A Cellular Automaton for Blocking Queen Games

TL;DR

This paper demonstrates that the complex patterns of winning positions in Blocking Wythoff Nim, a two-player game involving queen moves and blocking, can be effectively modeled and analyzed using cellular automata, revealing self-organized chaotic structures.

Contribution

It introduces a cellular automaton model to analyze the intricate patterns of winning positions in Blocking Wythoff Nim, highlighting self-organization and chaos at large parameter values.

Findings

Winning position patterns can be computed by cellular automata.

Large k patterns exhibit self-organization and chaos.

Patterns become independent of k as it grows large.

Abstract

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.