I found it helpful to imagine the picture after the smaller coin has made one full rotation. Let x be the angle that we have traveled along the larger coin. No

I found it helpful to imagine the picture after the smaller coin has made one full rotation. Let x be the angle that we have traveled along the larger coin. Now we notice there's some similar triangles, and x is also the angle that we have left to travel on the smaller coin. I.e. we traveled 2pi - x on the smaller coin. But we know the edges were touching, so the arclength of these two angles must be equal. 3x = 2pi - x x = pi/2 So going around the whole way, you can do 2pi/x = 4 rotations