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Two players Oh and Ex alternately choose points of a finite projective plane.
The first player (if any) to make a line in his/her chosen points is the winner.
Using the Erdos-Selfridge theorem, we can see that the game is a draw if the order of the projective plane is 5 or greater. The game is a trivial Oh win if the order is 2. Does Oh win if the order is 3 or 4?

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See "Tic-Tac-Toe on a Finite Plane", Maureen T. Carroll and Steven T. Dougherty, Mathematics Magazine, Vol. 77, No. 4 (Oct., 2004), pp. 260-274. (Preprint here: The second player can force a draw on the 3-by-3 and 4-by-4 projective planes.

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Thank you very much for the reference, which completely answers the question. – Martin Erickson Jun 28 '10 at 15:31

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