[masraum]

Originally Posted by Maffett

I think the answer is two times.

Sorry for the poor quality sketch. The contact point between the coins "travels" a distance of 6pi. The smaller coin has a circumference of 2pi. Therefore the smaller coin "rolls" three times during the trip. But point B only circumnavigates point A twice. It would circumnavigate three times if the trip had been in a straight line. But one doesn't happen because the distance traveled was a circle.

Nice try, but as has been said, your drawing is missing a couple of important points. I made two circles in a 3:1 diameter ratio and worked this out. Your rotation is off. If you start at the top of the big circle, then the spot where the two circles touch will be back at the bottom of the small circle when it is at the 3 o'clock position on the big circle. Each full rotation is at 12, 3, 6, and 9 o'clock positions.