## Book reviews: The Joy of SET and Elements of Mathematics

I have a couple of book reviews for you today! I finished both of these books recently and really enjoyed them. Though they are quite different, both gave me new ways to think about some topics I already knew, and in particular helped me make new connections between elementary and advanced concepts.

[Disclosure of Material Connection: Princeton Press kindly provided me with free review copies of these books. I was not required to write a positive review. The opinions expressed are my own.]

# The Joy of SET

Liz McMahon, Gary Gordon, Hannah Gordon, and Rebecca Gordon
Princeton University Press, 2016

Most people are probably familiar with the card game SET: each card has four attributes (number, color, shading, shape) each of which can have one of three values, for a total of $3^4 = 81$ cards. The goal is to find “sets”, which consist of three cards where each attribute is either identical on all three cards, or distinct on all three cards. It’s a fun game, and because it has to do with combinations of things and pattern recognition, many people probably have the intuitive sense that it’s a “mathy” sort of game, or the sort of game that people who enjoy math would also enjoy

Well, it turns out, as the authors convincingly demonstrate, that the mathematics behind SET actually goes very deep. For example, did you know that there are exactly $3^{n-1}(3^n - 1)/2$ distinct SETs in an $n$-dimensional version of the game? (The normal game that everyone plays has $n = 4$.) How about the fact that the SET deck is a concrete model of the four-dimensional affine geometry $AG(4,3)$? Did you know that the most cards you can have without a SET is 20, and that this is intimately connected to structures called maximal caps in affine geometries—and that no one knows how many cards you could have without a SET in a $7$-dimensional (or higher) version of the game?

The authors explain all this, and much more (with a lot of humor1 along the way!), ranging through probability, modular arithmetic, combinatorics, geometry, linear algebra, and a bunch of other topics. The book begins gently, but by the end it gets into some fairly deep mathematics, and there are lots of exercises and projects at the end of each chapter. This book would make a fantastic resource for a middle school, high school, or undergraduate math club. I could even see using it as the textbook for some sort of extra/special topics class with some motivated students.

# Elements of Mathematics

John Stillwell
Princeton University Press, 2016

I am a huge fan of Stillwell’s writing (almost six years ago I wrote a short review of another one of his books, Roads to Infinity) and I wasn’t disappointed. This book is definitely aimed at a more sophisticated audience than the SET book, but due to Stillwell’s lucid explanations it still manages to start out rather gently and holds many treasures even for the intrepid high school reader.

The book has two basic goals. The first is to simply lay out an overview of “elementary” mathematics, accessible in theory to anyone with a high school level mathematical background. “Elementary” mathematics refers not just to the sort of mathematics learned in grade school (arithmetic, fractions, and so on) but to the mathematics that would nowadays be viewed as “basic” by professional mathematicians—the sort of stuff that every professional mathematician is familiar with regardless of their specialty. In this respect the book is quite a tour de force, organized by areas of mathematics—arithmetic, computation, algebra, geometry, calculus, and so on—and in each area Stillwell manages to distill down the big ideas and the connections with other areas. He is a master expositor, and the text manages to be engaging and accessible without watering down the mathematics. I definitely learned new things from the book! One thing Stillwell does very well in particular is to explain not just the big ideas but the connections between them.

The other basic goal of the book is to explore the boundary between “elementary” and “advanced” mathematics. This sounds like it would be rather vague and amorphous—after all, aren’t the notions of “elementary” and “advanced” quite relative? Doesn’t it depend on how much background you have? Can’t math that is “elementary” to one person be “advanced” to someone else? This is all true, but Stillwell isn’t really talking about which areas of math are hard and which are easy. Professional mathematicians often talk about certain proofs being “elementary”, and it is often celebrated when someone finds an “elementary” proof of a theorem, even if that theorem had already been proved by “non-elementary” means, and even if the non-elementary proof was shorter. Stillwell is trying to pin down a precise meaning of this sense of “elementary”, and makes a well-reasoned case that it all comes down to infinity: something is non-elementary precisely when infinity enters into its proof in a fundamental way. This may seem rather arbitrary at first blush, but through a number of examples and surprising connections between different areas of mathematics, Stillwell makes it clear that this is an extremely “natural” place to draw a line in the sand. Not that having such a dividing line is in and of itself of any value—it’s simply fascinating to note that there is such a natural line at all, and by exploring it in depth we shed new light on the mathematics to either side of it.

1. They are extremely fond of footnotes. Reminds me of someone I know.