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Re: Hex Win Proof?
Posted:
Mar 23, 2004 10:17 PM
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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.
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