```Date: Mar 21, 2004 3:43 PM
Author: Arthur J. O'Dwyer
Subject: Re: Hex Win Proof?

On Sat, 20 Mar 2004, Chan-Ho Suh wrote:>> Tim Brauch <RnEeMwOs.pVoEst@tbrauch.cNOoSPAMm> wrote:> > j_welton@hotmail.com (Jonathan Welton) wrote...> > >> > > Neither of the proofs (which are basically the same) posted so far is> > > correct. Both would apparently conclude that a winning path would be> > > formed on a squared board, whereas this is not the case - a squared> > > board could end in a draw.> > >> > > An actual proof must use the hex nature of the board or,> > > alternatively, that 3 cells meet at each vertex. A proof is given in> > > Cameron Browne's book Hex Strategy, but whether it would convince an> > > intelligent layman is not clear.> > >> > > Maybe a simpler proof could be achieved by induction?  Almost certainly not.  Induction doesn't usually lead to clear proofs,the way "pouring water" does.  And in this case, it won't lead to ashort-but-obscure proof either: you'll get a long-and-obscure proof.> > I wasn't assuming a square board, I was imagining the board set up like> > a parallelogram.  At least, that is how I orientate the board when I> > play.  Then red goes top to bottom and blue goes left to right (red and> > blue because the board I made uses poker chips).>> What Jonathan is trying to point out is that you aren't using the fact> that there are hexagons.  If you took a checkerboard and squished it to> form a parallelogram (with angles not 90 degrees), then you would have> a board where every piece of the board looked like a little> parallelogram (instead of a hexagon).  Clearly we can color this> checkerboard without a winning path by the usual checkerboard coloring.  Both Tim's proof and Brian's proof assume that the reader knows whata hexagon is; how many sides it has; and how it differs from a square.I don't see what you're objecting to.> Your proof attempt makes no use of the specifics of the Hex board, and> so would apply to any board like the one above.  But it *does* make use of the specific topology of the Hex board.If it didn't, then (as you noted) it would prove a falsehood.  Sinceit does not prove a falsehood (which is impossible; falsehoods cannotbe proven), it must use the topology of the board.Q.E.D.  Bad analogy: I can prove that the speed at which a tennis ball hitsmy hand going down is the same speed at which it left my hand goingup, using a simple argument from conservation of energy.  You object:oh, but that proof is flawed, because what if there was a rocket engineattached to the tennis ball?  I respond: any fool can tell that thereis *not* a rocket engine attached to the tennis ball, so why would youeven think that could be a problem?-Arthur
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