>>14297660you can count the total paths by taking symmetry into account and working out the cases that are the same up to symmetry, then multiplying those base cases by the appropriate degrees of symmetry
there are only 3 unique starting points up to symmetry; vertex, edge, and center points.
note that you can start on a node and take a path that gets stuck. so you have to think enumerate all the possible paths up to symmetry.
consider the center point. each path leading away is symmetric, so there is only 1 choice up to symmetry with 6 degrees of symmetry. at the next node there are 3 choices, but 1 gets stuck on the next move, so really there are only two choices; follow the edge or go to the vertex. either choice constrains the rest of your moves. so the center point has only 2 paths up to symmetry, and 12 paths total starting/ending at it.
a similar analysis will yield paths starting at the edge and vertex.
it's important not to over count, so no other paths considered should start/end on the center node.
