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An ant starts at a given vertex of a cube. On each of its moves, it crawls along an edge to get to another vertex. After 7 moves, the ant has visited 7 different vertexes. The ant then discovers that it can't directly crawl to its starting position in a move. How many different paths could the ant have taken?
Source: AMC 10
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The only point the ant can end up on is the one across the diagonal of the cube. There are 3 points to get to which can go directly to the point from there. Since each of those points are in same placement relative to the origin, there will be the same number of paths to each of these points. There are 2 ways to each of those 3 points, so 2 x 3 = 6.