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Topic: Hex Win Proof?
Replies: 41   Last Post: Mar 24, 2004 6:39 PM

 Messages: [ Previous | Next ]
 Chan-Ho Suh Posts: 425 Registered: 12/10/04
Re: Hex Win Proof?
Posted: Mar 23, 2004 10:17 PM

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.

Date Subject Author
3/18/04 Bill Taylor
3/18/04 Tim Brauch
3/19/04 Brian Chandler
3/19/04 Jonathan Welton
3/19/04 Tim Brauch
3/19/04 Richard Henry
3/20/04 Chan-Ho Suh
3/21/04 Arthur J. O'Dwyer
3/19/04 Bob Harris
3/19/04 Tim Smith
3/19/04 Dvd Avins
3/20/04 Nate Smith
3/20/04 Chan-Ho Suh
3/20/04 G. A. Edgar
3/19/04 Richard Henry
3/19/04 Steven Meyers
3/20/04 Nate Smith
3/20/04 Larry Hammick
3/20/04 Tim Smith
3/21/04 Steven Meyers
3/22/04 Torben Mogensen
3/22/04 Chan-Ho Suh
3/22/04 Torben Mogensen
3/22/04 Chan-Ho Suh
3/23/04 Torben Mogensen
3/23/04 Robin Chapman
3/23/04 Chan-Ho Suh
3/24/04 Robin Chapman
3/24/04 Tim Smith
3/24/04 Robin Chapman
3/24/04 Tim Smith
3/24/04 Jon Haugsand
3/22/04 Andrzej Kolowski
3/23/04 Alexander Malkis
3/23/04 Chan-Ho Suh
3/23/04 Dr. Eric Wingler
3/24/04 Danny Purvis
3/24/04 Danny Purvis