# Rolling tetrahedron

**Discrete Mathematics**Level 4

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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