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:

> > (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 a
short-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 what
a 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. Since
it does not prove a falsehood (which is impossible; falsehoods cannot
be proven), it must use the topology of the board.

Bad analogy: I can prove that the speed at which a tennis ball hits
my hand going down is the same speed at which it left my hand going
up, using a simple argument from conservation of energy. You object:
oh, but that proof is flawed, because what if there was a rocket engine
attached to the tennis ball? I respond: any fool can tell that there
is *not* a rocket engine attached to the tennis ball, so why would you
even think that could be a problem?