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Transworld is part of the Penguin Random House group of companies whose addresses can be found at global.penguinrandomhouse.com

First published in Great Britain in 2016 by Doubleday
an imprint of Transworld Publishers

This edition published in Great Britain by Black Swan
an imprint of Transworld Publishers

Dr Hannah Fry and Dr Thomas Oléron Evans have asserted their right under the Copyright, Designs and Patents Act 1988 to be identified as the authors of this work.

Every effort has been made to obtain the necessary permissions with reference to copyright material, both illustrative and quoted. We apologize for any omissions in this respect and will be pleased to make the appropriate acknowledgements in any future edition.

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INTRODUCTION

I wish it could be Christmas every day

The nights are drawing in, the crisp winter air is filled with the inviting smells of roasted chestnuts and mulled wine, and children everywhere are struggling to contain their excitement about Santa’s imminent arrival. It can only be Christmas – the most magical time of the year.

True, your overdraft is straining under the weight of your generosity, you’re obliged to spend hours writing heartfelt festive wishes to close friends that you’ve accidentally forgotten about for the rest of the year and your daily alcohol consumption barely ever dips below the recommended weekly limit, but you wouldn’t have it any other way.

With so much goodwill everywhere, it’s no wonder glam-rockers Wizzard wished it could be Christmas every day.

But let’s ponder that for a moment and run the numbers. What would it be like if we decided, as a nation, to move towards a System for Christmas Repeating Every day At Midnight (SCREAM for short)?

One big winner of SCREAM would be the UK’s largest tinsel factory, Festive Productions in Cwmbran, south Wales, who would see a substantial increase in turnover. Since we Brits normally keep our decorations up for around a month, we’d need to replenish our tinsel supplies 12 times as frequently as we do now. That means Festive Productions, who currently make 600,000 metres of tinsel a day,* would have to up their daily output to a whopping 7.2 million metres.

Of course, all that tinsel would be useless without a steady supply of Christmas trees to dangle it on. Currently, the UK buys 8 million trees a year, compared to 40 million in the USA and 42 million in the rest of Europe.* Since each tree needs seven years to grow from a sapling before it’s worthy of becoming the dazzling centrepiece of your family home, switching to SCREAM could result in a shortfall that would take some years to rectify.

However, given that there are 350 million fir trees being grown on Christmas tree farms in North America alone, if we ration ourselves to one new tree a month we’ll be OK until at least May or so.

After that, we can utilize some of the other 3.04 trillion trees around the world* while we’re waiting for the fir farms to replenish. True, it might not have that pine-fresh smell, but if you hack up a willow and slap a bit of tinsel on it, it should do for a few years. But choose your tree wisely. A 120-metre redwood poking out of your living-room window probably won’t go down too well with the council.

The daily festive feast could cause even more upheaval. Each year the British public buys and eats 10 million turkeys for our Christmas dinners. From hatching to table, turkeys live for 6 months, so around 1.8 billion turkeys would need rearing and housing at any one time to meet the SCREAM demand.

The University of New Hampshire* suggests each turkey should have at least 6 ft² of covered shelter. That translates into a giant turkey coop of 1,003 km², around two-thirds the size of Greater London.

As Britain is a nation of animal lovers we’re certain most supporters of SCREAM would insist on all turkeys being free-range, which means we’d also need to factor in 100 ft² of pasture for each bird. This translates into a colossal turkey farm of 17,726 km² that would cover all of London and the home counties and no doubt be the envy of the world.

Sure, we’d be losing our nation’s capital, financial institutions, parliament and 2,000 years of history, but think how much joy and Christmas cheer we’d stand to gain. Overall, it’s a small sacrifice to make.

The benefits don’t stop there either. Those turkeys are going to generate a lot of poo: 4 billion kg a year, in fact.* As you might expect, this brown gold serves as an excellent fertilizer, but more impressively it can also be used to fuel power stations. Under SCREAM, then, all our energy concerns will be sorted and the flowers in Kew Gardens will flourish (though the resident turkeys may not appreciate them as much as the human visitors once did).

If we decide not to give up our entire south-east, we could scatter the turkey farms around the country, using buildings we no longer have a need for, like schools, stadiums and shopping centres.

We’ll want to keep the hospitals, though. If we’re all eating the Christmas average of 6,000 calories a day, 3,500 calories more than any of us need, we’ll each have put on around 6 st 12 lb by the end of March.* The increased prevalence of obesity might make our hospitals busy, but it’s great news for the manufacturers of reinforced wider beds and elasticated trousers.

SCREAM won’t necessarily be stress-free for everyone. The EastEnders scriptwriters will have to concoct a brand-new dramatic hour-long special every day, and Santa may well go into meltdown after a matter of months. But think how great it will be for the children who wake up to find a stocking full of presents every single morning.

Granted, our GDP might slip a bit if almost everyone stopped going to work. There’s also the small matter of having nowhere to shop for the presents, 1.8 billion turkeys running amok and a generation of children growing up with no work ethic or sense of discipline. But let’s not get bogged down by trivialities like that.

On balance, we can see hardly any convincing arguments against SCREAM. Unfortunately, the ‘experts’ of the nation are not to be persuaded. We’re stuck with Christmas as an annual event, and we’re going to have to make the best of it.

So, to really extract every possible ounce of fun and joy from the day, there’s only one sensible way to plan your festivities: using mathematics.

How else but with mathematics could you work out how to time cooking your Christmas turkey to perfection? Or determine the best way to wrap your presents? Or design a flawless Secret Santa system? Or guarantee you beat your family in the annual board-game argument?

Some people might try to accuse us of taking the magic out of Christmas by reducing it to a set of mathematical equations, but for us the exact opposite is true.

We believe that mathematics is so powerful that it has the potential to offer a new way of looking at anything – even something as warm and wonderful as Christmas. Maths can uncover hidden patterns behind the familiar festivities and provide unique insights into how to really get the most out of your traditional celebrations. All of which, we think, adds up to make this time of year even more magical.

We hope to persuade you by showing you how to use mathematics to dress your tree with flawless precision. We’ll give you statistics to predict what the Queen is going to say in her Christmas message. We’ll even offer ultimate proof, if ever it were needed, of Santa Claus’s existence.

So curl up in front of the fire, pour yourself a large warm glass of mulled wine, pop on a CD of Cliff Richard’s greatest Christmas hits, and enjoy the merriest mathematics of Christmas.

INTRODUCTION: I wish it could be Christmas every day

CHAPTER 1

The indisputable existence of Santa Claus

It is astonishing that some people still doubt the existence of Santa Claus. Despite the vast amount of photographic evidence, the hundreds of annual reports on Father Christmas’s activities from perfectly reputable news sources and the bulging stockings full of presents that reliably appear on Christmas morning, somehow the doubters remain unconvinced.

The conspiracy theorists have already tried turning to science to demonstrate their (clearly incorrect) position. They calculate that if Santa were to visit the 1.9 billion children in the world,* he would have to travel at 3,000 times the speed of sound while carrying around 300,000 tonnes of presents* (about the weight of six Titanics). Richard Dawkins, king of the sceptics, has insisted that the lack of any noticeable sonic booms from all that zipping about at supersonic speeds is more than enough evidence that Santa cannot possibly be real.*

Worse still, some claim that this astonishing weight of parcels travelling at such a remarkable speed would practically vaporize the leading reindeer, who would have to withstand the full hit of air resistance. Meanwhile, sitting in the back of his sleigh, Santa would be subjected to forces tens of thousands of times stronger than gravity, making it impossible for him to breathe or to retain any of the physical structure of his bones or internal organs, thus reducing him to a liquefied mess. While this would admittedly explain how he was able to slip down some of the narrower chimneys on his route, it probably wouldn’t make for very attractive Christmas cards.

Sure, all these scientific spoilsports sound convincing enough. Although their arguments totally depend on the assumption that Santa isn’t a macroscopic quantum object capable of being in two places at once. And that he’s unable to manipulate time (though how else do they think he manages not to age in photographs?). And that he hasn’t constructed a NASA-style heat shield to protect his reindeer. Or invented a device to suppress sonic booms.

They also assume that any of these simple explanations is more far-fetched than the idea that the vast majority of the adult world is participating in a massive conspiracy, with parents cheerily lying to their children on behalf of a mystical non-existent figure, postal services fiendishly filtering out letters to Santa rather than returning them to sender as they normally would, and news agencies annually publishing blatant falsehoods that go against all their journalistic ethics, simply to maintain the whole pointless charade. Riiiggght. Sure.

The sceptical scientists’ arguments also illustrate an important point. The great difference between scientific and mathematical proof.

The scientific method takes a theory – in our case that Santa is real – and sets about trying to prove that it is false. Although this may seem a little counterintuitive on the surface, it actually does make a lot of sense. If you go out looking for evidence that Santa doesn’t exist and don’t find any … well then, that is pretty revealing. The harder you try, and fail, to show that Santa cannot exist, the more support you have for your theory that he must. Eventually, when enough evidence has been gathered that all points in the same direction, your original theory is accepted as fact.

Mathematical proof is different. In mathematics, proving something ‘beyond all reasonable doubt’ isn’t good enough. You have to prove it beyond all unreasonable doubt as well. Mathematicians aren’t happy unless they have demonstrated the truth of a theory absolutely, irrefutably, irrevocably, categorically, indubitably, unequivocally and indisputably. In mathematics, proof really means proof, and once something is mathematically true, it is true for ever. Unlike, say, the theory of gravity – hey, Newton?

If we want to silence the doubters once and for all we have to turn to mathematical proof.^*

So here we go then. Let’s use some mathematical logic to see if we can prove the indisputable existence of Santa Claus, starting with the statement below …

Back with us? Good.

Those two statements – statement 1 and statement 2 – are all we need to prove the existence of Santa.

If statement 1 is true in its claim that everything on the two pages is false, then Santa doesn’t exist, the sceptics win and we’re not looking good for Christmas this year.

But if everything on those two pages is false, then statement 1 must itself be false. But that’s a contradiction, because we just supposed that it was true.

That means statement 1 can’t possibly be true. And if it is not the case that ‘Everything here and everything here is false’, then at least one of the two statements must be true.

But hang on, we’ve already decided that statement 1 can’t be true because it contains a contradiction, so the true statement must be statement 2. Hence Santa exists!¹

Not convinced? OK, how about a different approach.

Who’s to say there’s only one Santa, after all? Let’s try and determine if ‘Santas’ are real. Some sort of secret society of Santas, perhaps, passing their festive baton from generation to generation. We don’t really care about the dead ones, of course, only the ones that exist now. And in terms of existing Santas, there are only two possible statements that can be made:

(1) An existing Santa exists.

(2) An existing Santa does not exist.

We can all agree that one of those must be true, right?

But hold on, statement (2) is contradictory. How can an existing Santa not exist? A ‘Jolly Santa’ must be jolly by definition, a ‘Portly Santa’ must be a bit on the chubby side and so an ‘Existing Santa’ has to exist.

That means that statement (2) must be false – but since we agreed that one of the two statements was true, the only logical conclusion is that (1) is true and so at least one Santa must exist.²

Still need more?

All right, one more go.

We can surely all agree that at least one of these two statements is true:

(1)
Santa exists.

(2)
1 + 1 = 2.

Well let’s play with the second of those a little.

Let’s put a = 1, b = 1.

Then:

a = b

a² = b²

squaring both sides

a² - b² = 0

taking b² away from both sides

(a - b) (a + b) = 0

factorizing^* the left-hand side

dividing by (a – b)

cancelling the (a – b) on the left-hand side

a + b = 0

tidying everything up

1 + 1 = 0

putting a = 1 and b = 1 back into the left-hand side

Whoa there a minute! 1 + 1 = 0?!

Well, if 1 + 1 = 0 and not 2 as we thought, then statement (II) is false.

But if we agreed that at least one of the two statements was true – and we did, there’s no backing out of it now – then if it isn’t (II), it must be (I). Hence Santa exists.³