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If n is even ...

Posted By: Bob Koca
Date: Sunday, 16 May 2010, at 4:04 p.m.

In Response To: If n is even ... (Nack Ballard)

Yes, same as radial symmetry that you suggested.

There are other "matching" methods that work. For example if n is even form 4 quadrants (upper left, upper right, lower left, and lower right) of size n/2 by n/2 each. When your opponent plays in one of the quadrants you play in the same location in the radially symmetric quadrant. If n is a multiple of 4 then as before make 4 qaudrants. Play in same quadrant as opponent in a location radially symmetric in that quadrant.

I think these alternate strategies are related to the equivalence classes of positions mentioned earlier. At any point one can interchange two rows, interchange two columns, rotate the board, take a reflection about a diagonal, or take a reflection about the center, without changing the value of the game. I saw something similar in a discussion of the number of possible sudoku boards. I think it may be useful to define a canonical form of a position by doing those operations so that the line with maximum elements occur on top and then the played in spaces are pushed to the left as far as possible. More detail is needed from there. Doing that drastically cuts down on the number of positions. For example with a 3 by 3 grid with 4 spaces filled in, instead of 9C4 = 126 positions I see only 4:

1) (1,1) (1,2) (2,1) (2,2)

2) (1,1) (1,2) (2,1) (2,3)

3) (1,1) (1,2) (2,1) (3,3)

4) (1,1) (1,2) (2,3) (3,3)

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