.

Given a grid with n locations, the number of possible arrangements is given by n! × 4n. The number 4n is what is known as a function that grows exponentially with n. I explained the dangerous way that this function escalates in Chapter 1 when the king of India had to pay for the game of chess with grains of rice that grew exponentially across the board. The factorial n! (the product of all the numbers from 1 to n) is in fact a function that grows even faster than exponentially.

This is the mathematical definition of the long way round: an algorithm to solve a problem where the time it takes to calculate the solution ramps up in this exponential manner as the problem increases in size. This is going to be the sort of problem for which I would like to find a shortcut. But what will qualify as a good shortcut? This is the discovery of an algorithm that is still relatively quick to find a solution even as I increase the size of the problem: a so-called polynomial time algorithm.

Let’s say I have a random selection of words and I want to put them in alphabetical order. How long would this take, as the list of words gets longer and longer? A simple algorithm to do it could look through the original list of N words and pull out the word that is before all others in the dictionary. Once I have done that I just do the same again to the N − 1 words left. In this way I need to scan N + (N − 1) + (N − 2) + … + 1 words to sort them. But thanks to Gauss’s classroom shortcut, I know that this takes N × (N + 1)/2 = (N^2 + N)/2 scans in total.

This is an example of a polynomial time algorithm because as N, the number of words, increases, the number of scans needed just goes up as a quadratic equation in N (squaring N). In the case of the Travelling Salesman Problem I need an algorithm that, given N cities to make my way round, can find the shortest path by checking, say, only N^2, or a quadratic number of routes.

The algorithm that one first cooks up is unfortunately not polynomial. Essentially first choose one city to visit, then choose the next city … Given a map with N cities, this will mean checking N! routes. As I mentioned above, that is worse than exponential. The challenge is finding a better strategy than testing all routes.

A shortcut to a shortcut

To show that it’s not impossible that such an algorithm might exist, consider a problem that at first sight seems equally intractable. Choose two of the locations on the Travelling Salesman’s map of cities to visit. What is the shortest path between these two cities? At first sight it looks like there are still a lot of different options to consider. After all I could start by visiting any of the cities that are joined to our starting city, and then one of the cities joined to that city. I seem to be cranking up to something that is going to be exponential again in the number of cities.

But in 1956 Dutch computer scientist Edsger W. Dijkstra came up with a much cleverer strategy that would find the shortest path between the two cities in the same kind of time that it took to sort the words into alphabetical order. He’d been contemplating the practical problem of finding the fastest route between the two Dutch cities of Rotterdam and Groningen.

One morning I was shopping in Amsterdam with my young fiancée, and tired, we sat down on the café terrace to drink a cup of coffee and I was just thinking about whether I could do this, and I then designed the algorithm for the shortest path. It was a twenty-minute invention … One of the reasons that it is so nice was that I designed it without pencil and paper. I learned later that one of the advantages of designing without pencil and paper is that you are almost forced to avoid all avoidable complexities. Eventually that algorithm became, to my great amazement, one of the cornerstones of my fame.

Consider the following map:

10.3 What’s the shortest path between city 1 and city 5?

In Dijkstra’s algorithm I will set out from the start city, City 1. At each stage of my journey I am going to attach a running total to the cities on the way to help me to find the shortest route. First I label all the cities that are connected to the starting city with the distances to get to them. In this case City 2, City 3 and City 6 get labelled 7, 9 and 14 respectively. My first move will then be to move to the nearest of these cities. But be warned. Once the algorithm has worked its magic to solve the problem, this might not end up being the best city to move to first.

In this picture I begin by moving to City 2 since it is the shortest distance away from my starting point, City 1.

I then mark City 1 that I’ve just left as ‘visited’. From my new city, City 2, I am going to consider updating the labels on all the cities connected to it. So in this case I’ll possibly update the labels on City 3 and City 4. I start by calculating their distances from the start city, City 1, that go via my current city, City 2. If this new distance is shorter than the current label on that city then I update the label with this new distance. If the distance is longer then I keep the previous label. In the case of City 3, the new distance (7 + 10) is longer so I keep the original label, 9. Sometimes the city won’t yet have a label, like City 4, because it wasn’t connected to the previous cities. I label this new city I’ve reached with the distance I’ve just calculated to get to it. So in this case City 4 is labelled with 7 + 15 = 22.

I then again mark the city I am on as visited and move to the unvisited city that has the smallest current distance from my starting position. In the above, I now move to City 3. Here is an example where although it looked clever to set off to City 2, distances from here are large. So the algorithm is already guiding me to probably go via City 3.

Once again I update the labels on any cities unvisited that are connected to City 3. By continuing this process I will eventually get to the destination City 5 and it will have a label on it that represents the shortest distance from the city I started from. I can then trace back through the journeys to see which cities I went through to get there with this distance. In my example, notice that it wasn’t via City 2 in the end.

How long does this take me in term of steps I need to implement to find the shortest route? With N cities it is rather similar to putting words in alphabetical order. Each step removes a city that no longer needs to be considered. So the time it takes the algorithm to complete is N^2 or quadratic in N. A shortcut in our mathematical language!

But a shortcut in this mathematical language could still take a very long time to come up with the answer in reality. Mathematicians generally regard polynomial time algorithms as the ones we are after as the shortcuts. Quadratic algorithms are pretty fast. But even though cubic, quartic and quintic polynomials are regarded as mathematically fast, they can nonetheless still physically take a long time.

If a computer can do 100 million actions per second then for small N I won’t have too much problem. But there is a massive difference in time between an algorithm that finds the answer in N^2 steps and the algorithm that takes N^5 steps.

In one second the N^2 algorithm can check a network with 10,000 cities. The N^5 algorithm would need 31,710 years to check the same number of cities! Yet that’s still regarded as a mathematical shortcut. It’s undoubtedly a shortcut compared to the exponential algorithm that we currently have, which would require way beyond the lifetime of the universe to check a network with 10,000 cities. Indeed even 100 cities is beyond the lifetime of the universe for a 2N exponential algorithm.

For practical purposes it is still worth battling to find algorithms that require the fewest powers of N to run through. Some shortcuts are shorter than others.

Needles in haystacks

You might hope that if you aren’t a travelling salesman you won’t be affected by the lack of a shortcut to find the shortest path to get you round your customers. The trouble is that there are many problems out there that share the same complexity issues. In engineering, for example, you might want to wire a circuit board that has 100 different locations and you need to find the most efficient way to get your robot to lay down the wires to make the circuit. Given that the robot is going to be knocking out thousands of these things each day, just a few seconds’ decrease in the robot’s path round the network could save the company a huge sum of money. But it isn’t just travelling round a network that we’d love to find shortcuts for. Here is a selection of challenges that have the same quality as the Travelling Salesman Problem, problems that we believe may have no shortcuts to find a solution. It may simply be impossible to avoid the long slog, even for the great Gauss!

The car boot challenge. You have boxes of different dimensions that you want to transport in the boot of your car. The challenge is finding the selection of boxes that wastes the least amount of space. It turns out that there is no clever algorithm that by looking at the dimensions of the boxes will pick out the best combination. Suppose all the boxes are of the same height and width, exactly the same as the internal dimensions of your boot, but they are of different lengths. Your boot is 150 cm long, and the boxes available for packing have the following lengths: 16, 27, 37, 42, 52, 59, 65 and 95 cm. Is there a clever way to pick the combination of boxes that will fill the boot as efficiently as possible?

The school timetable challenge. Every school faces the challenge at the beginning of the school year of preparing a timetable for the students. But the choices that students make place restrictions on what can be timetabled when. Ada has chosen to do chemistry and music so these classes can’t be scheduled at the same time. While Alan has chosen chemistry and film studies. But there are only 8 periods in the day. The school has got to find some way of fitting them all in without an overlap. Given these sorts of constraints, putting together a timetable can sometimes feel like laying a carpet that doesn’t quite fit the room. As soon as you seem to have got it fitting in one corner, you find the carpet pops up in the other corner. Or it’s like doing a sudoku where you think you’ve got the solution only to find two 2s in the same row. Agghhh!

Sudoku. If you have tried solving some of the fiendish versions of this Japanese puzzle, you will often hit points where you just seem to have to make a guess at the next number, followed by working out the logical implications of this guess. If it turns out to be a wrong guess and you get a contradiction, you are forced to backtrack to the moment you made your guess and to try out a different pathway.

The dinner party challenge. A similar sort of challenge to school timetabling arises if you have friends you’d like to invite to dinner but there are certain of them that just don’t get on so you can’t invite them to the same dinner. Trying to find the minimal number of dinner parties you’ll need to ensure everyone comes to dinner but no one comes to a dinner with someone they don’t like requires checking every possible combination of guests.

Map colouring. If you take any map and try to colour countries so that no two with a common border have the same colour, it is always possible to achieve this with four colours. But can you get away with just three colours? Again the only algorithm that we have to determine if three colours might suffice is by attempting to go through all the different ways to colour maps. Just like the sudoku, you can start colouring and then a choice made early on forces two countries to have the same colour. With N countries there are 3N different ways to colour the countries with 3 colours, which means potentially checking an exponential number of different possibilities.

The fact that you need at most only four colours was one of the great theorems to be proved in the twentieth century. It had been shown in 1890 that five colours would do. The proof was not too complex, relying on a shortcut that mathematicians often use. Suppose there are maps that can’t be coloured with five colours. Take an example of them with the smallest number of countries. Then by some clever analysis you can show how to remove one country and still the resulting map can’t be completed with five colours. But this contradicts the fact that the map you started with was meant to be one with the smallest number of countries.

Here is a silly use of the shortcut of taking the smallest example of something to prove it can’t exist: a proof that there are no boring numbers. Suppose there are boring numbers. Let N be the smallest boring number. Suddenly it’s become interesting because it’s the smallest boring number.

The frustrating thing was that this clever shortcut didn’t seem to work if one tried to show that four colours suffice. Mathematicians couldn’t show why removing a country would still give you a map that couldn’t be coloured. And yet no one could come up with a counter example.

Eventually in 1976 a proof was found that four colours do suffice. But it certainly wasn’t regarded as a shortcut. Indeed the proof required the brute force of a computer to check thousands of cases that it would have been impossible for a human to sort through. This proof was a turning point in mathematics: it was the first time that a computer had been used to force a path through to the solution at the end of the proof. It was like we’d hit a mountain range and found no clever path through to the valley on the other side. So we used a machine to bore a hole through the mountain.

Many in the mathematical community felt a certain unease at the way a computer had been used to prove this theorem. Proofs were meant to give humans an understanding of why four colours sufficed, not simply whether it was true. The human brain is limited in the connections it can make, which is why it is absolutely essential for the brain to feel like it gets why the shortcut is the way it is. If the proof has been forced to take the long way round, it’s as if the proof can’t be loaded into the brain and we have the feeling that we have been denied true understanding.

A related problem to the map-colouring challenge is to take a network consisting of points joined together with a selection of lines. The lines are like the borders between the countries. The challenge is to know what is the fewest number of colours needed to colour the points so that no two points have the same colour if they are connected by a line.

Football. I think one of my favourite examples of a problem that we can’t find a shortcut to solve relates to football. Not the playing of it but that wonderful challenge that starts to rear its head towards the end of a season: is it still mathematically possible for my team to win the Premier League given where they currently stand in the table? You might think this is a simple task. I just make sure that my team wins all their games, getting three points for each win and then check if that is enough to come out top? However, it’s all the games between the other teams that I need to worry about. Clearly I want to make the team that is currently top lose lots of games. But that will have the effect of the teams they are playing winning and getting more points. What if I end up assigning them too many points and they come out on top?

This is another problem in which I’ve got to consider so many different combinations of matches and outcomes. As I assign wins, losses and draws, time and again I find myself backtracking like in a sudoku because one of the outcomes I’ve assigned has messed up my careful balancing act.

If there are N games left to play then each game could be a win, lose or draw for the home team making a total of 3N different outcomes, an exponential number of possibilities to consider. The challenge is to find a shortcut that will tell me quickly whether my team still has a mathematical chance of winning the league.

But I like this problem so much because when I was at school there was such an algorithm. What’s happened in the interim? It’s not that we lost the algorithm but that the way points are assigned has changed. In the past a team got only two points for a win and shared a point each if they drew. This was felt to cause teams to go for boring draws. So in 1981 a decision was made to try to incentivise teams to push for the win. Instead of two points, teams could earn three points for a win. It seemed an innocuous change but it had a dramatic effect on the challenge of working out if a team could still top the Premier League table.

Crucially, pre-1981 the total number of points shared out across the teams didn’t depend on who won, lost or drew. There are 20 teams that play each other twice, home and away, which means 20 × 19 matches. Each match in the old system had two points that were distributed according to the result, which meant that the total number of points at the end of the season was 2 × 20 × 19 = 760 points, shared between the 20 teams.

But now things are very different. Each match could result in three points being awarded to the winner or just one point each if the match is a draw. If all matches across the season were drawn, that would mean a total of 760 points again. But if there are no draws the total is 3 × 20 × 19 = 1140 points. This new variance in the total number of points meant that the previously successful algorithm that could tell me if my team still had a mathematical chance of winning the league no longer worked.

The fascinating thing with all these problems is that if you happen to stumble on a solution, it is quick to check if it really solves the challenge. I like to call these ‘needle in a haystack problems’: the initial challenge of finding the needle requires a long exhaustive search with little to help you home in on where the needle is, but as soon as your hand lands on the needle you know it! Cracking a safe can take a long time, trying one combination after another, but once you put in the right combination the door opens immediately.

There is a rather extraordinary feature of these needle in a haystack problems or, to give them their technical name, NP-complete problems. You might expect that each of them is going to require its own bespoke strategy to try to find an algorithm that will get to the solution in as short a time as possible. But it turns out that if you ever did discover a fast polynomial time algorithm that finds the shortest path in any map that the Travelling Salesman might be faced with, it means there must be such algorithms for every other such problem. This at least shortcuts the challenge of finding shortcuts. If there is a shortcut to one problem, it can be converted into a shortcut for any other challenge on our list. To misquote Tolkien: one shortcut to solve them all.

I can give a hint of why this might be true by seeing how it is possible to translate some of the problems I have described into each other. Take, for example, the school timetabling problem. I had classes and time slots and clashes between classes that I needed to avoid. Using this information, I can build a network where each class is a point in the network and the clashes between classes will correspond to lines that I draw between two classes that clash. Then assigning time slots becomes exactly the same challenge as colouring the points in the graph such that no two points with a line joining them have the same colour.

Exploiting the lack of shortcuts

There are some settings in which it is quite important that there is no shortcut. This is the case when it comes to making unbreakable codes. Code-makers want to exploit the fact that there doesn’t seem to be any way to crack a coded message beyond an exhaustive search of possibilities. Take a combination lock. One with 4 dials with ten numbers on each dial is going to require you to check 10,000 different numbers, from 0000 to 9999. Sometimes a badly manufactured lock may give away the positions where the lock will release thanks to a physical shift in the device when the first dial is set in place, but in general there is nothing a thief can do to shortcut trying all the combinations.

But other cryptosystems have revealed weaknesses that can be exploited to create shortcuts. Take the classic Caesar Cipher or substitution cipher. This is a code that systematically exchanges letters for alternative letters. For example, every instance of A gets replaced by a choice of another letter, G say. Then B is replaced by one of the remaining letters. In this way each letter of the alphabet gets assigned a new letter. There are a lot of different codes one can choose. There are 26! (1 × 2 × 3 … × 26) different ways to rearrange the letters of the alphabet. (In some of the rearrangements a letter might remain the same, X encoded by X, for example. An interesting challenge: how many codes are there where all the letters get changed?) To give a sense of the size of this number: 26! seconds translates into more time than there’s been since the Big Bang went off.

If a hacker intercepts a coded message, that’s a lot of different combinations that they will need to try to decode the message. But there is a weakness to this code that the ninth-century polymath Ya’qub al-Kindi spotted: some letters appear more frequently than others. For instance, in the English language ‘e’ is the most popular letter to appear in any text, occurring 13 per cent of the time, followed by the letter ‘t’ which crops up 9 per cent of the time. Letters also have their own personalities, which get reflected in other letters they like to be associated with. ‘Q’ is invariably followed by ‘u’, for example.

Al-Kindi realised that a hacker could exploit this as a shortcut to attacking a message that has been encoded with a substitution cipher. By doing a frequency analysis on the encoded text and matching the most frequent letters to those that occur more often in plain text, the hacker starts to make inroads into the message. It turns out that using a frequency analysis is an amazing shortcut to cracking these codes, which are far less secure than they first appear.

During the Second World War the Germans thought they’d found a cunning way to use a substitution cipher that could avoid this shortcut to cracking messages. Their idea was to use a different substitution cipher to encode each new letter in the message. This meant that if they were encoding EEEE then each E would be sent to a different letter, which would scupper any attack using al-Kindi’s frequency analysis. They built a machine to do this multiple substitution cipher encoding: it was called the Enigma machine.

You can still see one of these machines on display at Bletchley Park, the home of the UK’s codebreakers during the war. At first sight the machine looks like a conventional typewriter with a keyboard, but there is a second array of letters running above the keyboard. When I pressed one of the keys, one of the letters above the keyboard lit up. This would be how I would encode this letter. Essentially the wires in the machine are scrambling the letters in a classic substitution cipher. But at the same time as I pressed the key, I could also hear a clicking and saw one of three rotors sitting at the heart of the machine click on one step. When I pressed the same letter again, a different bulb lit up. This is because the wiring from the keyboard to the lights had been rearranged. The wires are connected via the rotors and when the rotors change alignment so does the wiring in the machine. In this way the clicking on of the rotors ensures that the machine uses a different substitution cipher for each letter it is encoding.

The whole thing looks uncrackable. There are 6 different rotors that I could use to set up the machine and each one could be started in 26 different settings. Plus there was a whole set of wires at the back that could add another fixed level of scrambling. It meant that there were 158 million million million different ways that the machine could be set up. Trying to find out which one an operator had used to encode a message looked like the ultimate task of searching for a needle in a haystack. The Germans were utterly confident that the machine was uncrackable.

But they hadn’t reckoned on the clever skills of mathematician Alan Turing, a twentieth-century Gauss, who sitting in Bletchley Park sniffed out a weakness in the system that could be exploited to shortcut an exhaustive search. The key was that the machine never encoded a letter by the same letter. The wiring would always send the letter to a different letter. It looks like an innocent fact about the machine. But Turing saw how this could be used to chase around the machine and squeeze out a much more limited set of possibilities for how a particular message had been encoded.

He still needed to employ a machine to do the final search. The huts at Bletchley Park would hum all night to the sound of the Bombes, the name the team gave to the machines that implemented Turing’s shortcut. But each night they would give the Allies access to the messages that the Germans thought they were sending securely.

Prime suspects

Today the codes that protect our credit cards as they fly around the internet are exploiting mathematical problems that we believe by their nature have no shortcuts. One of these ciphers, called RSA, relies on those enigmatic numbers: the primes. Each website secretly chooses two prime numbers of about 100 digits each in length which they multiply together. The result is a number with roughly 200 digits which is then made publicly available on the website. This is the website’s code number. When I visit a website, my computer receives this 200-digit number, which is then used to do a mathematical calculation involving my credit card. This scrambled number is sent across the internet. It is secure because to undo the calculation a hacker has to solve the following challenge: can they find the two prime numbers that when multiplied together give the website’s 200-digit code number? The reason this cryptography is regarded as secure is that it seems to be a needle in a haystack problem. The only way mathematicians know to find these primes is to try one after another, hoping that we suddenly hit on the needle where the number exactly divides the website’s code number.

Gauss himself wrote about the challenge of cracking numbers into primes in his great treatise on number theory Disquisitiones Arithmeticae: ‘The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length … Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.’

Little did he realise how important this problem would become in the age of the internet and e-commerce. No one to date has found a shortcut for finding the primes that divide large numbers, not even the great Gauss himself. The number of primes that one needs to check to crack a 200-digit number is so vast that such an attack would be totally ineffective. We think that the challenge of factorising – expressing a number as the product of smaller numbers – might be intrinsically difficult. That is one of the open problems that mathematicians are currently working on. Can we prove that there is no shortcut to finding the primes?

But hold on. How does the website decode the messages? The point is that it started by choosing the two primes of roughly 100 digits each, which it then multiplied together to generate the 200-digit public code number. The website is the only one in possession of the primes that will undo the calculation.

Yet finding primes is one of those problems that mathematicians haven’t solved yet. Cracking the secret to how the primes are arranged through the universe of numbers, called the Riemann hypothesis, is another of the seven Millennium Problems. But even though mathematicians don’t really understand exactly how primes are distributed, we do have an interesting shortcut to finding big primes for these internet codes. It depends on a discovery that the great seventeenth-century French mathematician Pierre de Fermat made about prime numbers. He proved that if p is a prime number and I take any number n less than p, then raising n to the power of p will leave a remainder of n when I divide by p. For example, 2^5 = 32 which has a remainder of 2 after I have divided it by 5.

This means that if I want to test whether a candidate number q, say, is prime, then if I can find a number less than q that fails this test I know q isn’t prime. For example, 2^6 = 64 which has a remainder of 4 not 2 on division by 6. This means 6 can’t be prime because it fails Fermat’s test. This would not be a very useful test if, say, there was only one number less than q that failed the test. It would mean testing potentially all of the numbers less than q, in which case I might just as well check indivisibility directly. The great advantage of the test is that if a potential prime number fails, it fails spectacularly. Using Fermat’s ruse, over half the numbers less than q will bear witness to q’s failure to be prime.

There is a fly in the ointment, however. There are some numbers that behave like primes, without any Fermat witnesses grassing them out, but that nonetheless are not prime. They are called pseudo-primes. Yet in the late 1980s two mathematicians, Gary Miller and Michael Rabin, were able to refine Fermat’s approach to produce a guaranteed test for primality that would run in polynomial time. The only caveat was that first the two mathematicians had to assume that they could climb to the top of a very tall mountain: the Riemann hypothesis (or a generalisation of the conjecture).

Miller and Rabin could prove that provided mathematicians found a way to clear this summit, they could guarantee a shortcut on the other side to finding primes. This summit is so important partly because so many mathematicians have shown that it would give access to a whole slew of shortcuts. I myself have several theorems that show why something is true provided I can first prove the Riemann hypothesis is true.

But one should never give up on the chance that there might be a sneakier way round the mountain. In 2002 the mathematical community was rocked by the exciting news that three Indian mathematicians, Manindra Agrawal, Neeraj Kayal and Nitin Saxena, based at the Indian Institute of Technology in Kanpur, had come up with a way to test whether or not a number was prime in polynomial time without having to go over Mount Riemann. Remarkably the second two authors of the discovery were undergraduate students working with Agrawal. Even Agrawal, the senior member of the team, was unknown to most number theorists in the mathematical community. It reminded many of the story of the great Ramanujan who had exploded onto the mathematical scene in the early twentieth century after writing to Cambridge mathematician G. H. Hardy about his mathematical discoveries.

Although the team’s breakthrough established a test for primality that worked in polynomial time without assuming that you could cross Mount Riemann, it was not an algorithm that was terribly practical in real terms. As I mentioned earlier, it’s important to know the degree of the polynomial that you are using. If it’s quadratic, things will run fast. Yet the original algorithm proposed by Agrawal, Kayal and Saxena had a polynomial with a degree of 12. This was cut down to 6, by American mathematician Carl Pomerance and Dutch mathematician Hendrik Lenstra, but as I’ve explained, although mathematically a shortcut, practically this slows down quite quickly. As the numbers you are testing get bigger the algorithm involving a polynomial of degree 6 takes substantially longer to come up with an answer.

Given that internet security depends on a healthy supply of big primes, how do websites find them quickly enough to run financial services efficiently? The answer is to use an algorithm that at least gives the website a high degree of confidence that the number is prime, while not actually guaranteeing that it is.

Remember that if a number is not a prime or a pseudo-prime, then half the numbers less than it will fail the Fermat test. But what if I was really unlucky and tested the half that passed the test? To guarantee finding a witness to a number’s non-primality seems to need testing half the numbers. But what are the chances of missing a witness? Say I did 100 tests and I didn’t get a witness. Either that means the number is a prime or a pseudo-prime or there is a one in 2^100 chance that I missed all the witnesses. That is a bet I’d be prepared to take! Those are very long odds.

Although we have great algorithms, both deterministic and probabilistic, for finding primes to make these codes, it seems that conventional algorithms don’t exist for breaking codes. What about something a little more unconventional?

Quantum shortcuts

One of the problems that conventional computers face when trying to solve something like finding the primes that divide a large crypto number is that they are required to carry out one trial before they can move on to the next test. (For clarity, in the following I will always mean we are trying to find exact division with no remainder.) What I would really like is to divide the computer up into bits and to get each bit to do a different test. Parallel processing is a very effective way to speed up actions. Take the case of building a house. In a competition held in Los Angeles to see which team of builders could erect a house the fastest, it was the one that had 200 builders working in parallel who won the race, building it in four hours. There are obviously some tasks that rely on things being done in sequence. Building a tower block or digging an underground car park relies on each floor being built before the next is added. But testing smaller numbers to see if they divide a larger number is perfect for doing in parallel. Each task does not rely on the outcome of any other check.

The trouble with parallel processing is that I still have a physical capacity problem. By dividing the problem in half, I have cut down the time to implement the check by two but I have doubled the amount of hardware that I need. This approach doesn’t really address the issue of finding the primes that divide a large number.

But what if I could do this parallel processing without the need for double the hardware? This was the idea of Peter Shor, a mathematician working at Bell Labs in the 1990s, who realised that one could exploit some rather unconventional computing to test things simultaneously. The idea was to use the bizarre physics of the quantum world. In quantum physics, it is possible to set up a particle like an electron so that before it is observed it essentially is located in two positions simultaneously. Let’s call these two positions 0 and 1. This is called being in quantum superposition. The advantage of this is that the hardware hasn’t doubled: it’s just one electron. But this electron is actually storing two pieces of information, not just one. This is called a qubit. Instead of a conventional computer, which has to set a bit in either the on or the off position, a 0 or a 1, this qubit splits into parallel quantum worlds, one in which the switch is set to 0, the other where it is in the 1 position.

The idea would therefore be to string a whole load of these qubits together. For example, if I were able to put 64 qubits in quantum superposition together, this bank could simultaneously represent all numbers from 0 to 2^64 − 1. A conventional computer would have to sequentially run through all these numbers putting each bit into 0 position or 1. But the quantum computer can do it simultaneously. It’s as if my conventional computer, like the electron, is suddenly living simultaneously in parallel universes. In each one the 64 qubits are set to a different number.

Now comes the smart bit. In each parallel world, I get the computer to check if this number that it is representing divides our crypto number. But how can I make sure that the quantum computer can pick out the one world where the number being tested does divide the crypto number successfully? This was Shor’s brilliant trick that he built into his quantum algorithm. When I observe a quantum superposition it has to make up its mind and collapse into one state or the other. Essentially it chooses either the 0 or the 1 position. There are probabilities that determine which way it will go.

What Shor managed to do was to rig up an algorithm that after doing the test division in each parallel universe, the probability would be overwhelming that it would collapse in the world where the number being tested successfully divided the crypto number. All the other worlds where this failed were sufficiently similar that they all cancelled each other out. The only one that stuck out was the world where division successfully happened.

Imagine twelve different directions pointing to the numbers on a clockface. If all of them are equal in length, then adding all these directions together would cancel them all out and bring me to the centre of the clockface. But what if one was twice as long as all the others? Now I would be left pointing in that direction. This is essentially what happens in the quantum observation of the test divisions.

Although Shor wrote the software as long ago as 1994, building a quantum computer that could implement the algorithm seemed a distant dream. One of the problems with quantum states is something called decoherence. The 64 qubits start observing each other and the thing collapses before it can be made to do the calculation. It is one reason that we think that Schrödinger’s cat – the quantum thought experiment where an unobserved cat can be both dead and alive at the same time – might not be possible. Sure, an electron can be put into superposition but how can all the atoms that make up a cat be put into a state of being dead and alive at the same time? The large number of atoms start interacting and decoherence means that the superposition collapses.

But in recent years there has been some amazing progress in isolating simultaneous quantum states. In October 2019, Nature published a paper by researchers at Google entitled: ‘Quantum Supremacy Using a Programmable Superconducting Processor’. In the paper the team reports being able to rig up 53 qubits in superposition, giving them the ability to represent simultaneously numbers up to about 10^16. The computer was able to perform a very bespoke task that would have taken a conventional computer 10,000 years to execute.

Although very exciting news, the task that the quantum computer was asked to carry out was not on a par with finding the primes that divide large numbers and was very much tuned to the hardware that was being used. Many felt that Google was hyping the ‘quantum supremacy’ headline a bit too much. The IBM quantum computing team were pretty scathing of the announcement and illustrated how the task the Google team was implementing could be done on a conventional computer in days not 10,000 years. It was still a fascinating result. But it seems like creating a quantum computer that could hack your credit card details remains some way off.

Biological computing

What about the Travelling Salesman Problem? Could I use unconventional means to find a shortcut? There is a challenge related to the Travelling Salesman Problem that researchers have solved using a very unusual sort of computer. Called the Hamiltonian Path Problem, the challenge is to find your way around a network of one-way streets connecting cities on a map.

10.4 Hamiltonian Path Problem: get from city A to city E visiting every other city once only.

You must discover a path that begins at one city, say City A, and ends at a second City E but the path must visit every other city on the way once and once only. Is such a path possible? This turns out to be as complex as the Travelling Salesman Problem. But it is again a problem that is ripe for parallel processing. Rather than exploiting the quantum world, mathematician Leonard Adleman cooked up an interesting use of biology to attack the problem. Adleman is the A in RSA, the name of the cryptography that exploits primes to keep online transactions secure.

At a seminar at MIT in 1994, Adleman announced the discovery of the supercomputer he’d built to attack the Hamiltonian Path Problem. He called it the TT-100, but the audience were rather perplexed when he pulled a test-tube from his jacket pocket. The TT stood for test-tube and the 100 for the 100 micro-litres that this small plastic vial contained. The micro-processors that were doing the work inside this test-tube were small strands of DNA.

Strands of DNA are made up from 4 bases labelled A, T, C and G. These bases like to bind to each other in pairs: A to T and C to G. If you cook up short single strands of the bases, called oligonucleotides, they will try to find another strand with bases that pair up. For example, a strand with ACA will try to find a strand with TGT to bind with to make a stable double strand of DNA.

Adleman’s idea was to give each city in a map that you were trying to navigate a label consisting of a string of 8 bases. Then if there was a one-way road between the two cities, he would create a strand of DNA with 16 bases such that the first 8 consisted of the code number of the origin city, and the second 8 were the complementary string to that of the city that the road ended at. If there was a road into City A and a road out, the two 16-base strands for these roads would join up along the last 8 bases of the road coming in and the first 8 bases of the road going out of City A.

Any route round the cities along these roads could actually be replicated in strands of DNA that were binding to each other each time a road went in and out of a city.

For example, City A could have the label ATGTACCA, City B the label GGTCCACG and City C the label TCGACCGG. The road from A to B would then be represented by

ATGTACCACCAGGTGC

And the road from B to C by

GGTCCACGAGCTGGCC

These two roads would then bind along the last 8 of the first route and the first 8 of the second route to show that there is a journey you can make from City A to City C.

The great thing is that it is possible to acquire such strands of DNA in huge amounts from commercial labs. Adleman ordered up enough to explore a network with 7 cities and then just filled his test-tube with the strands. In an act of parallel processing, the strands began binding together to create lots of different pathways through the network. Of course many of them violated the challenge of visiting cities only once. But Adleman realised the route he was after would be a strand of DNA of length

8 (the origin city) + 6 × 8 (for each road) + 8 (the destination city).

He could filter these out of the solution and then check for those strands that had each city occurring somewhere in the sequence in a similar process to genetic fingerprinting.

Although the whole process took over a week, it nonetheless opened up an intriguing possibility to exploit the world of biology to create machines that can parallel process efficiently. Chemists use a unit of measurement called the mole to quantify how many molecules are in a test tube. But a mole’s worth of a substance contains slightly more than 6 × 10^23 molecules. Adleman believes that exploiting the very small in the biological world can be a shortcut to navigating the very big in conventional computational challenges.

It may be that nature has already clocked this. It turns out that a strange organism called slime mould is rather good at finding the most efficient routes to navigate a map. Slime mould, or Physarum polycephalum, is a plasmodial, single-celled organism that grows outward from a single point, searching for food sources. The food it loves most is oat flakes.

A team in Oxford and Sapporo decided to set their slime mould the challenge of finding the shortest route around oat flakes laid out in the same way as stations on the Tokyo area’s rail network. Human engineers had spent years putting together the most efficient way to connect the cities. How would the slime mould do in comparison?

At the outset, the slime mould knows nothing about the locations of the oat flakes and so starts growing in all directions. But as it begins to encounter a food source, the many branches it has sent out that have failed to find food die back, leaving only the most efficient route to the food source. Within hours the slime mould is refining its structure and creating tunnels between new food sources that efficiently navigate the different locations.

What was remarkable to the team that built the experiment is that the resulting patterns of the slime mould resembled very closely the way humans had laid out the Tokyo area rail system. Humans had taken years. The slime mould did it in an afternoon. Does this single-celled slime mould know a shortcut that might help us to solve one of mathematics’ great unsolved problems?

Solution: Here is the shortest path round the Glastonbury Festival map. It took me a long time to check that there wasn’t a shorter path.

10.5 The shortest path around Glastonbury Festival.

Shortcut to the shortcut

Sometimes it’s just as important to know when there are no shortcuts to the problem you’re trying to crack. Realising that the long way round is the only way to your destination will stop you wasting time in the hope of finding the shortcut. And if you are going to do all the work, it’s worth knowing that you are not wasting your time. You can use the shortcut of changing one problem into a completely different problem to check if the challenge you are trying to crack is actually the Travelling Salesman in disguise. If there are no shortcuts then perhaps, like the cryptographers have done, this is something that you can exploit to your advantage.

Arrival

Human ingenuity has conjured up an extraordinary range of different shortcuts, which have accelerated the development of our species over the generations. We would never have arrived at the technologically advanced place we currently occupy without this portfolio of better ways of thinking. Without the shortcut of symbols for numbers, everything beyond three just looks like lots. Our physical journeys across the planet have been made more efficient by understanding its geometry. Although only 566 people have been into space and none further than the Moon, we have used the shortcut of trigonometry to navigate deep into the cosmos.

We have been able to shortcut our journey into the future using the power of pattern recognition and calculus to get a glimpse of what’s coming next before it happens. The shortcut of probability enables us to avoid having to repeat experiments hundreds of times to understand which outcome is more likely. Rather than wandering aimlessly round the internet looking for what we want, clever ways of analysing the connections allow us to shortcut our way to our destination. We’ve even come up with new numbers, like the square root of −1, to create a looking-glass world through which we can step to shortcut our way to a solution. Planes land safely thanks to a trip through this imaginary world.

Sidestepping tedious hard work was certainly the initial reason I set off on my mathematical journey. Avoiding mindless labour appealed to the lazy side of my teenage self. I am grateful to my maths teacher who rather than pushing the class into tedious repetition and calculation, instead showed me that mathematics was about thinking smart. But looking back I have also begun to see something of a paradox at the heart of shortcuts.

The job of a mathematician is to discover new ways of thinking smart, but coming up with these shortcuts is not easy. Doing mathematics still necessitates hours of meditation on a problem, thinking and seemingly getting nowhere. And then suddenly there’s this rush of understanding, the discovery of the shortcut through the wilds of the problem. But without laying down the hours of meditation and random scribbling on my yellow legal pads, I can’t achieve that rush of the discovery of the way through. It’s that thrill of the ‘aha’ moment that I crave. That’s the drug. And that hit is got by the discovery of a hidden passage, the shortcut, that gets you through to the other side.

In the end I realised it’s not actually because I’m lazy that I dedicated myself to the art of the shortcut. Almost the opposite. It’s the hard work of finding a shortcut that makes it so satisfying.

Faced with a mountain, you could take a helicopter to the top. You would enjoy the view but, as Robert Macfarlane explained to me, if you are a mountaineer this defeats the point. It is the satisfaction of achieving the peak that is the reason you put in the hard work. To walk the feet transparent.

I remember talking to a Harvard physicist about the intellectual challenges of tackling great unsolved problems. At one point she offered me a hypothetical button to press that would give me all the answers to all the problems that I was working on. As I reached out to press it, she grabbed my hand. ‘Are you sure you want to do that? Doesn’t it spoil the fun?’

Natalie Clein expressed the same reservation. If there was a shortcut to playing the cello maybe it would make performing less attractive. Achieving that ecstatic moment of psychological flow is about combining skill with a difficult challenge.

One of my favourite Hollywood movies is Good Will Hunting, partly because it represents one of popular culture’s first mentions of the Fields Medal, the mathematician’s Nobel Prize. But the film also illustrates the importance of spending frustrating hours working away at a problem as the foreplay to the moment when you discover the shortcut that cracks it. The central character in Good Will Hunting is a janitor in the maths department at MIT, played by Matt Damon, who sees a problem chalked up on the board and at once gets how to solve it. The maths professors are blown away when they come in next morning to see the solution scrawled on the board. But ultimately Damon’s character does not become a mathematician.

For me this is because he finds it too simple. It’s the girl he is trying to catch that is the complex problem without any obvious solution and that motivates his journey at the end of the movie. One of the important traits of the mathematical shortcut is that it should provide a moment of ecstatic release after all the hard grafting of trying to solve the problem head-on.

The shortcuts I’m after are not about looking up the answers in the back of the book. That isn’t a satisfying shortcut. The best shortcuts are those that emerge after the hard work of struggling with a problem. It almost has a musical quality to it where the tension of the music is finally resolved.

The paradox that has emerged is that although the motivation for the shortcut might be an initial reluctance to spend ages doing hard work, I might ultimately put in as much work to find the shortcut. Yet it is the nature of the curve describing the effort that perhaps reflects why I still enjoy the difficult work of finding the shortcut more. If I drew a graph of the effort I would expend adding up numbers from 1 to 100, it would probably look like a constant grind. Not varying much over time. The total effort creeping up in a linear fashion. The graph charting the effort to find the shortcut looks much more unpredictable. It has ups and downs. It probably spikes towards the end before swooping down to a low as the shortcut is implemented. But from this point on, the graph of effort never has to go above a baseline minimum because the shortcut does the work. All while the long-haul graph is still ploughing on at its constant grind.

Another of the curious paradoxes that emerged is one that the art curator Hans Ulrich Obrist highlighted. Detours are essential. You often arrive at the best shortcuts by beginning with the detour. The detour that the proof of Fermat’s Last Theorem took mathematicians on is worth all the strange highways and byways we encountered on the way. Those detours led to the discovery of many extraordinary shortcuts we were forced to conjure up on our journey.

The power of the shortcut is often that it allows those who follow the chance to reach their destination quicker. In 2016 the longest and deepest tunnel in the world was opened. The 57 kilometre Gotthard tunnel runs beneath the Alps, joining northern and southern Europe. It took 17 years to build but the trains that run through take 17 minutes to get from one end to the other.

One of Carl Friedrich Gauss’s last journeys was to attend the opening of a new railway link between Hanover and Göttingen. His health had deteriorated slowly, and he died in his sleep early in the morning of 23 February 1855.

Gauss had asked that one of the discoveries that inspired his journey to become a mathematician, the geometric construction of a 17-sided figure, be carved onto his grave. However, when the stonemason tasked with the carving saw the design he refused to include it. The construction might in theory produce a 17-sided polygon, but the stonecutter thought that it would just look like a big circle.

The shortcuts I spent my time learning as a student took their creators years of deep thought to carve out. But once the tunnel has been created, it allows those who follow the chance to get to the frontiers of knowledge as quickly as possible. Sitting there as a student having completed his task of adding the numbers from 1 to 100, Gauss had the chance to think about new things. And, for me, this is the point of the shortcut. If I spend time on mindless work, I deprive myself of the opportunity for self-exploration, new discoveries, expanding my horizons. The shortcut allows me to dedicate my effort to new exciting and rewarding ventures.

So I hope the journey we have been on provides you with the shortcuts to think smarter and to free up your time for new thoughts. The end of one shortcut is the chance to begin a new journey. Gauss summarised his views on the pursuit of knowledge in a letter to his friend Farkas Bolyai dated 2 September 1808 as follows:

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The never-satisfied man is so strange; if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.

A shortcut is not about a fast way to finish your journey but rather a stepping stone to beginning a new one. It is a pathway cleared, a tunnel dug, a bridge constructed to allow others to quickly reach the boundaries of knowledge so that they can make their own journeys into the darkness. Equipped with the tools that Gauss and his fellow mathematicians through the ages have honed, stretch out your arms for the next great conquest.

Acknowledgements

There aren’t many shortcuts to the epic task of writing a book but one of the best shortcuts is to have a great team supporting you. Like the best psychologist, Louise Haines, my editor at 4th Estate, has a wonderful way of asking that probing question that creates an environment in which you as an author discover the solutions to the problems you are having. Antony Topping, my agent at Greene and Heaton, has always been another important set of eyes that has ensured I don’t get lost down paths that lead nowhere. My copyeditor Iain Hunt very patiently wrestled with my mangled English grammar, knocking it into shape.

On the other side of the pond, my US editorial team Thomas Kelleher and Eric Henney at Basic Books have done a wonderful job of ensuring my shortcuts take US readers in the right direction.

Each of my contributors to the pitstops that punctuate this book were extremely generous with their time and ideas. I am deeply grateful to Natalie Clein, Brent Hoberman, Ed Cooke, Robert Macfarlane, Kate Raworth, Hans Ulrich Obrist, Conrad Shawcross, Fiona Kennedy, Susie Orbach, Helen Rodriguez and Ognjen Amidzic for fascinating discussions about their thoughts on shortcuts.

Thank you to the artists Sophia Al Maria, Tracy Emin, Alison Knowles and Yoko Ono for permission to include their do it instructions.

Writing such a book would be impossible without the time that my professorship provides me. Thank you to Charles Simonyi who endowed the chair and Oxford University for all the support I receive as the Professor for the Public Understanding of Science.

Both Newton and Shakespeare profited from being more productive during times of plague. The writing of this book overlapped with the pandemic that hit the planet at the beginning of 2020. This turned out to be a strange shortcut as it gutted my diary of distractions leaving me with time to sit and write. The result was that I finished the manuscript two months before my deadline. My editor Louise was in shock when the manuscript arrived. She is used to me being two years late! But it transpired I wasn’t the only author to submit early. Indeed Louise admitted she’d received novels from her authors she hadn’t even commissioned. Feedback might take a bit of time, she warned. While I waited I ended up using lockdown to write a new play. Probably a crazy project given that all the theatres had closed but I hope it will see the light of day at some point.

A day of writing during lockdown would end with each member of the family emerging from their rooms to share their day’s online adventures. The laughter and love that we shared together of an evening made the hard slog of bringing my book to the finishing line that much easier. Thanks to Shani, Tomer, Ina and Magaly. They were the best shortcut to completing the daunting journey of writing a book.

Index

The page numbers in this index relate to the printed version of this book; they do not match the pages of your ebook. You can use your ebook reader’s search tool to find a specific word or passage.

15 (game), 99

21 (Robert Luketic film, 2008), 251

Adleman, Leonard, 319–21

Agrawal, Manindra, 314
air traffic control, 68, 69–71, 325

Airbnb, 74
airport networks, 284

Al Maria, Sophia, 207
alan.eu, 78

Alexandria, Library of, 114–15, 116

Alfraganus (Persian geographer), 112
algebra: and ancient Babylon, 59, 62, 63; cubic equations, 63–5; developed at House of Wisdom, 84, 85; Fermat and calculus, 183–7; fitting equations to data, 93–4; fundamental theorem of, 66; geometry translated to, 8, 81, 91–5; as grammar underlying numbers, 7–8, 83–6, 91–5, 97; patterns of, 83–6, 92–4; quadratic equations, 62–3, 93, 94; translating to geometry, 8, 88–90
algorithms: Algorithmic Justice League, 229; and art production, 201–4, 207–8; Babylonians and quadratic equations, 62–3, 93; and calculus, 179–81, 201; Dijkstra’s algorithm, 299–301; Eratosthenes and prime numbers, 114–15; exponential, 295–7, 298, 301, 304; Google’s PageRank algorithm, 267–8, 277–80; and machine learning, 227–9; and music, 39; in natural world, 37–8; and navigating the internet, 5, 267–8, 277–80; polynomial time, 295, 297–8, 301, 307, 314; as recipe to reproduce pattern, 35, 42, 62–3, 201; Shor’s quantum algorithm, 317–18; and thinking behind CBT, 233, 238–9

Alhambra Palace, Granada, 96–7

Amazon (online retail company), 77, 127, 229

Amidzic, Ognjen, 290–2

Andre, Carl, 205
ant colonies, 198
antispheres, 130

‘Arabic numerals,’ 55

Archilochus (ancient Greek poet), 286

Archimedes, 181
architecture, 187–90, 198–9

Argand, Jean, 66

Aristarchus of Samos, 117

Aristotle, 14
arithmomania, 24–5
art, 154, 160–1, 163, 176, 189, 201–8

Asch, Solomon, 221–2
astronomical data: Copernicus’s heliocentric theory, 156–7; diagrams charting entire universe, 157; distance from the Earth to the Moon, 117–18; distance from the Earth to the Sun, 117–18; Einstein’s space-time, 132–4, 198; Galaxy Zoo project, 224–5, 227; geometry of space, 131–4, 198; maps at caves at Lascaux, 149–50; patterns in, 18–19, 20, 42, 210, 211; possible shapes of the universe, 168–9; and trigonometry, 117–18, 119; wormholes, 133–4

Aswan, 115–16

Babylonians, ancient, 53–5, 59, 62–3, 71, 93, 150

Bachelier, Louis, 261
badgers, 214

Baghdad, House of Wisdom, 84, 85

Bailson, Mr, 1–2, 3, 44, 83, 326

Barkham, Michael, 234

Baroque art, 176, 189

Barrie, J. M., 126

Bartels, Johann Martin, 16, 36–7

Basel, 270–1

Bayes, Thomas, 257, 258

A Beautiful Mind (Ron Howard film, 2001), 98–9

Beck, Aaron, 233

Beck, Harry, 153–4
bees, 6

Benjamin, Arthur, 86, 108

Bennet, Craig, 229–30
benzene molecule, 158–9

Berger, John, Ways of Seeing, 172

Berlin, Isaiah, 286

Bernoulli family of Basel, 251, 270–1; Daniel Bernoulli, 253, 271; Jakob Bernoulli, 251–2; Johann Bernoulli, 192–3; Nicolaus Bernoulli, 252–3
binary numbers, 72, 101–3
binomial coefficients, 36–7

Bletchley Park, 309–11

Bohr, Niels, 162

Boltanski, Christian, 201

Bolyai, János, 130–1

Bombelli, Rafael, 63, 64–5, 74–5

Boole, George, 165

Borda repeating circle, 122, 123

Bosch, Marc ten, 98

Bourgeois, Louise, 203

Bourjaily, Jacob, 163
bowling machines, cricket, 165

Box, George, 173
brain, human, 282, 283, 284–6; neuroscience, 13, 290–1 see also memory, human; thought, human

Branson, Richard, 76

Briggs, Henry, 59

Brin, Sergey, 277–80

Bristol’s walkable bridges route, 273–4

Brunswick, Duke of, 16, 82

Bunyan, John, The Pilgrim’s Progress, 10

C. elegans nematode worm, 282, 283

Cage, John, 202

Cairn Gorm, 136
calculations and counting: and air traffic control, 68, 69–71; ancient use of tables, 58–9; arithmomania, 24–5; cubic equations, 63–5; decimal numbers, 53, 71; and development of language, 51–2; early human methods, 52–3; finding the best notation, 72–3; imaginary numbers (complex numbers), 51, 61, 64–6, 67, 68, 70–1, 73, 143; Leibniz’s machines, 60–1; logarithms, 56, 57–8, 59; place-value system, 52, 53–6; power of machines, 60–1; powers of sixty, 53–5, 71; powers of twenty, 55, 71; quadratic equations, 62–3, 93, 94; ternary numbers/powers of three, 72–3; use of 60 for time, 54
calculus, 9, 175, 199–200, 325; and architecture, 187–90; and dogs, 196–8; and Leibniz, 182–3, 189, 190, 193–4, 251; and marginal cost economics, 180–1; multiple uses of, 175–6, 187–8, 190–1, 201; and Newton, 177–82, 193–4; Newton-Leibniz dispute, 193–4; and Pierre de Fermat, 183–7, 196; and rollercoasters, 191–2; and space travel, 175, 176–7, 178, 198

El Capitan (Yosemite National Park), 139

Cardano, Gerolamo, 63, 64

Carroll, Lewis, Sylvie and Bruno Concluded, 149, 156

Cartesian coordinates, 92–3, 95–6
cartography: choices about what to leave out, 149; compromises in mapping the planet, 152; conformal mapping, 151–2; earliest maps of the Earth, 150; Gall-Peters map, 152; London Underground map, 8, 153–4, 155, 167, 272; Mercator’s projection, 150–2; north-south orientation, 152–3; and political choices, 152–3; topological maps, 154–5, 288

Casals, Pablo, 49

Casanova, Manuel, 286
catenary curves, 189–90
cats, 212–13, 214, 215

Cawelti, John, Adventure, Mystery and Romance, 138

Ceres (asteroid), 18–19, 42, 210, 211

Chan, Priscilla, 78

Chan Zuckerberg Initiative (CZI), 78
chemistry, 157–61
chess, 11, 22, 221, 290–2

Chevalier de Méré (Antoine Gombaud), 242–3

China, ancient, 40

Christie, Agatha, 14
cities, universal scaling across, 25–30

Clark, David, 236

Clein, Natalie, 42, 45–6, 47, 48–9, 327
climate change, 42
cognitive behavioural therapy (CBT), 233–6, 238–9

Columbus, Christopher, 20, 111–13

Comici, Emilio, 140
complex numbers (imaginary numbers), 51, 61, 64–6, 67, 68, 70–1, 73, 143

Computer Misuse of Information Act, 75

Conquest, Stuart, 290–1

Contagion (Steven Soderbergh film, 2011), 146–7

Cooke, Ed, 104–9, 155

Copernicus, Nicolaus, De revolutionibus orbium coelestium, 156–7

Covid-19 pandemic, 24, 108, 210
credit rating agencies, 264

Crick, Francis, 160, 161

Crick, Odile, 160–1

Crimean War, 143–6, 147, 149

Crowe, Russell, 98–9
cryptography, 308–18, 319, 323

Csikszentmihalyi, Mihaly, 46
cubic equations, 63–5
cycloid curve, 193

Dalí, Salvador, 98, 161

Daly, Herman, 170
data: 1 in 20 random possibility threshold, 212–13, 216–18; amount needed for informed inference, 228, 231; census information, 210, 211; computer vision, 228; Gaussian distribution (bell curve), 211–12; Gauss’s love of, 210–11; Gauss’s method of least squares, 211; levels of daily creation of, 42, 209; limitations of big data, 94; need for diversity, 229; optimising of decision making, 230–1; small sample sizes, 215–16; storage, 33, 34–5; tracking too many variables, 229–30; understanding of sources of, 32, 229; use of statistical tools, 212–20, 228–30; using the crowd, 224–7; wisdom of the crowd theory, 218–24, 231

Delambre, Jean-Baptiste, 121–3, 124, 131
demographic data, 213–14, 215

Descartes, Rene, 65, 91–2, 95–6
diagrams: abuse of by news organisations, 147–8; amplituhedron, 163; Argand diagram, 66; Copernicus’s heliocentric theory, 156–7; Herman Daly’s circle, 170–1; dangers of, 169, 172–3; discarding of the extraneous in, 8–9, 149, 173, 274, 288; double helix structure of DNA, 160–1; doughnut economic diagram, 168–9, 171–3; Feynman diagrams, 161–3; mind maps, 155; misuse of bar charts, 147–9; of molecular world, 157–61; and new ways of seeing the world, 155–7; Nightingale’s rose diagram, 145–6, 147, 149; Penrose’s theory of twistors, 163; as powerful shortcut, 8–9, 66, 142–3; Rockstrom’s diagram, 170–1; of today’s cosmologists, 157; as tools in psychotherapy, 235–6; Venn diagrams, 164–7 see also cartography
digital technology: advances of AI and robotics, 15; arrival of internet, 67, 76; and binary numbers, 72; cryptography for internet/e-commerce, 311–18, 319; data storage, 33, 34–5; digital desire paths, 127–8; digital footprints, 43; dot-com boom, 76; extraordinary speed of computers, 11–12; Google’s PageRank algorithm, 267–8, 277–80; and landing of planes, 68; levels of daily data creation, 42, 209; limits on power of computers, 51, 61, 301; machine learning, 227–8; navigating the internet, 5, 267–8, 277–80; parallel processing, 315–16, 319

Dijkstra, Edsger W., 298–301

Dirac, Paul, 162

DNA, structure of, 160–1, 319–20
do it (art algorithm), 201–4, 207–8
dogs, 196–8

Duchamp, Marcel, 202

Dyson, Freeman, 162
the Earth: distance to the Moon, 117–18; distance to the Sun, 117–18; earliest maps of, 150; estimates of circumference of, 112, 114, 115–16; Mercator’s projection, 150–2
earthquakes, 73
eclipses, 118, 133, 225–6
economics: assumption of limitless growth, 170; Herman Daly’s circle, 170–1; data with patterns, 18; doughnut economic diagram, 168–9, 171–3; language of, 81; marginal cost, 180–1; market equilibrium point, 93, 169; measurement using dollars, 170–1; Rockstrom’s diagram, 170–1; standard diagrams, 169

Eddington, Arthur, 133

Egypt, ancient, 5, 53, 84, 120
eigenvalue of a matrix, 279–80

Einstein, Albert, 132–4

Eisenstein, Gotthold, 181
electromagnetic waves, 68–71, 268
emergent phenomena, notion of, 81

Emin, Tracey, 207
engineering, 5, 9, 175–6, 198–9, 200, 302

Enigma machine, 309–11
environmental science, 42, 169, 170, 171–3; capture-recapture strategy, 214–15

Eratosthenes, 112, 114–16, 117, 121

Erdös, Paul, 255

Escher, M. C., 163

Euclid, 75, 129

Euler, Leonhard, 70, 230, 270–4, 276–7, 279

Everest, Mount, 140
exponential growth, 21–2; algorithms, 295–7, 298, 301, 304; exponential constant (e = 2.71828), 70, 230–1; exponential functions, 70–1, 297; grains of rice on chessboard, 22–3, 297; remembering things by rote, 107; universal scaling across cities, 25–30

Facebook, 75–6, 77–8, 110, 127, 226, 267

Faraday, Michael, 158
farmers, 20

Fermat, Pierre de, 183–7, 196, 241–5, 251, 260, 312–13

Fermat’s Last Theorem, 138, 168, 328

Feynman, Richard, 161–3, 169–70, 176

Fibonacci numbers, 37–9, 41, 164, 207

Fields Medal, 327
financial crash (2008), 18
financial markets, 39, 43, 261–2; high-yield (junk) bonds, 263–6; short selling, 262–3
the flow (in the zone), 46–7
fMRI (functional magnetic resonance imaging) scanners, 229–30, 285

Fontana, Niccolo (Tartaglia), 64
football, 222–3, 245–7, 305–7

Founders Factory, 74, 75

Free Solo (Elizabeth Chai Vasarhely film, 2018), 139

French Revolution, 121, 123
frequency analysis, 309

Fuchs, Conrad Heinrich, 285
furniture industry, 76

Galaxy Zoo project, 224–5, 227

Galileo Galilei, 86–8, 94, 121, 177–8

Gall-Peters map, 152

Galois, Évariste, 97

Galton, Francis, 218
game theory, 98
games, 98–103

Gaudí, Antoni, 190

Gauss, Carl Friedrich: on the act of learning, 329; and binomial coefficients, 36–7; on calculus, 199; death of (1855), 284–5, 328–9; discovery of curved geometries, 131–2, 134, 152; Einstein on, 134; on Euler, 270; finds asteroid Ceres, 18–19, 42, 210, 211; Gaussian distribution (bell curve), 211–12; and imaginary numbers, 65–6, 143; and languages, 82, 104; leaves his brain to science, 284–5; and logarithms, 210; love of data, 210–12; mapping of physical shortcuts, 111, 119, 131–2; as nervous of sharing incomplete ideas, 66, 78; and networks, 154, 268, 286; on the Newton apple story, 181; as prince of the shortcut, 6, 16, 181; schoolroom shortcut story, 1–3, 4, 6, 7, 11, 14, 16, 21, 51, 89; solves 17-sided figure challenge, 19, 328–9; Theorema Egregium, 152; and topology, 154–5; and triangular numbers, 21, 29, 36, 89; Disquisitiones Arithmeticae, 311–12

Geis, Irving, 160

General Electric, 13
geodesics, 128–34, 139–40
geometry: algebra translating to, 8, 88–90; curved geometries, 130–4, 152; Descartes’s dictionary, 91–2, 95–6; Einstein’s space-time, 132–4, 198; Gauss solves 17-sided figure challenge, 19, 328–9; mapping physical shortcuts, 110–16, 119–20, 121–3, 125–7, 128–32, 134; non-Euclidean, 75, 129–30; odd numbers and square numbers, 87–8; positive and negative curvature, 128–30; and surfaces of spheres, 128–30; translating to algebra, 81, 91–5

George II, King, 226

GetYourGuide, 78

Ghebreyesus, Tedros Adhanom, 210

Gladwell, Malcolm, 15, 45

Glenn, John, 174–5, 178, 198

Good Will Hunting (Gus Van Sant film, 1997), 327

Google, 110, 127, 226, 267–8, 277–80, 318

Gotthard tunnel, 328

Gottingen, University of, 16, 211, 268, 284–6

Gove, Michael, 221

GPS, 92

Grand National (Blackpool rollercoaster), 191–2

La Grande Arche (La Defense, Paris), 97

Graunt, John, 210, 213
gravitational constant (g), 93
gravity, 93, 132, 163, 177, 178, 198

Greece, ancient, 19, 75, 84, 112, 114–19, 205, 213, 286

Gross, Thilo, 273–4
group theory, 97

Guare, John, Six Degrees of Separation, 282
haemoglobin, 159–60

Halley, Edmond, 225–6

Hamiltonian Path Problem, 318–19

Handel, George Frideric, 226

Hanover, Kingdom of, 111, 119, 131, 211

Hardy, G.H., 84, 105–6, 314
health and medicine, 42–3, 76, 78, 216, 232–9, 258–9
hedge funds, 262–3

Heifetz, Jascha, 45

Heine, Heinrich, 82

Henry I, King of England, 120–1, 123

Her (Spike Jonze film, 2013), 11–12

Hertz, Heinrich, 68
heuristics, 3–4, 274–6

Hidden Figures (Theodore Melfi film, 2016), 175

Hilbert, David, 269, 276

Hipparchus, 117–18

Hoberman, Brent, 67, 74, 75, 76, 77, 78–9

Hockney, David, 203

Honnold, Alex, 139

Hooke, Robert, 60

Hulsmeyer, Christian, 68–9
human rights, 171–2
hyperdimensional space, 94–8

IBM quantum computing team, 318

Illinois Institute of Technology, Chicago, 127
imaginary numbers (complex numbers), 51, 61, 64–6, 67, 68, 70–1, 73, 143

India, 39, 55

Ingram, Charles, 220–1

Instagram, 143
interest on loans, 23–4

James, William, 81–2

Johnson, Katherine, 175

Johnson, Samuel, 14

Kahneman, Daniel, 275

Kaliningrad, 269–70, 271–3, 274, 276–7

Kant, Immanuel, 269, 276

Karinthy, Frigyes, ‘Chain Links,’ 280–1

Kayal, Neeraj, 314

Kekulé, August, 12, 158–9

Kendrew, John, 159–60

Kennedy, Fiona, 232–6

Kholodenko, Vadym, 48
al-Khwarizmi, Muhammad ibn Musa, 84
al-Kindi, Ya’qub, 309

Kirchner, Cristina, 265

Knowles, Alison, 208

Königsberg, 269–70, 271–3, 274, 276–7

Koolhaas, Rem, 127

Lane Fox, Martha, 74
language: algebra as grammar underlying numbers, 7–8, 62, 83–6, 91–5, 97; of George Boole, 165; and concept of number, 52; conceptual/symbolic nature of, 8, 51–2; creating and changing of, 7–8; Descartes’s dictionary, 91–2, 95–6; and frequency analysis, 309; Galileo on mathematics, 86–7; and games, 98–103; and Gauss, 82, 104; gendered nouns, 82; Leibniz’s calculus, 182–3, 194; mnemonics, 109; of psychology, 81–2; ‘repetition and testing,’ 109; shortcuts aiding memory, 104–9; and social construction of reality, 82; and symmetry, 96–7; translating into a new idiom, 7–8, 80–2, 85–90, 91–8, 99–103

Laplace, Pierre-Simon, 59, 213–14, 215, 270

Lascaux, caves at, 149–50
lastminute. com, 74, 75, 76, 77

Lavier, Bertrand, 201

Layard, Richard, 234

Leibniz, Gottfried, 60–1, 65, 86, 176, 182–3, 189, 190, 192–4, 251, 271

Lenstra, Hendrik, 314

Leonardo da Vinci, 142–3, 155

Let’s Make a Deal (US game show), 253–5
light, 5–6, 131–3, 134, 196, 198

Lipkin, Ian, 146

Listing, Johann Benedict, 154–5
literature, 10–11

Little Langdale (Cumbria), 137

Little Red Riding Hood, 10

Llull, Ramon, 165

Lobachevsky, Nikolai Ivanovich, 130–1
logarithms, 56, 57–8, 59, 73, 210, 284

London Underground map, 8, 153–4, 155, 167, 272

Louisville, University of, 286

Macfarlane, Robert, 135–6, 137, 138–9, 140–1, 326
machine learning, 227–8
made.com, 76

Mallory, George, 140

Manhattan, lay-out of, 124, 125–6

Marx, Karl, 15
mathematics: ‘8 choose 2,’ 36; birth of in Babylonian era, 62–3; breaking of Enigma code, 309–11; calculation as still central to, 50–1; capacity for abstract thinking, 4–5; definition of short cut, 295–8; of the doughnut, 168–9; in educational curriculum, 7; eigenvalue of a matrix, 279–80; and fear, 77; flashes of insight, 289–90; functions, 58; Galileo on language of, 86–7; hyperdimensional space, 94–8; language of, 80–7, 90–103; long cuts in, 137, 138; and lure of the shortcut, 1, 3; odd numbers and square numbers, 87–8; patterns at heart of, 7, 18, 19, 33–5, 41–2, 266; and perfectionism, 78; π (pi), 70, 230; problems without shortcuts, 294–8, 302–8, 311–15, 323; during the Renaissance, 5, 63; rock climbing analogy, 138–9; role of in antiquity, 5, 62–3, 112, 114–19; square roots of negative numbers, 64–6, 143; tension between ease and complexity, 137–8; Travelling Salesman Problem, 294–5, 298, 318; universal scaling across cities, 25–30

Maupertuis, Pierre-Louis, 6, 199

Mayan culture, 52–3, 55, 71

McArthur, Stuart, 153
measurement, units of: defining of a metre, 121–3; historical differences, 112; pre-metric, 120–1; use of body parts, 120–1

Méchain, Pierre, 121–3, 124, 131
medicine, 42–3, 76, 78, 216, 232–9, 258–9
memory, human, 33–5, 47–8, 104–6; memory palace concept, 104, 106–8, 109, 155; mnemonics, 109; spatial memory, 106–8, 109

Memrise, 104, 109
mental health, 42–3, 232–8, 239; Improving Access to Psychological Therapies (IAPT), 234, 238–9

Mercator, Gerardus, 150–2

Mersenne, Marin, 242

Messina, Chris, 127–8

Michigan State University, 127

Miegakure (computer game), 98

Milgram, Stanley, 281–2

Millennium Prize Problems, 295, 312

Miller, Gary, 313
mind maps, 155
mindfulness, 13

Möbius strip, 191–2
molecular world, 157–61

Monbiot, George, Out of the Wreckage, 169

Monet, Claude, 205
mountaineering, 140–1

Munich Olympic stadium (1972), 198–9
the Muppets, 25

Murray, W. H., 136
music, 42, 44–6, 48–9, 67, 327; Indian tabla players, 39; memorising a piece of, 47–8; and practice, 45–6, 47; Schenkerian analysis, 47

Napier, John, 56–8, 59, 284

Napoleon, 113

NASA, 174–5

Nash, John, 98–9

Nasser, Gamal Abdel, 113
natural world: Adleman’s parallel processing DNA, 319–21; capture-recapture strategy, 214–15; Fibonacci numbers in, 37–9; and low-energy optimal solutions, 196–9; migration patterns of animals, 20; shortcuts in, 5–6, 37–9, 196–9; slime mould and rail networks, 321–2
navigation, maritime: Mercator’s projection, 150–2; rhumb lines, 150, 151
network theory, 277
networks: analysis of, 124–5; Chan Zuckerberg Initiative, 78; in cities, 29–30; and collaboration, 286–7; Gauss and Weber’s work on, 268, 286; and Google’s algorithm, 278–80; in human brain, 282, 283, 284–6; and journey through modern world, 267–8; Königsberg bridge puzzle, 269–70, 271–3, 274, 276–7; London Underground map, 8, 153–4, 155, 167; Milgram experiment, 281–2; and psychotherapy, 233, 235; six degrees of separation, 282; small worlds, 282–4, 288; of social connections, 28–30, 280–2
neuroscience, 13, 290–2

New College, Oxford, 59

HMS Newport, 113

Newton, Isaac, 78, 94, 132, 155, 176, 177–81, 192–4, 199–200, 260, 261, 266; Principia Mathematica (1687), 181–2

Nightingale, Florence, 143–6, 147, 149

Nim (game), 99–103

Ning Li, 76

Obrist, Hans Ulrich, 201–4, 328

Ono, Yoko, 202, 208

Orbach, Susie, 42–3, 236–9

Otto, Frei, 198–9

Oxfam, 170, 171–2

Page, Larry, 277–80

Painlevé, Paul, Analyse des travaux scientifiques, 66–7, 71

Panama Canal, 114
pandemics, responses to, 146–7, 210
participatory budgeting experiments, 223–4

Pascal, Blaise, 40–2, 241–5, 251, 260

Pascal’s triangle, 40–2
patterns: of algebra, 83–6, 92–4; algorithms as recipes to reproduce, 35, 42, 62–3, 201; in astronomical data, 18–19, 20, 42, 210, 211; and data storage, 33, 34–5; desire paths, 125–30, 134; and digital code, 33; economic data, 18, 261–6; exponential growth, 21–3, 25–30, 70–1, 107, 230–1, 295–7, 298, 301, 304; extra 15 per cent figure, 26, 28, 30; farmers’ use of seasons, 20; at heart of mathematics, 7, 18, 19, 33–5, 41–2, 266; and heuristics, 275; and human behaviour, 42–3, 237; and human memory, 33–5, 47–8, 104–9, 275; limitations of big data, 94; memorising a piece of music, 47–8; migration patterns of animals, 20; misleading, 30–2, 229–30; nature and Fibonacci numbers, 37–9; ‘next number in this sequence’ challenges, 20–2, 25, 30–1; odd numbers and square numbers, 87–8; ‘pattern’ in French language, 82; as powerful shortcut, 7, 17–18, 325; in psychotherapy, 42–3, 237–8, 239; and random numbers, 32; triangular numbers, 20–1, 29–30, 36, 41–2, 89–90, 325; understanding of data sources, 32

Patterson, Simon, 154
payday loan companies, 24
pedometers, 60

Pennings, Tim, 196–7

Penrose, Roger, 163
pension schemes, 211

Pepys, Samuel, 241, 243, 244, 259, 260
philology, 82
philosophy, 60–1, 182–3
physical landscapes: desire paths in, 125–7, 128–30, 134; desire paths on spheres, 128–9; diagonal shortcuts, 125–6; estimating circumference of Earth, 112, 114, 115–16; Gauss’s land survey, 111, 119, 131, 134; geodesics, 128–34, 139–41; Neolithic pathways, 136–7; Panama Canal shortcut, 114; Suez Canal shortcut, 113–14; and trigonometry, 119–20, 121–3, 134; underlying geometry of, 110, 116, 119–20, 121–3; units of measurement, 112, 120–3
physics: Feynman diagrams, 161–3; and Gauss, 268; gravity, 198; laws of, 5–6; Penrose’s theory of twistors, 163; quantum mechanics, 68, 161–3, 198, 316–18

Playfair, William, The Commercial and Political Atlas, 147

Pocono Manor Inn (Pennsylvania), 162

Poincaré, Henri, 289

Polanyi, Michael, 289

Pomerance, Carl, 314
post-colonial politics, 152
prime numbers, 114–15, 164, 210, 311–15, 319
prisoner’s dilemma, 98
probability: Bayes’ theorem, 257, 258; Jakob Bernoulli’s formula, 251–3; card counting in blackjack, 250–1, 262; and casinos, 241–2, 243, 247–8; considering the opposite/negative, 245–7, 259–60; and court cases, 257–8; and craps (dice game), 247–9; and decision making, 240–1, 260; explored by throwing dice, 241–5; Fermat and Pascal’s work on, 241–5, 251, 260; and financial markets, 261–6; and medicine, 258–9; as powerful shortcut, 9, 241, 325; quantifying epistemic uncertainty, 256–7; and roulette, 249–50; St Petersburg Paradox, 253; three doors in Let’s Make a Deal, 253–5
pseudo-primes, 313
psychology, 81–2; therapy, 42–3, 232–9
quadratic equations, 62–3, 93, 94
quantum computing, 316–18
quantum mechanics, 68, 161–3, 198, 316–17; Schrodinger’s cat, 317–18

Rabin, Michael, 313
radar (radio detection and ranging), 68–71

Ramanujan, Srinivasa, 12–13, 84, 105–6, 314

Rawlinson, Robert, 144

Raworth, Kate, Doughnut Economics, 168–73

Reck, Johannes, 78
regression analysis, 228

Renaissance, 5

Reservoir Dogs (Tarantino film, 1992), 142

Richter scale, 73

Riemann, Bernhard, 96

Riemann hypothesis, 77, 138, 139, 312, 313
risk analysis, 240

Road Trip (Todd Phillips film, 2000), 11

Robins, Benjamin, 226–7
rock climbing, 139

Rockström, Johan, 170–1

Rodriguez, Helen, 263–5
rollercoasters, 191–2

Roman numerals, 56

Rousseau, Jean-Jacques, Confessions, 135

Rowling, J. K., 154

Royal Academy of Arts, 205

Royal Opera House, London, 206–7

Royal Society, 60, 193–4, 226

Russian language, 82, 83, 104, 109

Ruth, Babe, 14

Sagrada Familia, Barcelona, 190

Santa Fe Institute, 27

Saxena, Nitin, 314

Schenker, Henrich, 47

Schrodinger’s cat, 317–18

Schwinger, Julian, 162
science: calculating speed of falling objects, 87–8, 177–80; citizen science projects, 224–7; data generation, 210; electromagnetic waves, 68–71, 268; evidence-based approach to psychotherapy, 42–3, 232–9; Galileo’s experiments, 87–8, 94, 121, 177–8; language of, 81; and laziness, 12; limitations of big data, 94; and logarithms, 59; molecular world, 157–61; research community, 78; scientific revolution (seventeenth century), 176, 210; thresholds for significance (P values), 216–18; understanding of consciousness, 245; West’s work at Santa Fe Institute, 27–8, 29

Scottish Highlands, 136

Seacole, Mary, 144

Serpentine Gallery, 106–7, 201

Shafran, Daniil, 48

Shakespeare, William, 177

Shawcross, Conrad, 204–6

Sheldonian Theatre, Oxford, 188

Shepherd, Nan, The Living Mountain, 136

Shipman, Dr Harold, 217

Shor, Peter, 316, 317
shortcuts: abilities needed to find, 4–5; approximations, 8; art production, 201–8; definition of mathematical short cut, 295–8; different sorts of, 9–10; discarding of extraneous information, 8–9, 149, 173, 274–5, 288; Gauss schoolroom story, 1–3, 4, 6, 7, 11, 14, 16, 21, 51, 89; and gravity, 132; human body as limiting factor, 15–16; and human cunning, 12; and human laziness, 1, 6, 12–14; in literature, 10–11; mapping of physical shortcuts, 110–16, 119–20, 121–3, 125–7, 128–32, 134; missing the point, 10, 135, 136, 326; in natural world, 5–6, 37–9, 196–9; in other professions/fields, 15–16; and our relationship to time, 10; paradoxes at the heart of, 326, 327–8; as path to meaningful work, 14, 15; problems for which there are none, 294–8, 302–8, 311–15, 323; pushing legal limits, 74, 75–6; as stepping stones to new journeys, 329–30; as transferable, 7; used by start-ups, 74, 75–9; to winning games, 99–103

The Simpsons, 11
slime mould (Physarum polycephalum), 321–2
snowfall, 127
social media, 110, 127–8, 143, 222, 227, 267, 282
social networks, 28–30, 280–2
space travel, 174–5, 176–7, 178, 198, 325
spatial memory, 106–8, 109

Spiegelhalter, David, 217

Spreckelsen, Johan Otto von, 97

St Paul’s Cathedral, London, 187–90

St Petersburg Academy, 253, 270–1
start-ups, 74, 75–7, 78
statistics, 9, 212–20, 228, 231 see also data; probability

Strogatz, Steve, 283, 284

‘Stuck in the Middle with You’ (Stealers Wheel song), 142, 167

Sudeley Castle, 205–6
sudoku, 294, 303

Suez Canal, 113–14
surveying of land, 121–3

Sutherland, John, 144
symmetry, 96–7, 204–7
tabla players, 39

Tesla, Nikola, 25
the tetrahedron, 205–6

Thorp, Edward, 250, 262, 263, 265
thought, human: abstract thinking, 4–5; and digital machines, 11–12; and laziness, 12–14; cognitive processes/abilities, 13, 233–9; default mode of neural processing, 13; heuristics, 3–4, 274–6; letting the mind wander, 67, 289–90; while walking, 134–7 see also memory, human

Tolkien, J.R.R., The Lord of the Rings, 10–11, 307
topology, 8, 154–5, 168–9, 288
trade routes, 111–12

Travelling Salesman Problem, 294–5, 298, 318
triangular numbers, 20–1, 29–30, 36, 41–2, 89–90, 325
triangulation, science of, 121–3
trigonometry, 69, 117–19, 134; and defining of a metre, 121–3; sines and cosines, 70, 117, 119, 134; trigonometrical tables, 118, 119

Tropic of Cancer, 115–16

Turing, Alan, 290, 310–11
turning turtles (game), 102–3

Tversky, Amos, 275

Twain, Mark, 224–5, 231

Twitter, 127–8, 143, 267, 282

Uber, 74
urban planners, 125–7, 134
vampires, 24–5

Venn diagrams, 164–7

Viner, Andrew, Venn That Tune, 167
viruses, 38

Wadham College, Oxford, 188

Wagner, Richard, Ring cycle, 77

Wagner, Rudolf, 284–5
walking, 135–7

Wallis, John, 188

Watson, James, 160

Watson-Watt, Robert, 69

Watts, Duncan, 283, 284
wave functions, 70–1

Weber, Wilhelm, 268, 286

Welch, Jack, 13

Wessel, Caspar, 66

West, Franz, 203

West, Geoffrey, 27–8, 29

Whiskas cat food advert, 212–13

Who Wants to Be a Millionaire? (game show), 219–21

Wilde, Oscar, 13

Winterson, Jeanette, 236

Wittgenstein, Ludwig, 135

Woolsthorpe Manor, Lincolnshire, 177, 181
work, 13–16

World Health Organization, 210
wormholes, 133–4

Wren, Christopher, 188–90, 198
zeta function, 67

Zuckerberg, Mark, 75–6, 77–8

About the Author

Marcus du Sautoy is Professor of Mathematics at the University of Oxford and a Fellow of the Royal Society. In 2008 he was appointed to the university’s prestigious professorship as the Simonyi Chair for the Public Understanding of Science, a post previously held by Richard Dawkins. He has presented numerous radio and TV programmes, including a four-part landmark TV series for the BBC called The Story of Maths. He works extensively with a range of arts organisations bringing science alive for the public, from the Royal Opera House to the Glastonbury Festival.

Also by Marcus du Sautoy

The Music of the Primes

Finding Moonshine

The Number Mysteries

What We Cannot Know

How to Count to Infinity

The Creativity Code

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