Thinking Better

The Art of the Shortcut

Marcus du Sautoy

• Publisher: Fourth Estate
• Year: 2021
• ISBN:
  9780008393939
• Topic: Business
• Subtopic: Business Skills
• Index: Business

Thinking Better is a celebration of the art of the short cut – and an encouragement to all of us, in our lives and maybe particularly in our business lives, to realise that thinking better is often more successful than working faster.

A new invention is often born of someone who can’t be bothered to do things the hard way. Laziness doesn’t mean that you do nothing; often it means you prefer to play rather than work. But play is frequently the place to foster creativity and new ideas rather than the dull mechanistic world of work – it’s one of the reasons that the offices of start-ups are often filled will pool tables and board games as much as desks and computers.

There is evidence that humans working in conjunction with computers, literally working as a team, can achieve more than computers can achieve on their own. We may not be able to rely on computers to solve all of our problems – whether personal, or business, or even on a planetary level – but Thinking Better explores how together we just might be able build a successful future together.

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 + … + 2⁶² + 2⁶³)

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

= 2 + 4 + 8 + 16 + 32 + … + 2⁶³ + 2⁶⁴

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 + … + 2⁶³ + 2⁶⁴) − (1 + 2 + 4 + 8 + 16 + … + 2⁶² + 2⁶³)

Most of these terms cancel! There is just the 2⁶⁴ 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 = 2⁶⁴ −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...