> In article <email@example.com>, 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 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 of the equilateral triangle lattice. If both sides "won" we would get two such paths, not crossing with endpoints in order blue, red, blue, red on the outer boundary. We should be able to prove that the system containing the board with marked blue path is homeomorphic to the corresponding system with a straight blue path, by a stepwise process which I don't have the patience to fully explain, but would flatten out 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_