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David Gale's subset take-away game

J. Daniel Christensen·Mark Tilford
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Abstract

Subset take-away is a two-player game involving a fixed finite set A. Players alternate choosing a proper, non-empty subset of A, with the condition that one may not name a set containing a set that was named earlier. A player unable to move loses. It was conjectured by David Gale that this game is always a second player win, and this was known to hold if A has no more than 5 elements. In this paper, we describe a technique called "binary star reduction" that often allows one to dramatically reduce the complexity of a position. Using this tool and some computer search we show that Gale's conjecture holds when A has six elements. We also show how this game can be interpreted geometrically.

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J. Daniel Christensen, Mark Tilford. 2016-05-04. David Gale's subset take-away game. https://arxiv.org/abs/math/0207185

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