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4 Removed incorrect remark.

Let $R$ be a rectangular region of the integer lattice $\mathbb{Z}^2$, each of whose unit squares is labeled with a number in $\lbrace 1, 2, 3, 4, 5, 6 \rbrace$. Say that such a labeled $R$ is die-rolling Hamiltonian, or simply rollable, if there is a Hamiltonian cycle obtained by rolling a unit die cube over its edges so that, for each square $s \in R$, the cube lands on $s$ precisely once, and when it does so, the top face of the cube matches the number in $s$. For example, the $4 \times 4$ "board" shown below is rollable.

Q. Is it true that, if $R$ is die-rolling Hamiltonian, then the Hamiltonian cycle is unique, i.e., there are never two distinct die-rolling Hamiltonian cycles on $R$?

This "unique-rollability" question arose out of a problem I posed in 2005, and was largely solved two years later, in a paper entitled, "On rolling cube puzzles" (complete citation below; the $4 \times 4$ example above is from Fig. 17 of that paper). Although the original question involved computational complexity, the possible uniqueness of Hamiltonian cycles is independent of those computational issues, so I thought it might be useful to expose it to a different community, who might bring different tools to bear. It is known to hold for $R$ with side lengths at most 8. If not every cell of $R$ is labeled, and unlabeled cells are forbidden to the die, then there are examples with more than one Hamiltonian cycle.

Edit1. Rolling a regular tetrahedron on the equilateral triangular (hexagonal) lattice is not as interesting. See the Trigg article cited below.

Edit2. Serendipitously, gordon-royle posted a perhaps(?) relevantly related question: "Uniquely Hamiltonian graphs with minimum degree 4."Note my rollable graphs are regular of degree 4, except around the boundary.

• The computational version is Open Problem 68 at The Open Problems Project.
• "On rolling cube puzzles." Buchin, Buchin, Demaine, Demaine, El-Khechen, Fekete, Knauer, Schulz, Taslakian. Proceedings of the 19th Canadian Conference on Computational Geometry, Pages 141–144, 2007. PDF link to full paper.
• Charles W. Trigg. "Tetrahedron rolled onto a plane." J. Recreational Mathematics, 3(2):82–87, 1970.
3 added 297 characters in body; deleted 12 characters in body

Let $R$ be a rectangular region of the integer lattice $\mathbb{Z}^2$, each of whose unit squares is labeled with a number in $\lbrace 1, 2, 3, 4, 5, 6 \rbrace$. Say that such a labeled $R$ is die-rolling Hamiltonian, or simply rollable, if there is a Hamiltonian cycle obtained by rolling a unit die cube over its edges so that, for each square $s \in R$, the cube lands on $s$ precisely once, and when it does so, the top face of the cube matches the number in $s$. For example, the $4 \times 4$ "board" shown below is rollable.

Q. Is it true that, if $R$ is die-rolling Hamiltonian, then the Hamiltonian cycle is unique, i.e., there are never two distinct die-rolling Hamiltonian cycles on $R$?

This "unique-rollability" question arose out of a problem I posed in 2005, and was largely solved two years later, in a paper entitled, "On rolling cube puzzles" (complete citation below; the $4 \times 4$ example above is from Fig. 17 of that paper). Although the original question involved computational complexity, the possible uniqueness of Hamiltonian cycles is independent of those computational issues, so I thought it might be useful to expose it to a different community, who might bring different tools to bear. It is known to hold for $R$ with side lengths at most 8. If not every cell of $R$ is labeled, and unlabeled cells are forbidden to the die, then there are examples with more than one Hamiltonian cycle.

Edit1. Rolling a regular tetrahedron on the equilateral triangular (hexagonal) lattice is not as interesting. See the Trigg article cited below.

Edit2. Serendipitously, gordon-royle posted a perhaps(?) relevantly related question: "Uniquely Hamiltonian graphs with minimum degree 4." Note my rollable graphs are regular of degree 4, except around the boundary.

• The computational version is Open Problem 68 at The Open Problems Project.
• "On rolling cube puzzles." Buchin, Buchin, Demaine, Demaine, El-Khechen, Fekete, Knauer, Schulz, Taslakian. Proceedings of the 19th Canadian Conference on Computational Geometry, Pages 141–144, 2007. PDF link to full paper.
• Charles W. Trigg. "Tetrahedron rolled onto a plane." J. Recreational Mathematics, 3(2):82–87, 1970.
2 Dates

Let $R$ be a rectangular region of the integer lattice $\mathbb{Z}^2$, each of whose unit squares is labeled with a number in $\lbrace 1, 2, 3, 4, 5, 6 \rbrace$. Say that such a labeled $R$ is die-rolling Hamiltonian, or simply rollable, if there is a Hamiltonian cycle obtained by rolling a unit die cube over its edges so that, for each square $s \in R$, the cube lands on $s$ precisely once, and when it does so, the top face of the cube matches the number in $s$. For example, the $4 \times 4$ "board" shown below is rollable.

Q. Is it true that, if $R$ is die-rolling Hamiltonian, then the Hamiltonian cycle is unique, i.e., there are never two distinct die-rolling Hamiltonian cycles on $R$?

This "unique-rollability" question arose out of a problem I posed in 20062005, and was largely solved a year two years later, in a paper entitled, "On rolling cube puzzles" (complete citation below; the $4 \times 4$ example above is from Fig. 17 of that paper). Although the original question involved computational complexity, the possible uniqueness of Hamiltonian cycles is independent of those computational issues, so I thought it might be useful to expose it to a different community, who might bring different tools to bear. It is known to hold for $R$ with side lengths at most 8. If not every cell of $R$ is labeled, and unlabeled cells are forbidden to the die, then there are examples with more than one Hamiltonian cycle.

Addendum. Rolling a regular tetrahedron on the equilateral triangular (hexagonal) lattice is not as interesting. See the Trigg article cited below.

• The computational version is Open Problem 68 at The Open Problems Project.
• "On rolling cube puzzles." Buchin, Buchin, Demaine, Demaine, El-Khechen, Fekete, Knauer, Schulz, Taslakian. Proceedings of the 19th Canadian Conference on Computational Geometry, Pages 141–144, 2007. PDF link to full paper.
• Charles W. Trigg. "Tetrahedron rolled onto a plane." J. Recreational Mathematics, 3(2):82–87, 1970.
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