All right, I admit – I’m talking about the secret of a happy life, rather than the full-on ‘core truth of the cosmos and everything in it’, the goal of philosophers down the ages. But it’s absolutely true that I did discover it as a schoolboy. In fact I held it in my hands – literally. Because the secret of a happy life is nothing more, or less, than a Rubik’s Cube.

I’m not talking about the pleasure to be had from playing with a Rubik’s Cube. Rather it’s the consideration of the cube’s mathematical qualities that leads to a happy life. A Rubik’s Cube, you see, has more combinations than light travels inches in a century. And once you get your head round that fact, you can draw a very important conclusion from it.

 

When I first read the statistic, I simply couldn’t believe it. Someone must have misplaced a decimal point somewhere, and misplaced it very badly. The speed of light is 186,000 miles per second. How many inches is that? And how many seconds are there in a day, never mind a century? No, there was no way a toy, measuring just three squares in each of its three dimensions, could have that many combinations.

Then I worked it out. The light first. 12 inches to a foot, 5280 feet to a mile, 186,000 miles per second. Multiply that by 60 (seconds in a minute), 60 again (minutes in an hour) 24 (hours in a day), 365 (days in a year – let’s not bother about leap years), and finally 100 years in a century. This gives a grand total of 37,165,049,856,000,000,000. Or, if you want the correct technical term, 37 quintillion and change.

The number of permutations into which a Rubik’s Cube can be arranged is 43,252,003,274,489,856,000. That’s 43 quintillion and change. The cube does indeed have more combinations than light travels inches in a century. (This, incidentally, is just the number achieveable by turning the cube as it is meant to be turned. By breaking it apart and rearranging the pieces – in other words, by cheating – you can achieve 12 times as many.)

It seems impossible. The cube has such a tiny number of factors – three lots of three, and even then the middle square on each side stays the same. Yet it produces an unfeasibly large number of possibilities. Forget holding a Rubik’s cube in your hand, you feel as though you’ve entered William Blake territory and are holding infinity in your hand.

Which is where the ‘happy life’ realisation occurred. So much of our time on this planet is spent planning, scheming, analysing, predicting, jockeying, preparing and assessing. We survey the cluttered landscape of people and events that make up our existence, and we try to work out how we can control it. How we can move through it, from a ‘here’ with which we are, to a lesser or greater extent, dissatisfied, to a ‘there’ which will make us content. We spend our whole life trying to plan our life, in all its myriad aspects: work, leisure, relationships, family, friendships. Peace of mind, we tell ourselves, is simply a matter of putting in the hours, weighing up the odds, placing our bets and making our moves.

But stop for a second. Survey that landscape once more, with a steady eye and a disinterested gaze. How many people do you see? There are your nearest and dearest, which depending on your definition of those two words may be one, two, a dozen individuals. Then there are your friends – perhaps another dozen. And acquaintances – a hundred, two hundred? Work colleagues push the total up still further.

And this is before we’ve got to the hundreds, thousands – no, let’s be honest, it’s millions – of people whose presence in your life may only be temporary, or indeed tangential, but can have a dramatic effect on it. The driver whose wing mirror remains unchecked as he crosses your bicycle’s path. The goalkeeper who saves the penalty that keeps your team in the FA Cup.  And that’s just the people. The events? Well, we all know what that butterfly flapping its wings on the other side of the world can do …

You think you can plan this landscape called Life? You think you can subvert it to your will, arrange its pieces to your satisfaction, much less your happiness? Look how many elements there are in it. Think of all the ways those elements can interact. The Rubik’s Cube, with just three times three times three elements (some of which don’t move), has 43 quintillion possibilities. And you are in complete control of that cube. You can take as long as you like to decide your next move. Imagine how many possibilities can result from the factors in your life, most of which are outside your control, and all of which change in real-time, giving you no chance to pause the game while you try and work things out. Or rather don’t. Don’t imagine how many, because you can’t.

And that’s what the Rubik’s Cube alerted me to: the futility of trying to make plans. Having alerted me to it, the cube freed me from it. Not completely, of course. It would be absurd to make no plans, to live your life totally at the whim of chance, as the title character does in the novel The Dice Man. We all do our best to look ahead, to spot and avoid danger, to spot and take opportunities. But in doing so we should remember how much the odds are stacked against us, how little chance we have of playing Mystic Meg when the number of variables is so huge. The next time you formulate a plan and put it into action, picture a Rubik’s Cube in the palm of your hand. That way if the plan fails you’ll be comforted by how unlikely it was to succeed in the first place. Conversely if it does succeed, you’ll be all the happier.

‘Man makes plans,’ goes the Yiddish proverb, ‘and God laughs.’ Maybe He does, if He exists. The beauty of my realisation (could we call it the Tao of Rubik?) is that it works even for those of us who don’t believe in God. Maths takes His (or his, or its) place. I’d always enjoyed maths, in a Sudoku-ish way; playing with numbers, trying to solve the conundrums that appear on newspapers’ puzzle pages, that sort of thing. But since hearing that stat about the Rubik’s Cube, and investigating its truth, the way in which little numbers can get astoundingly large has come to fascinate me. Maths has taken on a spiritual dimension. (As I say, that’s ‘spiritual’ in its widest sense, certainly far wider than conventional religion.) Example after example has appeared of the speed with which a seemingly minuscule number of elements can spiral out of control, certainly the degree of control achievable by a human.

Some of these relate to the human body itself, indeed the very process by which that human body is made. Sperm meets egg, cell is produced, cell divides into two, each of those divides into two, and so on. A fully-formed body contains a hundred trillion cells. (That’s a hundred thousand billion.) How many times does that initial cell have to sub-divide before you reach a hundred trillion? The answer is just 47. Imagine you’re standing at a photocopier. You make two copies of a document, then two copies of each of those copies ... Do that 47 times and you would have produced 100 trillion copies.

Other examples relate to the pursuits in which we humans engage. Chess, for instance. This is particularly satisfying, as chess is the game to which life is so often compared. The board measures just eight squares by eight; unlike the Rubik’s Cube it doesn’t even have a third dimension. Each player has a mere 16 pieces, all of which have to obey very specific rules about how they can move around the board. Given these restrictions, and the hundreds upon hundreds of years that mankind has been playing the game, you’d think that we’d have got everything mapped out by now, that every possibility which could occur on those 64 little squares would have occurred. But think again. Even on the very lowest estimates (yes, ‘estimates’ – we can’t even be sure of the exact number), the number of different possible games of chess is 10 to the power of 120 (10 with 120 zeroes after it). This is greater than the number of atoms in the observable universe. Pick the bones out of that, Bobby Fischer.

Or indeed Deep Blue. Surely computers – such as the ones that now defeat chess grandmasters – can help us ‘do the math’, evaluating the options in all sorts of life scenarios? Well, as the chess example shows, not necessarily. They can beat grandmasters. But they don’t always do so. And they certainly can’t cope with a list of possibilities numbering 10 to the 120. They can’t even help us with the ‘Travelling Salesman’ problem, as it’s known to mathematicians. This is the task of finding the shortest route between a given number of points. If the salesman has to visit three locations, the task is simple – there are only six possible routes. (This assumes that there is only one road between each point, which of course in the real world there won’t be, but let’s be kind to ourselves for once.) Add another location and you’ve got 24 routes. By the time you get to ten locations, the number of routes has shot up to 3,628,800. Fine, a computer could handle that. But 20 locations means more than 10 quadrillion routes; even the world’s most powerful computers will struggle. Just a few more appointments in the book and they’ll be completely stumped.

The most beautiful case of small numbers getting big, however, is the ‘Birthday Paradox’. This is the question of how many people you have to gather together in a room before it becomes more likely than not that two of them will share a birthday. What would your answer be? Many people opt for 183, as this is slightly more than half of 365. The correct answer, though, is 23. How can that be? How can such a small gathering make it odds-on that two people will have the same birthday?

The key to understanding it is to imagine that you yourself are the first person in the room. When the second person walks in, there is a 1 in 365 chance that they will share your birthday. The third person also has a 1 in 365 chance, doubling the overall odds to 2 in 365. But those are the odds on the other people sharing your birthday. And that’s not what we’re asking here. We’re asking whether any two people will share a birthday. We haven’t yet factored in the possibility that the second and third people could have been born on the same day.

If you prefer – and I do prefer this, because it gets the point across intuitively, without the need for complicated maths – you can think of the three people in the room as the three points of a triangle. It has three sides, representing the three connections between you, the three chances of a birthday being shared. When a fourth person walks in they make the shape a square. And a square doesn’t just have four sides, it also has two diagonals: six connections between you. Five people (a pentagon) means 10 connections. See how the numbers are getting big again? Now imagine 23 people in the room. Imagine the lines joining each one to all the others. You wouldn’t be able to move for lines. All of a sudden the answer seems all too plausible.

And that, in the end, is the icing on the mathematical cake of life’s happiness. Not only does an awareness of how complicated things can get with a tiny number of factors give a reminder of our powerlessness, of how unlikely it is that our plans will come off. It also reminds us how inter-connected our lives are with those of other people. Instead of resenting that fact we should be celebrating it. Instead of feeling annoyed that other people get in the way of our happiness, we should be glad they’re there at all.

So the next time you find yourself in a room with 22 other people, go and look for the one whose birthday you share. Maybe when that birthday comes round you can buy each other a Rubik’s Cube.