TITLE: Puzzle: A coin rolls without slipping around another coin
QUESTION [8 upvotes]: If a coin rolls without slipping around another coin of the same or
  different size, how many times will it rotate while making one
  revolution?

The proof given is like this:

Cut the curve open at some point and uncoil it into a straight line
  segment. Rolling the circle along this segment it will rotate (length
  curve)/(length circle) times.
Keeping the circle attached to one end of the segment, we then recoil
  the segment back into thecurve, which contributes the final rotation.

I can understand the first part of the proof, but I couldn't understand the second part, any ideas? Also, I am also interested in different approaches for proving this one.
Problem/Solution source.

REPLY [2 votes]: It is maybe easiest if you imagine an "axis" joining the two centers of the circles all the time (so one end of the axis doesn't advance at all an the other end turns a full circe), and a camera mounted on this axis filming the circles.
Since the camera makes a full tour in one direction (call it positive), it will see the fixed circle make a complete tour in the negative direction. If that fixed circle has radius $R$ and the other radius $r$, then at the point of contact this negative rotation produces a displacement of $2\pi R$ in the backwards direction, which will cause the other circle to make $R/r$ tours in the positive direction (so that it also produces a backward displacement of $(R/r)2\pi r=2\pi R$ at the point of contact, corresponding to "no slipping"). But since the camera also made a complete tour in the positive direction while filming, it is clear that the mobile circle made $(R/r)+1$ tours in reality (i.e., with repect to a fixed observer).