the screech made by your hubcap as it slips (and rolls) on the curb while your tire merely rolls on the pavement. The smaller the small circle relative to the large circle, the more the small one slips. Of course the center of the two circles does not rotate at all, so it slides the whole way. Alternatively, one can reject the assumption that the smaller circle is independent of the larger circle. Imagine a tire as the larger circle, and imagine the smaller circle as the interior circumference of the tire and not as the rim. The movement of the inner circle is dependent on the larger circle. Thus its movement from any point to another can be calculated by using an inverse of their ratio. Analysis and solutions The paradox is that the smaller inner circle moves 2πR, the circumference of the larger outer circle with radius R, rather than its own circumference. If the inner circle were rolled separately, it would move 2πr, its own circumference with radius r. The inner circle is not separate but rigidly connected to the larger. So 2πr is a red herring. The inner circle's center and radius are both relevant, but its circumference is not. First solution If the smaller circle depends on the larger one (Case I), the larger circle's motion forces the smaller to traverse the larger’s circumference. If the larger circle depends on the smaller one (Case II), then the smaller circle's motion forces the larger circle to traverse the smaller circle’s circumference. This is the simplest solution. Second solution This solution considers the transition from the starting to ending positions. Let Pb be a point on the bigger circle and Ps be a point on the smaller circle, both on the same radius. For convenience, assume they are both directly below the center, analogous to both hands of a clock pointing towards six. Both Pb and Ps travel in a cycloid path as they roll together one revolution. While each travels 2πR horizontally from start to end, Ps's cycloid path is shorter and more efficient than Pb's. Pb travels farther above and farther below the center's path – the only straight one – than does Ps. If Pb and Ps were anywhere else on their respective circles, the curved paths would be the same length. Summarizing, the smaller circle moves horizontally 2πR because any point on the