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. This paper extends a previous study (Cook et al. in Cellular automata and discrete complex systems, AUTOMATA 2015, Lecture Notes in Computer Science, vol 9099, pp 71–84, 2015), containing further analysis and new insights into the long term behaviour and structures generated by our blocking queen cellular automaton.
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The periods \((5,1)\) and \((-1,2)\) are for the épaulette below and to the left of the main diagonal which is of course symmetric with the épaulette above and to the right of the main diagonal.
We thank Michal Szabados for pointing this out to us.
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Urban Larsson is supported by the Killam Trust. Turlough Neary is supported by Swiss National Science Foundation Grants 200021-141029 and 200021-153295.
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Cook, M., Larsson, U. & Neary, T. A cellular automaton for blocking queen games. Nat Comput 16, 397–410 (2017). https://doi.org/10.1007/s11047-016-9581-2
- Wythoff Nim
- Blocking Wythoff Nim
- Cellular automata