```Date: Mar 23, 2004 10:17 PM
Author: Chan-Ho Suh
Subject: Re: Hex Win Proof?

In article <c3q50e\$19lq\$1@news.wplus.net>, Alex Malkis<alexloeschediesmalk@line.cs.uni-sb.de> wrote:> I heard there is a proof of some fixpoint theorem (Brouwer's fixpoint > theorem, maybe?) with the help of the hex game.> > Does anyone know?> > Best regards,> Alex.> > PS. To email me, remove "loeschedies" from the email address given.The Brouwer fixed point theorem is equivalent to the Hex theorem thatno game can end in a draw.  So the two dimensional Brouwer fixed point theorem is equivalent to thetwo dimensional Hex theorem, etc.  You can consider higher-dimensionalversions of Hex to make sense of this. There are several higher-dimenional variants, but the one you want forthis equivalence with Brouwer is to to consider the 2d Hex board asbeing a lattice with each square having a diagonal drawn in (make allthe diagonals the same).   Then the pieces are played on the latticepoints and while normally there would be four adjacent lattice pointsto a lattice point, since you drew in the diagonals, each point has sixneighbors.   To make a 3D Hex board, you would draw a 3D lattice made of cubes anddraw a diagonal in each cube going across the cube, making sure youdraw the same diagonal in each cube.  And so on and so forth for higherdimensions.
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