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Walks by an ant on a cube.

Source: 2015 AMC 10B #20

February 26, 2015
geometry3D geometryrotationcountingdistinguishabilityAMC

Problem Statement

Erin the ant starts at a given corner of a cube and crawls along exactly 77 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?
<spanclass=′latex−bold′>(A)</span>6<spanclass=′latex−bold′>(B)</span>9<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>18<spanclass=′latex−bold′>(E)</span>24 <span class='latex-bold'>(A) </span>\text{6}\qquad<span class='latex-bold'>(B) </span>\text{9}\qquad<span class='latex-bold'>(C) </span>\text{12}\qquad<span class='latex-bold'>(D) </span>\text{18}\qquad<span class='latex-bold'>(E) </span>\text{24}
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