On Thursday, May 1, 2003, at 06:00 PM, Dan Hoey wrote:
Bill Thurston writes:
Does anyone have some thoughts about generalizations of hex to various manifolds? ... For CP^2, a good rule seems to be to declare the winner to be anyone such that H^2(their colors) is non-zero.
Forgive me if this is exactly what you said, but I suggested playing hex on the projective plane a few years ago. Bill Taylor pointed out the natural winning configuration: a path that is not contractible to a point. (If that isn't what you said, what's CP^2?)
No, your example of RP^2 is not one I had thought about. CP^2 means complex projective 2-space, which is as a real manifold is 4-dimensional. I'm not familiar with the "Y" board I think the ASCII art didn't come out right on my screen, but I'll look into it. The winning configuration in CP^2 would be the one containing a surface that is not homologous to 0. But let's stick to 2-dimensional surfaces---there is lots I don't understand. For instance, is it possible to invent a balanced version of hex on a Klein bottle? i.e. where the only asymetry between white and black has to do with who goes first? Bill
You can play on a "Y" board, which is graph-isomorphic to a hemi-icosahedron with the sides triangulated. Edge moves are made by playing two antipodal stones. On the commercial board (available from Kadon), you play on the vertices of the triangulation. If you prefer ASCII art, I wrote a program that produces drawings where you play in the spaces. Here are three sizes of board. For the smallest, there's only one starting move, only one response, and then two of the four third moves are winners. The next size up has two openers--are they both winners? ____ ____ / \__ ____/ \ / __/ \__ __/ \__ \ \__/ \ \__ __/ __/ \__/ / \ \____/ \ __/ \____/ / \ \ \__/ \ \ __/ / \__ \ \ ____ / / \____/ / __/ \____/ __/ \__/ / / \____ \__/ / \__/ / / \__/ / \ \ / __/ \__ / \__/ __/ __/ __/ __/ \ \ \__/ \__/ \__ \____ \ \__/ / \__/ \__/ \ \____/ / \ / \ \____/ \__ \____/ / / / \__/ \__/ / / / __/ \__ \____ \ \__ \__ / \____/ \ \ \ \__/ \__ \____/ \__ \__/ \__/ \__/ / \ \__/ / / \ \____/ \__ \__ \ \ \__/ / / \ \__/ / __/ \__ \____/ \__ \__ \____/ \__ \__/ / / \ \__/ \__ \____/ \ \__ \__/ \ \__/ \ \ / / / \ \____/ \ \____/ \__ \__ \____/ / / \ \__/ / __/ \ \____/ \ \ \__/ \__ \__/ / / \ \__/ \__ \____/ \ \__ \__ \__ \__/ \ \__/ / / \ \__/ \ \____/ \__/ \__ \____/ / / \ \__/ / / \ \____/ \ \ \ \____/ / / / \ \____/ \____/ \ \__ \__ \__ \ \__/ \__/ \____/ \ \____/ \__/ \__/ \__ / / \ / \ / \ \____/ \ \ \ \ \__/ \__/ \__/ \____/ \__ \__ \__ \__ \__ / \ / \ / \ / \ / \__/ \__/ \__/ \__/ \ \ \__/ \__/ \____/ \__/ \ \ \ \ \ / / \ / \ / \ / \ __/ __/ __/ __/ / \__/ \__/ \__/ \____/ \__/ \__/ \__/ \__/ \__/ / \ / \ / \ / \____/ / / / / \ \__/ \__/ \____/ / \____/ __/ __/ __/ / / \ / \____/ \____/ / \__/ \__/ \__/ \ \ \__/ / \ / \____/ / / / \__/ / \ \ \__/ / \____/ __/ __/ / \ \ \__/ __/ \____/ / \__/ \__/ \ \__/ / \__/ / \____/ / / / / \ \ \____/ / \____/ __/ \ \ \__/ __/ \____/ / \__/ \__/ / \__/ __/ \____/ __/ / \ \ \____/ __/ \__/ \ \__/ __/ \____/ __/ / / \__/ __/ \____/ \ \ \____/ __/ \__/ __/ \____/ / \__/ __/ \ \____/ \____/
By the way, did anyone figure out toral 4x4 tic-tac-toe?
Dan Hoey@AIC.NRL.Navy.Mil
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