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Challenge 312: Circle Time

Thanks to Thomas, Y12, for supplying this problem.

n chairs are arranged in a circle, and each is numbered with one of the integers from 0 to n - 1 inclusive, but not necessarily in numerical order. At 12 noon, n children arrive, and one child sits on each chair. At the end of every minute that passes after they first sit down, every child moves k chairs to their left, where k is the number of the chair they are sitting on. Given that no two children ever end up sitting on the same chair, prove that eventually there will be a time when every child is sitting in the chair they had at the start.

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