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


In article <c3q50e$19lq$1@news.wplus.net>, Alex Malkis <alexloeschediesmalk@line.cs.unisb.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 higherdimensional versions of Hex to make sense of this.
There are several higherdimenional 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.

