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

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 F3, 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.

SET (game), combinatorics, finite geometry, cap
Mathematical Subject Classification 2010
Primary: 97A20, 51EXX
Secondary: 51E15, 51E22
Received: 13 October 2014
Revised: 29 January 2015
Accepted: 6 February 2015
Published: 2 March 2016

Communicated by Kenneth S. Berenhaut
David Clark
Department of Mathematics
Grand Valley State University
1 Campus Drive
Allendale, MI 49401
United States
George Fisk
Department of Mathematics
University of Minnesota
Minneapolis, MN 55455
United States
Nurullah Goren
Department of Mathematics
Pomona College
Claremont, CA 91711
United States