How do you prove Santa Claus exists using mathematical logic? What’s an optimal algorithm for wrapping Christmas gifts? Does the heat equation give insights on how to best cook a turkey?

These are a few of the questions addressed in the book *The Indisputable Existence of Santa Claus: The Mathematics of Christmas* by Hannah Fry and Thomas Oléron Evans.

The new hardcover edition is out now, just in time for the holidays (the book was first published last year). Both Fry and Evans are mathematicians, and bring an equal measure of rigour and humour to the text. Fry frequently presents on BBC radio and television, working to popularize mathematics. She has a knack for bringing math to the masses: Her TEDx talk on the mathematics of love garnered over four million views.

Mathematics is not necessarily the first thing we think of over the holidays, or even the second, third or *n*^{th} thing (where *n* is an arbitrarily large positive integer).

For a decent proportion of the population, equations, calculations and geometry are things to avoid altogether. With all the gift buying, card sending and holiday parties with friends, family and co-workers, math may be the farthest thing from our minds.

## Math is everywhere

What the authors set out to do is to glide readers gently towards the deductive, quantitative mindset of a mathematician. They succeed in doing this by breaking the book into bite-sized chapters written in an engaging and accessible fashion. The hardcover edition is petite. It fits nicely in stockings.

The authors also know a trade secret in mathematics: The subject is everywhere. More importantly, math can be downright fun — if presented in the right way.

The topics are familiar but have a fresh, mathematical take. In one chapter, they tackle the problem of decorating a Christmas tree. Mathematics appears when they compute the optimal length of a garland based on whether the tree is either cylindrical shaped (the easier case) or conical (the tougher but more realistic case).

An application of platonic solids in a step-by-step guide leads to the creation of a 20-pointed stellation of the icosahedron — in plain language, a pretty, star-like Christmas ornament.

The applications of mathematics to Christmas discussed in the book are diverse — from the economics of gift buying, to game theory as a strategy for winning at Monopoly, to optimizing ways to cut a cake into even pieces.

Stressed out by Secret Santa, where co-workers buy each other randomly-assigned gifts? The theory of combinatorics and derangements gives the reader a fair way to operate Secret Santa, so no one ends up buying themselves a present.

One of my favourite parts of the work (and one that is decidedly tongue-in-cheek) is the application of two-step Markov chains to simulate the Queen’s annual Christmas speech.

The authors do this by studying the Queen’s past holiday elocutions and presenting simple, probabilistic rules for how she composes her sentences. The output is a vaguely familiar but nonsensical take on the Queen’s Christmas speech. However, as the authors point out “…it may make her sound like she’s overdone it on eggnog, but hey — isn’t that what Christmas is all about.”

Anyone with a basic grasp of mathematics at a post-high school level will find the book simple enough to follow. If you are a fan of irreverent, Monty Pythonesque humor, then this is for you.

No other mathematical background or sophistication is needed, but if the sight of an equation or derivation sends you flying up the chimney, then you might skip this one on your reading list. Precocious kids and teens will enjoy it, as will STEM geeks. Mathematicians are also happily accepting Christmas gifts.

## Proving the existence of Santa Claus

How do the authors prove the existence of Santa Claus? After all, Santa Claus is commonly considered a kids fairy tale; something that parents make up to convince their kids to behave or go to bed early.

To do so, they begin with a statement to consider: “Everything on this page and everything on page 152 is false….” Turning to page 152, you’ll find the statement: “Santa Claus exists….” These two statements are all you need to prove Santa Clause exists, according to the authors.

As this review is spoiler-free, you’ll have to read the book to get the full explanation. One hint is that the truth or falsity of self-referential statements traces back to the Liar paradox and Kurt Gödel’s incompleteness theorems.

After reading about all the applications of mathematics to Christmas, it’s evident to me that if Santa Claus exists, he loves mathematics.

Prove me wrong!