  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.
