On Sat, 20 Mar 2004, Chan-Ho Suh wrote: > > Tim Brauch <RnEeMwOs.pVoEst@tbrauch.cNOoSPAMm> wrote: > > email@example.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 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. Q.E.D.
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?