An ant crawls carefully around the edges of a cube, starting at point $P$ and in the direction of the arrow.

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.