A bug on a cube
A bug starts on a vertex of a cube.
If he only moves along the edges, how many ways can he travel to the opposite vertex visiting each vertex along the way no more than once (including the one he starts at which once he leaves he is not allowed to return to)?
Note : He doesn't necessarily need to visit all the vertices.
Image credit : https://commons.wikimedia.org
The answer is 18.
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1 solution
Initially there are three paths he can take (along any of the three edges available to him).
Then he has two choices that don't lead him back to where he started.
At this point, he only has one choice for every move that won't lead him either (a) onto a vertex already visited, or (b) onto the opposite vertex from where he started.
So essentially, he is left with only 3 choices as to when to move to the final (opposite) vertex.
3 × 2 × 3 = 1 8