# The Joy of SET

SET is a successful card game that is commercialized since 1990 and it is all about pattern recognition. The game has 81 cards. Each card has symbols with four different attributes: (1) a number (1, 2, or 3 identical symbols). (2) a shape (oval, squiggle, diamond), (3) a colour (red, green purple), and (4) a shading (empty, striped, solid). A set of 3 cards forms a SET if for each attribute the cards have either the same or all different values. For example 3 cards described by the quadruples (1 oval red empty), (1 oval green striped), and (1 oval purple empty) form a SET with number and shape the same while color and shading are all different. Clearly if number, shape, color, and shading are represented by digits 0,1,2, then the 81 cards can be described as vertices on a 3 x 3 x 3 x 3 grid by 4 coordinates. For example the previous SET example could be (0,0,0,0), (0,0,1,1), and (0,0,2,2).

The game starts by putting 12 cards, face up on the table. When a participant is first to detect a SET, he or she can take the 3 cards of the SET and they are replaced by 3 new ones. The winner is the one who collected the most SETs.

It is clear that with this structure, there are a lot of combinatorial problems that can be asked. This is exactly what the authors of this book do. Solving these questions may not help to play the game better though. You may only become a better player, i.e., faster in recognizing the SET patterns, with a lot of experience and training. So the combinatorics and the statistics that are needed to answer the questions or solve the exercises are mainly for the fun of the mathematics, and possibly for the fun of learning new mathematical structures.

Possible problems involve the number of possible SETs in the whole card deck, the probability of having 0, 1, 2, etc. SETs in the initial 12 cards, the maximal number of cards on the table before there must be at least one SET in it, the probability to end the game with a certain number of cards on the table that does not contain a SET, the number of intersets (if 5 cards form 2 SETs with one card in common, then an interset is the set of 4 cards obtained by removing the common card), the number of planes (a plane is a 3 x 3 grid of cards like a magic square where each row, column or (modulo 3) diagonal forms a SET), etc. Clearly this suggests to introduce combinatorics, modulo 3 arithmetic, probability, and finite affine geometry. These topics allow to ask questions well outside the SET game. Thus the reader will benefit by learning in a playful way several mathematical topics such as finite geometry, combinatorics, linear algebra, including vector spaces, and even error correcting codes. Conversely, in some sense also the mathematics benefit from it, because finite geometry properties can be discovered that one would not have cared for, if not for the card game. On the other hand, it will increase the fascination for the game, (and the authors of this book certainly are fascinated) if you realize what mathematical structures and problems can be discovered underneath.

The book has two main parts. The first is relatively elementary and covers the kind of questions that were mentioned above. The material is presented via a trialogue among 3 characters discussing the questions and the solutions in the tradition of Socrates. There are also exercises with answers given in an appendix and there extra projects (without solution) usually asking to prove certain results.

After part 1, there is a short interruption with some advise on how to improve your skill in playing the game. The main advise is however, as I mentioned above, to play it as much as possible. So the authors move quickly on to the second part which is basically a continuation of part 1, but the dialogues are now left out and some of the previous questions and problems are reprised and perhaps somewhat extended. Also questions that are more involved are tackled, and the reader is exposed to q-binomials, hyperplanes, more involved statistics and probability questions, central limit theorem, normal distribution, vector spaces, matrices and matrix-vector product, parallel and orthogonal planes, Hamming weights and error correcting codes, affine transformations, and the projective plane. As one gets away from the actual game and starts asking questions on a more mathematical level, it gets more interesting of course. Finally there are some questions that do not allow to compute the answer exactly. In those cases one has to fall back on computers and simulate the problem to find the probability that such or such answer will hold. As in the first part there is ample possibility after each chapter to solve exercises, engage in projects, or run simulations.

The authors used a lot of witty puns in their presentation of the mathematics. For example the triples of characters in the first part having their conservation have names that start with S, E, and T and they are varying in each chapter. They reflect on the fact that they are in a book and can speak signs and symbols like * and © and all the formulas, until they realize that this is just because they say what is printed and not the other way around. When the second triplet is starting in chapter 2, they wonder what happened to the characters from the previous chapter, and then they realize, they were removed because they formed a SET, etc. The mathematics have derivations and proofs, but these are kept really low profile, not at all boring, abstract or formal. The subtitle of the book is *The many mathematical dimensions of a seemingly simple card game*, which of course is well reflecting the content, but actually a more accurate title (but less appealing to potential buyers) could have been: *The joy of learning mathematics inspired by the game SET*.

**Submitted by Adhemar Bultheel |

**26 / Dec / 2016