```Date: Mar 24, 2004 3:41 AM
Author: Robin Chapman
Subject: Re: Hex Win Proof?

Chan-Ho Suh wrote:> In article <c3p0bk\$2fl7ms\$1@athena.ex.ac.uk>, Robin Chapman> <rjc@ivorynospamtower.freeserve.co.uk> wrote:> >> Torben ÃÂÃÂÃÂÃÂ¸ÃÂÃÂidius Mogensen wrote:>> >> > >> > Indeed, if we want to prove it to a mathematician who does not already>> > accept the intermediate value theorem (of which the intersection>> > property is a simple consequence).>> >> Is it?>> >> Now I presume this "intersection property" can be paraphrased as>> "a path with endpoints at two opposite vertices of a square with>> all other points in the interior of the square must meet a path>> with endpoints at the other two vertices of the square with>> all other points in the interior of there square".>> >> That's a simple consequence of the intermediate value theorem, is it?>> >> I must be stupid, since the only way I can see to prove that is>> using the Jordan Curve Theorem. :-(> > I don't think you're being stupid, unless I'm being stupid also :-)> > 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 stepwiseand end us with a nice "theta" shape which we can apply the IVT to. > I'm very skeptical of this, because if you could show the paths must> intersect with the intermediate value theorem, then I think you have a> proof of Jordan separation with just a little extra work.  So this> would be a much simpler proof than I've ever seen of that.-- 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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