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The theorem is equivalent to the determinacy of the Hex game. That's a very famous `application'.

The details can be found in [David Gale (1979). "The Game of Hex and Brouwer Fixed-Point Theorem". The American Mathematical Monthly 86: 818–827], a beautiful paper, which JSTOR serves at http://www.jstor.org/stable/2320146.

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The theorem is equivalent to the determinacy of the Hex game. That's a very famous `application'.