Date: Mar 24, 2004 5:15 AM
Author: Tim Smith
Subject: Re: Hex Win Proof?
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.