Futility Closet posted a puzzle that might make your eyes glaze over, or could spur some of you to compete over who has the best, clearest explanation.
In the top figure, one coin rolls around another coin of equal size.
In the bottom figure, the same coin rolls along a straight line.
In each case the rolling coin has made one complete rotation. But the red arc at the top is half the length of the red line at the bottom. Why?
I look forward to any explanation you may have, and later I’ll add some I found elsewhere. -via Boing Boing
(Image credit: Lymantria)
Now let's disregard the rolling and see where the difference comes from. Assume the moving coin is not free standing; instead, it's stuck to a stick, like a lollipop, in such a way that it can't roll around itself. Assume the stick's protruding length is equal to the coin's radius. In the first scenario, we position the end of the stick in the center of the second coin, and we rotate it half a turn. In the second scenario, we just translate it horizontally for whatever distance we want. We notice that in the first scenario, the coin ends up upside down, whereas in the second scenario it doesn't rotate at all. So in the first scenario we gain half a turn simply by going around the second coin for half a turn, as compared to the second scenario where the coin doesn't rotate at all.
I know it looks like a long explanation, but if you can visualize it then it's really easy to follow.
The distance traveled by the outer rim is the same when you average the edge we're measuring and the opposite edge. The center of each coin travels the same distance.