A quite different proof given by [[David Gale]] is based on the game of [[Hex (board game)|Hex]]. The basic theorem aboutregarding Hex, first proven by John Nash, is that no game of Hex can end in a draw.; the Thisfirst player always has a winning strategy (although this theorem is nonconstructive, and explicit strategies have not been fully developed for board sizes of dimensions 10 x 10 or greater). This turns out to be equivalent to the Brouwer fixed-point theorem for dimension 2. By considering ''n''-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the [[determinacy]] theorem for Hex.<ref>{{cite journal|author=David Gale |year=1979|title=The Game of Hex and Brouwer Fixed-Point Theorem | journal=The American Mathematical Monthly | volume=86 | pages=818–827|doi=10.2307/2320146|jstor=2320146|issue=10}}</ref>
===A proof using the Lefschetz fixed-point theorem===
