The Game of Tumbleweed is PSPACE-complete

06/19/2022

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by   Lear Bahack, et al.

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Tumbleweed is a popular two-player perfect-information new territorial game played at the prestigious Mind Sport Olympiad. We define a generalized version of the game, where the board size is arbitrary and so is the possible number of neutral stones. Our result: the complexity of deciding for a given configuration which of the players has a winning strategy is PSPACE-complete. The proof is by a log-space reduction from a Boolean formula game of T.J. Schaefer, known to be PSPACE-complete. We embed the non-planar Schaefer game within the planar Tumbleweed board without using proper "bridges", that are impossible due to the board's topology. Instead, our new technique uses a one-move tight race that forces the players to move only according to the protocol of playing the embedded 4-CNF game.