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 that
no game can end in a draw.

So the two dimensional Brouwer fixed point theorem is equivalent to the
two dimensional Hex theorem, etc. You can consider higher-dimensional
versions of Hex to make sense of this.

There are several higher-dimenional variants, but the one you want for
this equivalence with Brouwer is to to consider the 2d Hex board as
being a lattice with each square having a diagonal drawn in (make all
the diagonals the same). Then the pieces are played on the lattice
points and while normally there would be four adjacent lattice points
to a lattice point, since you drew in the diagonals, each point has six
neighbors.

To make a 3D Hex board, you would draw a 3D lattice made of cubes and
draw a diagonal in each cube going across the cube, making sure you
draw the same diagonal in each cube. And so on and so forth for higher
dimensions.