Crawl around the cube
Weekly Problem 37 - 2010
An ant is crawling around the edges of a cube. From the description of his path, can you predict when he will return to his starting point?
An ant is crawling around the edges of a cube. From the description of his path, can you predict when he will return to his starting point?
Age
11 to 14
Challenge level
Problem
An ant crawls carefully around the edges of a cube, starting at point $P$ and in the direction of the arrow.
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At the end of the first edge it chooses whether to go either left or right. It then turns the other way at the end of the next edge and continues like this, turning right or left alternately at the end of each successive edge.
After how many edges does the ant return to point $P$ for the first time?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
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At Q the ant can choose first to go left to T, then right to W.
Otherwise, at Q it can go right to R and then left to W.
W is the corner diagonally opposite to P and is reached by either route after three edges (and no fewer).
So after exactly three more edges, the ant must reach the corner opposite W, that is, P.
So the ant will return to P after travelling along 6 edges.