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Topic: Geometry of hex
Replies: 11   Last Post: Dec 18, 2002 3:34 PM

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Bill Taylor

Posts: 1,909
Registered: 12/8/04
Re: Geometry of hex
Posted: Dec 18, 2002 3:53 AM
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There have been many responses to this, it is a good question; and many of the
answers have been good answers. But no-one (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 continuous-path Brouwer
cases) or local geometry (in the Hex-game 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 4-sided 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
Each game is unique and this one is no different to any other.

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