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A circle of radius 1 rolls without slipping round the inside of a square of side length 4. Find an expression involving $\pi$ for the exact number of revolutions the circle makes by the time it returns to its original position.

Circle (radius 1) inside square (side length 4)


If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.


This problem is taken from the UKMT Mathematical Challenges.