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'Roll On' printed from http://nrich.maths.org/
A circular disc of diameter $d$ rolls without slipping around the inside of a ring of internal diameter $3d$, as shown in the diagram. By the time that the centre of the inner disc returns to its original position for the first time, how many times will the inner disc have turned about its centre? What if the disc rolls around the outside of the ring?
This problem is taken from the UKMT Mathematical Challenges.