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 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.

--

--Tim Smith