Bent Hamiltonian Paths in Toroidal Grid Graphs and Applications to the Snake Cube Puzzle

Document Type

Article

Publication Date

1-8-2025

Abstract

The snake cube puzzle is a popular toy that can be represented by an encoding of a Hamiltonian path in a grid graph. We computationally determine all possible solvable puzzle configurations and counts of solutions of each configuration up to symmetry, finding agreement with prior analysis. This allows us to introduce a measure of the difficulty of a particular snake cube puzzle, revealing that the commonly sold puzzle configuration has maximal difficulty. Then we extend the work to other topological spaces by considering various toroidal grid graphs and determine all encodings and unique solutions. It is known that there do not exist bent Hamiltonian paths in 3x3x3 grid graphs; we prove that there cannot exist a bent Hamiltonian path in a 3x3x3 toroidal grid graph with one or two pairs of faces identified, however, there is such a path when three pairs of faces are identified. Finally, we discuss the minimum number of non-bent portions of a Hamiltonian path/cycle in an nxnxn (toroidal) grid graph more generally.

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