David Gale's subset take-away game
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.
Cite
@article{arxiv.math/0207185,
title = {David Gale's subset take-away game},
author = {J. Daniel Christensen and Mark Tilford},
journal= {arXiv preprint arXiv:math/0207185},
year = {2016}
}
Comments
6 pages. v2 corrects a typo, updates the bibliography, and improves the figures