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 {\em 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.