A pencil AB lying on a table is given a half turn about the end B (so that A moves to A*) and then a half turn about A* (so that B moves to B*). The point C on the pencil is one third of the way from A to B.

What is the ratio of the total distances moved by A and by C?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.