A Cellular Automaton for Blocking Queen Games
We show that the winning positions of a certain type of two-player game form interesting patterns which often defy analysis, yet can be computed by a cellular automaton. The game, known as Blocking Wythoff Nim, consists of moving a queen as in chess, but always towards (0,0), and it may not be moved to any of \(k-1\) temporarily “blocked” positions specified on the previous turn by the other player. The game ends when a player wins by blocking all possible moves of the other player. The value of \(k\) is a parameter that defines the game, and the pattern of winning positions can be very sensitive to \(k\). As \(k\) becomes large, parts of the pattern of winning positions converge to recurring chaotic patterns that are independent of \(k\). The patterns for large \(k\) display an unprecedented amount of self-organization at many scales, and here we attempt to describe the self-organized structure that appears.
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