#### Vol. 9, No. 2, 2016

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A variation on the game Set

### David Clark, George Fisk and Nurullah Goren

Vol. 9 (2016), No. 2, 249–264
##### Abstract

Set is a very popular card game with strong mathematical structure. In this paper, we describe “anti-Set”, a variation on Set in which we reverse the objective of the game by trying to avoid drawing “sets”. In anti-Set, two players take turns selecting cards from the Set deck into their hands. The first player to hold a set loses the game.

By examining the geometric structure behind Set, we determine a winning strategy for the first player. We extend this winning strategy to all nontrivial affine geometries over ${\mathbb{F}}_{3}$, of which Set is only one example. Thus we find a winning strategy for an infinite class of games and prove this winning strategy in geometric terms. We also describe a strategy for the second player which allows her to lengthen the game. This strategy demonstrates a connection between strategies in anti-Set and maximal caps in affine geometries.

##### Keywords
SET (game), combinatorics, finite geometry, cap
##### Mathematical Subject Classification 2010
Primary: 97A20, 51EXX
Secondary: 51E15, 51E22