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Topic: Hex Win Proof?
Replies: 41   Last Post: Mar 24, 2004 6:39 PM

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Robin Chapman

Posts: 3,289
Registered: 12/6/04
Re: Hex Win Proof?
Posted: Mar 24, 2004 5:43 AM
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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 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_



Date Subject Author
3/18/04
Read Hex Win Proof?
Bill Taylor
3/18/04
Read Re: Hex Win Proof?
Tim Brauch
3/19/04
Read Re: Hex Win Proof?
Brian Chandler
3/19/04
Read Re: Hex Win Proof?
Jonathan Welton
3/19/04
Read Re: Hex Win Proof?
Tim Brauch
3/19/04
Read Re: Hex Win Proof?
Richard Henry
3/20/04
Read Re: Hex Win Proof?
Glenn C. Rhoads
3/20/04
Read Re: Hex Win Proof?
Chan-Ho Suh
3/21/04
Read Re: Hex Win Proof?
Arthur J. O'Dwyer
3/19/04
Read Re: Hex Win Proof?
Bob Harris
3/19/04
Read Re: Hex Win Proof?
Tim Smith
3/19/04
Read Re: Hex Win Proof?
Dvd Avins
3/20/04
Read Re: Hex Win Proof?
Nate Smith
3/20/04
Read Re: Hex Win Proof?
Chan-Ho Suh
3/20/04
Read Re: Hex Win Proof?
G. A. Edgar
3/19/04
Read Re: Hex Win Proof?
Richard Henry
3/19/04
Read Re: Hex Win Proof?
Steven Meyers
3/20/04
Read Re: Hex Win Proof?
Nate Smith
3/20/04
Read Re: Hex Win Proof?
Larry Hammick
3/20/04
Read Re: Hex Win Proof?
Tim Smith
3/20/04
Read Re: Hex Win Proof?
Glenn C. Rhoads
3/20/04
Read Re: Hex Win Proof?
Glenn C. Rhoads
3/21/04
Read Re: Hex Win Proof?
Steven Meyers
3/22/04
Read Re: Hex Win Proof?
Glenn C. Rhoads
3/22/04
Read Re: Hex Win Proof?
Torben Mogensen
3/22/04
Read Re: Hex Win Proof?
Chan-Ho Suh
3/22/04
Read Re: Hex Win Proof?
Torben Mogensen
3/22/04
Read Re: Hex Win Proof?
Chan-Ho Suh
3/23/04
Read Re: Hex Win Proof?
Torben Mogensen
3/23/04
Read Re: Hex Win Proof?
Robin Chapman
3/23/04
Read Re: Hex Win Proof?
Chan-Ho Suh
3/24/04
Read Re: Hex Win Proof?
Robin Chapman
3/24/04
Read Re: Hex Win Proof?
Tim Smith
3/24/04
Read Re: Hex Win Proof?
Robin Chapman
3/24/04
Read Re: Hex Win Proof?
Tim Smith
3/24/04
Read Re: Hex Win Proof?
Jon Haugsand
3/22/04
Read Re: Hex Win Proof?
Andrzej Kolowski
3/23/04
Read Re: Hex Win Proof?
Alexander Malkis
3/23/04
Read Re: Hex Win Proof?
Chan-Ho Suh
3/23/04
Read Re: Hex Win Proof?
Dr. Eric Wingler
3/24/04
Read Re: Hex Win Proof?
Danny Purvis
3/24/04
Read Re: Hex Win Proof?
Danny Purvis

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