AMC10 2015 B

Answer (A): The first two edges of Erin’s crawl can be chosen in $3\cdot 2=6$ ways. These edges share a unique face of the cube, called the initial face. At this point, Erin is standing at a vertex $u$ and there is only one unvisited vertex $v$ of the initial face. If $v$ is not visited right after $u$, then Erin visits all vertices adjacent to $v$ before $v$. This means that once Erin reaches $v$, she cannot continue her crawl to any unvisited vertex, and $v$ cannot be her last visited vertex because $v$ is adjacent to her starting point. Thus $v$ must be visited right after $u$. There are only two ways to visit the remaining four vertices (clockwise or counterclockwise around the face opposite to the initial face) and exactly one of them cannot be followed by a return to the starting vertex. Therefore there are exactly $6$ paths in all.