About five years ago, Jason Rosenhouse and Laura Taalman wrote a book, *Taking Sudoku Seriously*, that explored various mathematical aspects of that popular number puzzle. I got a lot of mileage out of that book — not only the pleasure of reading it in the first place, but also the use of it as a text for an honors seminar, a source of examples for a course in combinatorics, and, just a few weeks ago, its use as a source of information for a talk to my university’s Math Club. The book now under review strikes me as somewhat similar to the Rosenhouse and Taalman text: it does for the card game SET what the earlier book does for Sudoku. Like the Rosenhouse/Taalman book, it made for very enjoyable reading, and I fully expect it to be as useful in the future.

The game of SET, which can be played competitively or individually, is easy to describe. It is played with cards, on each one of which appears one, two or three symbols. Each symbol is either a diamond, oval, or squiggle shape; each is either red, green or purple; the interior of each one is either filled-in solid, left blank, or shaded. Thus, each card is characterized by four attributes (number, color, shape, interior), for each one of which there are three choices. A collection of three cards is called a *Set* if, for each of these four attributes, the three objects shown either all agree or are all different. So, for example, three cards, one showing a single purple oval whose interior is blank, one showing a single green diamond with solid interior, and one showing a single red squiggle with shaded interior, form a Set: they agree in one attribute (number), and disagree in the remaining three. Helpful color pictures can be found, among many other places, at http://www.setgame.com/set/puzzle, which also offers a daily puzzle that can be played online: 12 cards, containing within them six different Sets, are displayed, and the player is offered an opportunity to find all six while being timed by the computer.

Certain mathematical questions — How many possible cards are there? Assuming there is one card in the deck for each possibility, how many Sets can be made? If three cards are chosen at random, what is the probability that a Set is formed? — come readily to mind. In addition to questions like this (that would make great examples in classes on combinatorics or probability) there are other areas of mathematics that are touched on by the game. Somewhat less obviously, for example, SET turns out to be related to finite geometry (think of each card as a point, and a Set as a line with three points). I first learned about these geometric connections in a nice article by Davis and Maclagan (http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/set.pdf) but the book uder review is to my knowledge the first textbook that addresses them.

*The Joy of SET* addresses the mathematical issues discussed above, and many more. It starts with a preliminary chapter that talks briefly about the game and its history, and which poses a number of mathematical questions, including the ones above, suggested by the game. Some of these questions are answered immediately, but discussion of a number of the others is deferred until the next four chapters, which discuss, respectively, combinatorics, probability, modular arithmetic and geometry. No prior background in any of these areas is assumed; all the basic principles are developed from scratch as needed.

This is then followed by what the authors call an “interlude”: a brief description of strategies for the game, some of them intended to be humorous (“[S]tart hovering over whichever cards are closest to you. This will have the effect of both intimidating others as well as preventing them from seeing certain cards.”), as well as variations on the game.

The second half of the book revisits some of the mathematical topics covered in the first half, but at a somewhat greater level of sophistication, both in the mathematics and the style of exposition. (For example, in the first half of the book, use was made of imaginary conversations between people as a way of advancing the exposition. This device is not used in the second half of the book.) In addition to somewhat more demanding chapters on combinatorics, probability and statistics, linear algebra and geometry, there is also a chapter on computer simulations. Even in this part of the book, however, the authors have taken pains to make the text broadly accessible, although some prior background in college-level mathematics would be useful on occasion.

With the possible slight exception of what I thought was a tendency to overuse footnotes, I found this book to be a model of mathematical exposition. The quality of writing is consistently high: clear but not condescending, humorous, chatty, and a genuine pleasure to read. I have to confess to a certain amount of trepidation when I first saw the dialog device used in the first half of the book; I once reviewed another book for this column where I found the use of these made-up dialogs to be frequently irritating. But the authors here do a better job of things: the dialogs (one of which is between Euclid, Socrates and Theano) were better written and not grating — and I learned from them, because, if nothing else, I had never heard of Theano prior to reading this book.

Other useful pedagogical devices are the inclusion of both a good bibliography of books, journal articles and websites (although I did not see a website listed for this book, although one was referenced in the text) and exercises at the end of every chapter, with solutions provided at the back of the book. The exercises are interesting and fun. For example, in the very first exercise, the authors talk about how to label a card by using a number, followed by three letters to denote the attributes; the card with three red open squiggles, for example, would be denoted 3ROS. They then ask “what card has a code that forms the basis of many Western religions?” (The answer is “1GOD”.)

The authors (a husband and wife and their two daughters) state in the preface that they enjoyed writing the book, and I don’t doubt that for an instant: they are obviously enthusiastic about the subject and enjoy talking about it, and their enthusiasm translates nicely to the printed page. Also adding to the fun of reading the book is the inclusion of lots of multi-color pictures illustrating various combinations of SET cards. (There are color pictures of other things, as well, including a couple of photos of cats staring at a SET layout. As I said, the authors have a sense of humor.)

It’s always nice to find a book that, while largely comprehensible to a layperson, can also be read with profit and pleasure by a professional. I doubt it will be very long before I find something from this book to use in one of my classes, and I am already imagining what an interesting honors course could be based on it.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.