smaller circle travels a shorter, and thus more direct path than any point on the larger circle. Third solution This solution only compares the starting and ending positions. The larger circle and the smaller circle have the same center. If said center is moved, both circles move the same distance, which is a necessary property of translation and equals 2πR in the experiment. Also, every other point on both circles has the same position relative to the center before and after rolling one revolution (or any other integer count of revolutions). For a wheel with multiple concentric inner circles, each circle's translation movement is identical because all have an identical center. This further proves the circumference of any inner circle is entirely irrelevant (Case 1). See also Trochoid Brachistochrone curve References Further reading Rota Aristotelica, The Archimedes Project, Digital Research Library Wheel paradox Circles Physical paradoxes Wheels