```Date: Mar 24, 2004 6:39 PM
Author: Tim Smith
Subject: Re: Hex Win Proof?

In article <c3ropb\$2fmjnb\$1@athena.ex.ac.uk>, Robin Chapman wrote:>> I'd be suspicious of any use of well-known curve theorems without going>> over their proofs and making sure they apply to paths on the Hex board,>> because a path on the Hex board can, without intersecting itself, close>> off a region of the board.> > I don't see that this is relevant. One replaces the path of pieces on the> hex board by a curve built from line segments joining the centres of the> hexagons in question. These paths are between verticesYeah, you're right.  For what we need here, that's going in the "safe"direction.  That is, take a curve derived from the placement of hexes, andthat curve has to satisfy all the general theorems about 2D curves.The "unsafe" direction would be taking a general property of 2D curves, andtrying to apply that property to paths of hexes.  E.g., a simple closedcurve has an inside and an outside, and the inside is connected.  However, asimple closed path of hexes doesn't necessarily have an inside, and if itdoes, the inside is not necessarily connected--it can be pinched off intomultiple disconnected regions.-- --Tim Smith
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