TITLE: Paths on a dodecahedron
QUESTION [7 upvotes]: 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 $p_1+p_2+\dots+p_{12}$.

REPLY [4 votes]: It is not necessary to set up the full adjacency matrix of the dodecahedron. Drawing the edge graph with the starting vertex $v_0$ at the center and the opposite vertex at infinity one realizes that due to symmetry there are just six classes $C_i$ $(0\leq i\leq5)$ of vertices. 

It is therefore sufficient to consider the numbers $p_i(n)$, whereby $p_i(n)$ denotes the number of ways to reach a vertex of class $C_i$ in exactly $n$ steps, starting at $v_0$. Looking at my drawing I then obtain the following equations:
$${\bf p}(n+1)=\left[\matrix
{0&3&0&0&0&0\cr
1&0&2&0&0&0\cr
0&1&1&1&0&0\cr
0&0&1&1&1&0\cr
0&0&0&2&0&1\cr
0&0&0&0&3&0\cr}\right]\>{\bf p}(n)\ ,$$
whereby ${\bf p}(n)$ denotes the column vector of the $p_i(n)$.
Since there are so many treatments of the problem already I stop here.