Two small disks sit on the rim of a large stationary disk, touching each other. They roll without slipping in opposite directions, one spinning at ω, the other at 2ω. They drift apart, travel all the way round, and the question is the time τ until they touch again. Watch the two of them go.
Because the big disk is fixed, the point of contact is momentarily still. A small disk spinning at ω therefore creeps around the rim at a much slower orbital rate Ω.
One disk goes clockwise, the other anticlockwise, so their orbital speeds add. The gap between them closes at the sum of the two rates.
They begin only a sliver Δθ apart, so to touch again the centres must sweep through everything except that sliver on each side: 2π − 2Δθ.