Algorithms for finding knight's tours on Aztec diamonds

Abstract

A knight’s tour is a sequence of knight’s moves such that each square on the board is visited exactly once. An Aztec diamond is a square board of size $2n$ where triangular regions of side length $n-1$ have been removed from all four corners.

We show that the existence of knight’s tours on Aztec diamonds cannot be proved inductively via smaller Aztec diamonds, and explain why a divide-and-conquer approach is also not promising. We then describe two algorithms that aim to efficiently find knight’s tours on Aztec diamonds. The first is based on random walks, a straightforward but limited technique that yielded tours on Aztec diamonds for all $n\ne 22$ apart from $n=17,21$. The second is a path-conversion algorithm that finds a solution for all $n\le 100$. We then apply the path-conversion algorithm to random graphs to test the robustness of our algorithm. Online supplements provide source code, output and more details about these algorithms.