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Puzzled: Paths and Matchings


Puzzled: Paths and Matchings

Alice (red) beats Bob (blue) in a Path Game.

The Path Game. Alice and Bob alternately place a checker on an unoccupied square of an initially empty eight-by-eight checkerboard (see the figure here). The rule is that after Alice places her initial checker, every new checker must be (orthogonally) adjacent to the most recently placed checker. The players are in effect constructing a path of checkers. The last player to make a legal move wins the game. Your mission: find a winning strategy for Bob.

The Match Game. Suppose that Alice, when it is her turn, is no longer obliged to play next to Bob's last checker and may instead place her checker on any unoccupied square. Bob is still constrained to "match" Alice, that is, to play next to Alice's last move. As in the Path Game, the last player to make a legal move wins the game. So who wins now, with best play?


 

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