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