```Date: Mar 24, 2004 5:43 AM
Author: Robin Chapman
Subject: Re: Hex Win Proof?

Tim Smith wrote:> In article <c3rhkr\$289gvb\$2@athena.ex.ac.uk>, Robin Chapman wrote:>>> I can't see a way to prove this without Jordan separation.  It's not>>> just>>> a matter of the intermediate value theorem.  If one path can be>>> straightened out, then one can apply the intermediate value theorem, but>>> saying that you can straighten out a path is essentially the content of>>> the Jordan curve theorem.>> >> More than that --- it's almost the Schoenflies theorem.  On the other>> hand, if one is dealing with a path on a lattice, like we are doing here,>> then one can do the straightening stepwise and end us with a nice "theta">> shape which we can apply the IVT to.> > 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 pieceson the hex board by a curve built from line segments joiningthe centres of the hexagons in question. These paths are between verticesof the equilateral triangle lattice. If both sides "won" we would gettwo such paths, not crossing with endpoints in order blue, red, blue,red on the outer boundary. We should be able to prove that the systemcontaining the board with marked blue path is homeomorphic tothe corresponding system with a straight blue path, by a stepwise processwhich I don't have the patience to fully explain, but would flattenout an "ear" on the blue path at each stage.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"Francis Wheen, _How Mumbo-Jumbo Conquered the World_
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