Rolling tetrahedron

A regular tetrahedron is resting on a surface. Every second, the tetrahedron rolls over onto a different face. The direction that it rolls in is uniformly random and independent (i.e. when the tetrahedron is resting, there is a 1/3 chance that it will roll in any one of three possible directions during that second).

Let \(a_n\) be the probability that after \(n\) rolls (seconds), the tetrahedron will be resting on the same face that it started on.



Can anyone tell me if it is possible to generalize this result for other polyhedra? After some labor, I was able to derive simple explicit formulae for \(a_n\) on the other Platonic solids (sans icosahedron).


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