# 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.

Find:

$-\log_3(4a_{98}-1)$

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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