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 where triangular regions of side length 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 apart from . The second is a path-conversion algorithm that finds a solution for all . 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.
Citation
Samantha Davies. Chenxiao Xue. Carl Yerger. "Algorithms for finding knight's tours on Aztec diamonds." Involve 10 (5) 721 - 734, 2017. https://doi.org/10.2140/involve.2017.10.721
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