In article <firstname.lastname@example.org>, Alex Malkis <email@example.com> 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.