Date: Mar 24, 2004 6:39 PM
Author: Tim Smith
Subject: Re: Hex Win Proof?
In article <firstname.lastname@example.org>, 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 vertices
Yeah, 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, and
that curve has to satisfy all the general theorems about 2D curves.
The "unsafe" direction would be taking a general property of 2D curves, and
trying to apply that property to paths of hexes. E.g., a simple closed
curve has an inside and an outside, and the inside is connected. However, a
simple closed path of hexes doesn't necessarily have an inside, and if it
does, the inside is not necessarily connected--it can be pinched off into
multiple disconnected regions.