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 ÃÂÃÂÃÂÃÂ¸ÃÂÃÂidius 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 stepwise

and 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_