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There is also an almost elementary [[combinatorial proof]]. Its main step consists in establishing [[Sperner's lemma]] in ''n'' dimensions.
A quite different proof can be given based on the game of [[Hex_(board_game)|Hex]]. The basic theorem about Hex is that no game can end in a draw. This is equivalent to the Brouwer fixed point theorem for dimension 2. By considering ''n''-dimensional versions of Hex, one can prove that in general that Brouwer's theorem is equivalent to the "no draw" theorem for Hex.
== Generalizations ==
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