Bunbury the rabbit starts at vertex A on triangle APS. Each move, he hops to a vertex other than the one he currently occupies. In how many different ways can he start and end at A after 10 moves?
This was posted here, and it has a very good answer that is understandable.
https://math.stackexchange.com/questions/3598274/ways-to-end-at-the-same-vertex-on-triangle