Problem:
Erin the ant starts at a given corner of a cube and crawls along exactly 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?
Answer Choices:
A.
B.
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D.
E.
Solution:
The first two edges of Erin's crawl can be chosen in ways. These edges share a unique face of the cube, called the initial face. At this point, Erin is standing at a vertex and there is only one unvisited vertex of the initial face. If is not visited right after , then Erin visits all vertices adjacent to before . This means that once Erin reaches , she cannot continue her crawl to any unvisited vertex, and cannot be her last visited vertex because is adjacent to her starting point. Thus must be visited right after . 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 paths in all.
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The problems on this page are the property of the MAA's American Mathematics Competitions