Solution: Quarter Quandary
The reason for the unexpected outcome in Quarter Quandary is that the two quarters have TWO circumferences that are interacting. When the revolving quarter goes around halfway, it has traversed the equivalent of one circumference (half of its own and half of the stationary quarter's). This produces one complete rotation for the moving quarter. The rest of its trip around the stationary quarter traverses the equivalent of another circumference (again, half on each quarter) which produces the second rotation.