Paths on a dodecahedron Looking at this question, I misread "dodecagon" as "dodecahedron".  I think the latter is a cool problem, so I'm posing it as a question of its own :)

Starting from one vertex of a dodecahedron, an ant wants to reach the opposite vertex of the dodecahedron, moving to adjacent vertices. If $p_n$ is the number of such paths with length $n$, compute $p_1+p_2+\dots+p_{12}$.


Let $A$ be the adjacency matrix. Then $A^k$ gives you the number of paths of length $k$ between corresponding vertices. (This works for any graph.) This can computed very quickly, and you can also get asymptotics and so on, by computing eigenvalues.
For example, there are
171619248
such paths of length 20 (not too far from MJD's guess), and
25768876036573921452762172776956776774411837488
such paths of length 100.
