College Mathematics Journal

Vol48-N4

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UFOs in the game SET: Looking for Airplanes and Spaceships Jonathan Needleman and Felicia Sciortino Jonathan Needleman (needlejs@lemoyne.edu) is an associate professor of mathematics at Le Moyne College. He loves using recreational mathematics to excite people about the world of mathematics, and then, when they are least expecting it, showing them what they have been working on has connections to research mathematics. In his free time Needleman plays a lot of board games, but does not see this as any different from his day job. Felicia Sciortino (sciortfm@stu.lemoyne.edu) is a recent graduate from Le Moyne College in Syracuse, New York. While studying mathematics with minors in philosophy and biology, she kept herself busy with classes, club meetings, and work, but always found time for a game of SET. Since graduating, Sciortino cannot play as much as she used to, but chooses to spend her time as a Jesuit Volunteer at St. Charles Mission School in Pryor, Montana. In her free time, she enjoys reading, writing, and walking. The game SET is a fun and addictive pattern matching card game rich with mathemati- cal structure. Created by Marsha Jean Falco in 1974, the game has grown in popularity since it was first published in 1990. It is now played around the world and its rule book has been officially translated into 24 languages [6]. The game is based on the four attributes number, color, shading, and shape, where each attribute has three possible phenotypes. For example, color can be red, green, or purple. The game uses a special deck of 81 cards where each card represents a unique collection of four phenotypes. For a complete list of phenotypes see Table 1. In standard game play the objective is to find collections of three cards that form a set before your opponents. To determine if three cards are a set, examine the cards one attribute at a time. If three cards are either all the same color or are all different colors then the three cards are a color-set, that is "all or none, not two and one." A set is a collection of three cards which form an attribute-set for all four attributes. An example of a set is shown in Figure 1: The three cards are all different color, number, and shading, but all three have the same shape, diamonds. The mathematics of SET has been studied extensively. One of the first questions studied focused on how many cards are needed to guarantee the existence of a set. For a full deck there can be at most 20 cards without a set [5]. Interestingly, this result predates the game SET by looking at a geometry that happens to be related to SET. A more recent expository paper reproves this result using the language of the game [2]. In this geometry, lines happen to be sets. A 13-author article from this JOURNAL [1] introduces an object that lives in two dimensions, planets (plane-sets), which come http://dx.doi.org/10.4169/college.math.j.48.4.249 MSC: 97A20, 51E15 VOL. 48, NO. 4, SEPTEMBER 2017 THE COLLEGE MATHEMATICS JOURNAL 249

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