Hopefully I don't ruin the question this time.
Assume that we have two coins that are not identical in size but are both perfect circles. We want to roll a coin along the edge of another coin where the moving coin starts at 1 side and ends up on the exact opposite side (only traveling half of the non-moving coin) and still end up in the same starting position (the moving-coin's orientation ends in the same position it started).
What size(s) must the moving coin have relative to the non-moving coin?
If we want the moving coin to travel 3/4's around the non-moving coin and still end up in its starting position, what sizes for the moving coin work?