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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)

It's all about how far the center point travels. The center point of the coin travels the same distance (2 * Pi * radius) in both scenarios. A more extreme example: if you drove a nail anywhere in the edge of the coin and spun it around, the arrow would be pointing up again, the red line would have a length of 0, but the center still would have traveled 2*Pi*r.
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The center always travels a distance that is an average of the outside edge and the inner edge, just as if you moved a bar around instead of looking at the circle's diameter, the center moves a distance that is an average of the ends.

And that the center moves the same distance is more of a coincidence in this case. You can easily make an S shaped curve, or even an ellipse, where the center will move the same distance and you won't get the circle pointing the same way.

This works because if you roll around a circle (any sized circle), the spinning circle swings around. And it turns out that the shortening of the path on the surface of the circle works out to the exact same amount that the circle is spun around.

Worked out: If the center of moving circle of radius r goes in a circular path of radius R, and length L, then the angle the center moves around the fixed circle is L/R. The length the path where the circles touch then has length L(R-r)/R. The angle the moving circle would roll on a flat surface that length is L(R-r)/Rr=L/r - L/R, but if you add to that the L/R it also swung around, you get a total angle of L/r, which does not care about the size of the fixed circle at all, and for a path length of 2pi*r you get that the circle always ends up pointing in the same direction again. But this only works for circles because it depends on the angle and the distance going around being the same.
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That's a really neat observation, and it works in other scenarios as well (I checked with a circle inside another circle twice its size, and the logic still holds), but I don't get why it works.
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