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.
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="http://cs.smith.edu/~orourke/MathOverflow/DiceRolling.jpg" alt="Dice Rolling" />
<br />
> <b>Q</b>. 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.

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

<hr />
<ul>
<li> The computational version is
<a href="http://cs.smith.edu/~orourke/TOPP/P68.html#Problem.68">
Open Problem 68</a> at
<a href="http://cs.smith.edu/~orourke/TOPP/">The Open Problems Project</a>.
</li>
<li>
"On rolling cube puzzles."
Buchin, Buchin, Demaine, Demaine, El-Khechen, Fekete, Knauer, Schulz, Taslakian.
<em>Proceedings of the 19th Canadian Conference on Computational Geometry</em>, Pages 141–144, 2007.
<a href="http://people.csail.mit.edu/schulz/papers/RollingFull.pdf">PDF link to full paper.</a>
</li>
<li>
Charles W. Trigg. "Tetrahedron rolled onto a plane." <em>J. Recreational Mathematics</em>, 3(2):82–87, 1970.
</li>
</ul>