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Topic: Tic-tac-toe on Mobius strip has something counter-intuitive. Please
help!

Replies: 8   Last Post: Aug 21, 2008 12:57 PM

 Messages: [ Previous | Next ]
 Larry Hammick Posts: 1,876 From: Vancouver Registered: 12/6/04
Posted: Aug 21, 2008 8:24 AM

About the real projective plane, it is a space in which
-- each "point" is a pair of opposite points on a Euclidean sphere
-- each "line" is the set of "points" on a great circle of that sphere.
So, on the projective plane, any two lines intersect in /one/ point.

A finite projective plane can be constructed having qq + q + 1 points, and
the same number of lines, where q may be any prime power. The number of
points on a line, and the number of lines through a point, turns out to be
q+1. E.g. when q=2 there is a projective plane with 7 points 0,1,2,3,4,5,6
and the seven lines
0 1 4
0 2 5
0 3 6
1 2 3
1 5 6
2 4 6
3 4 5
You can verify that any two points are on one common line, and any two lines
have one common point.
http://en.wikipedia.org/wiki/Projective_plane

Back to the game, I wouldn't get too cosmic about the Mobius strip. Here:
a b c g h i ...
d e f d e f ...
g h i a b c ...
The figure represents a matrix of 9 points on a Mobius strip. Each point of
the Mobius strip is represented by a /pair/ of points on a torus. If you
follow the row abcgh... to the right, it wraps back into the 3*3 square on
the left in the desired way. When a player makes a move, he takes /both/ c's
or /both/ e's or whatever. Some construction of this sort might yield an
interesting game. I wouldn't worry about how to draw the /grid/ as we do in
tic-tac-toe; it is the cells of the grid, and the list of sets of cells
which comprise winning "rows", that matter.

LH