This work shows new algorithms and findings on permutation groups in graphs, highlighting Wilson's theorem's applications.
The 15-puzzle is a classic sliding puzzle consisting of a 4x4 grid with 15 numbered square tiles and one empty space. In 1974, Wilson generalized the 15-puzzle to find the group of permutations on graphs. In this work, we provide a variation of a proof of Wilsons theorem, propose a result for 1-connected and disconnected graphs, find a new manual algorithm for solving sliding graph puzzles, and extend existing computer algorithms on the 15-puzzle to solve any sliding graph puzzle.
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Wong et al. (2025) studied this question.