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:Each player has an opposite pair of sides as before, but now some sides belong to neither player. The game is played the same way as before. It doesn't appear to be so interesting to play, but people have devised other higher dimensional versions which are probably more fun and interesting mathematically. In any case, in the version I described, there can never be a draw, and this no-draw result is equivalent to the Brouwer fixed point theorem. --[[User:C S|C S]][[User talk:C S| (Talk)]] 07:03, 15 January 2006 (UTC)
::Does the number of players remain 2? What exactly is the object on these higher dimensional grids? For example, in 3 dimensions, a "side" refers to a 2-dimensional face of the lattice, yes? If the object for both sides is to connect any point on one of your border faces with a point on the opposite border face via a continuous path, then the blocking nature of the game disappears, since both sides can achieve their object without having to block the other. If the lattice were filled with stones, most likely there would be multiple winning paths for both sides, which would be an illegal position, inasmuch as the game ends the moment either side achieves the objective. It may well be the case that a draw is impossible, but is this truly all you need to prove BFPT?
::Maybe the object on a 3D grid needs to be as follows: One player need only construct a path connecting his opposite faces as described, but the other player must '''block all possible paths by the opponent''', by building a membrane which stretches through the grid, connecting an entire path which crosses one of his border faces to a path across his other border face, with no holes. Of course this is an unfair game, but now it is impossible for both sides to achieve a winning path on a filled grid.--[[User:Twixter|Twixter]] ([[User talk:Twixter|talk]]) 00:56, 13 July 2012 (UTC)
== elemantary proof with stokes' theorem ==
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