David asks: << Can anybody tell me who is the winner of the following two games. Game of Ramsey. Two players alternately pick edges from a complete graph on 6 vertices until one of them has collected three edges that form a triangle at which point his Opponent is the winner. Game of van der Waerden. Two players alternately pick numbers from the set {1,2, . .,9} until one of them has collected three numbers which form an arithmetic progression at which point his Opponent is the winner. The point is, of course, that these games cannot end in a draw.
Darned if I know, but very interesting questions! I'd like to add (two variants of) another game, which Martin Gardner and I (and probably innumerable others) invented independently, or at least close variants of the same idea: Who wins in the Game of Squares: Players take turns placing a marker of their own color (say white or black) on the vertices of an 8x8 square grid until there is some square (tilted or not) with all four vertices having the same-colored markers on them. The player whose color it is loses. Same question for the slightly more symmetrical Game of Toral Squares: Same game but on the 64 vertices of GxG, where G is a regular octagon. (Microchallenge: How many distinct squares are there in the GxG case, where a square is defined by its set of vertices? Better, generalize to the case of G = the regular N-gon) --Dan