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Re: Geometry of hex
Posted:
Dec 18, 2002 3:53 AM


There have been many responses to this, it is a good question; and many of the answers have been good answers. But noone (that I've noticed) has made the following observation, which is surprising in view of the references to Brouwer's theorem and other *topological* matters.
This question cannot be answered fully without noticing that there are both LOCAL and GLOBAL concerns to be addressed.
The LOCAL concerns are to do with local topology (in the continuouspath Brouwer cases) or local geometry (in the Hexgame and similar cases). No answer can be complete unless it speaks to this matter, in particular managing to distinguish between Hex games and similar chessboard games. That is a local matter.
But equally no answer can be complete unless it also addresses (as almost all do) the global topology, which is purely combinatoric. This will explain why there is always one winner and one loser in Hex, and in the game "Y"; but not in (say) the game on a 4sided Hex board but with "Y" rules  that the winner must touch all four sides (may be no winner); or where a player only has to touch ANY two opposite sides (may be "two winners"  merely a race game).
So, hopeful applicants must be sure to address both matters in their prospectus!
 Bill Taylor W.Taylor@math.canterbury.ac.nz  Each game is unique and this one is no different to any other. 



