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Select a vertex $V$ of a cube. How many paths begin at $V$, traverse exactly $3$ edges of the cube, and end at the vertex furthest from $V$?

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What have you tried? If you just try listing them, do you run into any problems? – Robert Mastragostino Dec 18 '12 at 0:05

Hint: Draw a cube, making sure to include the "invisible" edges. Now take advantage of symmetry. The first step is to any one of $3$ vertices. Pick one of these vertices, say $W$, and count the number of paths of length $2$ that get you where you want to go. Then multiply your count by $3$.

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