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 ErdosSelfridge 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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1 Answer
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See "TicTacToe on a Finite Plane", Maureen T. Carroll and Steven T. Dougherty, Mathematics Magazine, Vol. 77, No. 4 (Oct., 2004), pp. 260274. (Preprint here: http://academic.scranton.edu/faculty/carrollm1/tictac.pdf) The second player can force a draw on the 3by3 and 4by4 projective planes.

$\begingroup$ Thank you very much for the reference, which completely answers the question. $\endgroup$ Commented Jun 28, 2010 at 15:31