Involve: A Journal of Mathematics

  • Involve
  • Volume 9, Number 2 (2016), 249-264.

A variation on the game Set

David Clark, George Fisk, and Nurullah Goren

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Involve, Volume 9, Number 2 (2016), 249-264.

Received: 13 October 2014
Revised: 29 January 2015
Accepted: 6 February 2015
First available in Project Euclid: 22 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 97A20: Recreational mathematics, games [See also 00A08] 51EXX
Secondary: 51E15: Affine and projective planes 51E22: Linear codes and caps in Galois spaces [See also 94B05]

SET (game) combinatorics finite geometry cap


Clark, David; Fisk, George; Goren, Nurullah. A variation on the game Set. Involve 9 (2016), no. 2, 249--264. doi:10.2140/involve.2016.9.249.

Export citation


  • T. Beth, D. Jungnickel, and H. Lenz, Design theory, Cambridge University Press, 1986.
  • M. T. Carroll and S. T. Dougherty, “Tic-tac-toe on a finite plane”, Math. Mag. 77:4 (2004), 260–274.
  • B. L. Davis and D. Maclagan, “The card game SET”, Math. Intelligencer 25:3 (2003), 33–40.
  • P. Dembowski, Finite geometries, Classics in Mathematics 44, Springer, Berlin, 1997.
  • G. Pellegrino, “Sul massimo ordine delle calotte in $S\sb{4,3}$”, Matematiche $($Catania$)$ 25 (1970), 149–157.
  • A. Potechin, “Maximal caps in ${\rm AG}(6,3)$”, Des. Codes Cryptogr. 46:3 (2008), 243–259.