## Exploring the Mathematics of the Card Game Set

Posted on September 28, 2013 by Ivars Peterson

Contemplation by a team of mathematics teachers of the four-dimensional structure of the game Set has given rise to some intriguing mathematical research worthy of publication in a mathematics journal—and a novel variant of the fast-paced pattern-recognition game.

A Set deck has 81 cards, each of which displays a design with four attributes: shape, number, shading, and color. Each attribute has three possible values. For example, a card may display one, two, or three identical shapes, and those shapes may be red, green, or purple.

A player’s object in a traditional game of Set is to pick out from 12-card arrays groupings of three cards that qualify as sets. Three cards are a set if, with respect to each of the four attributes, the cards are either all the same or all different.

Invented in 1974 by population geneticist Marsha Jean Falco, Set has not only become a popular pastime for both children and adults but also attracted mathematical attention (see “SET Math“).

An addition to the Set-related mathematics literature appears in the September 2013 issue of The College Mathematics Journal (CMJ). “Sets, Planets, and Comets” begins with the observation that sets in the game of Set can be considered lines in a certain four-dimensional space. The authors—members of the American Institute of Mathematics Math Teachers’ Circle—then introduce planes into the game, and a new variant—Planet—is born.

The primary difference between Planet and Set is that players are not restricted to claiming only sets; they may also identify and take planets or comets—four-card and nine-card groupings, respectively, that are defined in the paper.

The authors observe that Planet “proceeds at a more leisurely pace than the typical frenzy of a Set game” and note that the rule changes help to level the playing field between Set novices and experts. They also point out that, unlike in Set, additional cards never need to be dealt in Planet. Cards removed from the original array of nine are replaced, but the nine cards visible at any time will always contain a set, a planet, or a comet. The authors verified this with a computer search.

Brian Conrey, executive director of the American Institute of Mathematics, led the meeting of the Math Teachers’ Circle from which “Sets, Planets, and Comets” grew. He posed an open question for the group to tackle and marveled as his eventual co-authors—many of them middle and high school math teachers—dug in with gusto. Conrey sees the resulting paper as an illustration of the approachability of mathematics.

“There are plenty of good, unsolved problems that anyone can work on,” he says. “You don’t necessarily need specialized training in mathematical research so much as a willingness to try something and to be persistent.”

And games, Conrey says, are fertile ground for interesting and accessible questions: “Games invite variations, extensions, and new inventions.”

Of that at least, Planet is proof positive.–Katharine Merow