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The Game Of Kayles

 Answer : To win at this game you must, sooner or later, leave your opponent an even number of similar groups.  Then whatever he does in one group you repeat in a similar group. Suppose, for example, that you leave him these groups: o.o.ooo.ooo.  Now, if he knocks down a single, you knock down a single; if he knocks down two in one triplet, you knock down two in the other triplet; if he knocks down the central kayle in a triplet, you knock down the central one in the other triplet. In this way you must eventually win.  As the game is started with the arrangement o.ooooooooooo, the first player can always win, but only by knocking down the sixth or tenth kayle (counting the one already fallen as the second), and this leaves in either case o.ooo.ooooooo, as the order of the groups is of no importance.  Whatever the second player now does, this can always be resolved into an even number of equal groups.  Let us suppose that he knocks down the single one; then we play to leave him oo.ooooooo.  Now, whatever he does we can afterwards leave him either ooo.ooo or o.oo.ooo.  We know why the former wins, and the latter wins also; because, however he may play, we can always leave him either o.o, or o.o.o.o, or oo.oo, as the case may be.  The complete analysis I can now leave for the amusement of the reader.

Medieval Brain Teasers