This is an interesting and somewhat counter-intuitive problem that illustrates important concepts about rotations and gears. Students will enjoy trying it out with coins.

One possible starting point is the question: if you roll a coin half way around another coin of the same type, what position will it end in — upside down or the right way up? Clearly it rolls a distance of half the circumference, and if this was a straight line distance it is easy to see it would have rotated one half revolution. Then imagine dragging the straight line into a circle so that the point the rolling coin is touching is at the bottom (thus making a circle of the same size as the coin). This action takes the upside down coin and makes it the right way up!

Considering the curve described by a point on the outer circle leads to the Epicycloid and Cardioid curves, which themselves can open up interesting explorations of locus, polar coordinates, cyclic curves and even Ptolemaic astronomy and epicycles.