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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?

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*This problem is taken from the UKMT Mathematical Challenges.*

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