Evans Thomas Oleron - The indisputable existence of Santa Claus the mathematics of Christmas

Dr Hannah Fry is a mathematician from University College London. In her day job she uses mathematical models to study patterns in human behaviour, from riots and terrorism to trade and shopping. Youll also find her on BBC radio and television, where she regularly presents science programmes, or on YouTubes Numberphile channel.

Dr Thomas Olron Evans describes himself as a mathematician and a writer, though how others would describe him is anyones guess. He used to be a teacher, but managed to escape and now has a research and lecturing job in London. While broadly agreeable, this does make life a little complicated, since he lives in Strasbourg, France.

For more information about the authors, visit their websites, www. hannahfry .co.uk and www. mathistopheles .co.uk, or follow them on Twitter @FryRsquared and @mathistopheles .

How do you apply game theory to select who should be on your Christmas shopping list?

Can you predict Her Majestys Christmas message?

Will calculations show that Santa is getting steadily thinner, shimmying up and down chimneys for a whole night, or fatter, as he tucks into a mince pie and a glass of sherry in billions of houses across the world?

A dazzling, magical mathematical tour complete with graphs, diagrams and sketches, The Indisputable Existence of Santa Claus will brighten up the bleakest midwinter with a stockingful of elegant solutions to all your Christmas conundrums.

Maths has never been merrier.

Our little book had some big help from a number of wonderful people: Susanna Wadeson and the whole team at Transworld, Claire Conrad from Janklow & Nesbit, Gillian Somerscales, Julia Lloyd, Lis Adlington and Anna Gregson. Thanks too to Andy Hudson-Smith and the team at CASA for their continued support.

Hannah: Thanks to Tracy, Natalie, Marge & Parge, Mr P, Miss McGee and mini-P-to-be. Im very lucky indeed to have you on my team.

Thomas: Thanks to my fantastic family Emily, Gareth and Catherine Evans for all manner of reasons, and to the wonderful Emilie Olron Evans, for very graciously sharing me with Santa over the past few months. Thanks also to Dennis Barnett and Elizabeth Swan, two tremendous maths teachers, who had a very significant influence on my life and the lives of so many others. Finally, back in 1996, I promised to dedicate my first book to my uncle. Im finally able to make good on that promise, so my share of this text is dedicated to Michael Evans; thanks for your imagination and for all those Target novelizations.

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 Christmass activities from perfectly reputable news sources and the bulging stockings full of presents that reliably appear on Christmas morning, somehow the doubters remain unconvinced.

Thankfully mathematics can help.

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,

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 wouldnt make for very attractive Christmas cards.

Sure. All these scientific spoilsports sound convincing enough. Although their arguments totally depend on the assumption that Santa isnt a macro-scopic quantum object capable of being in two places at once. And that hes unable to manipulate time (though how else do they think he manages not to age in photographs?). And that he hasnt 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 doesnt exist and dont 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 isnt good enough. You have to prove it beyond all unreasonable doubt as well. Mathematicians arent happy unless they have demonstrated the truth of a theory absolutely, irrefutably, irrevocably, categoric-ally, 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. Lets use some mathematical logic to see if we can prove the indisputable existence of Santa Claus, starting with the statement

Back with us? Good.

Those two statements the one are all we need to prove the existence of Santa.

If the statement is true in its claim that everything on the two pages is false, then Santa doesnt exist, the sceptics win and were not looking good for Christmas this year.

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

That means the statement is false, then at least one of the two statements must be true.

But hang on, weve already decided that the statement

Not convinced? OK, how about a different approach.

Whos to say theres only one Santa, after all? Lets try and determine if Santas are real. Some sort of secret society of Santas, perhaps, passing their festive baton from generation to generation. We dont 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: