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Random-turn hex and other selection games. (English) Zbl 1153.91012

The game of hex has two players who take turns placing stones of their respective colors on the hexagons of a rhomsub-shaped hexagonal grid. A player wins by completing a path connecting the two opposite sides of his or her color. Random-turn hex is the same as ordinary hex, except that instead of alternating turns, players toss out to be very simple. In many cases, it is more tractable than hex and exhibits more symmetry. In one general class of games called selection games, which includes hex, the probability that player I wins when both players play optimally is equal to the probability, that player I wins when both players play randomly. It is shown that in a certain “fine lattice limit” winning probabilities of random-turn hex are a “conformal invariant” of the board shape. Random-turn games are models of conflicts where opposing agents do not alternate turns.

MSC:

91A46 Combinatorial games
00A08 Recreational mathematics
60C05 Combinatorial probability
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