Answer format: Find an upper bound for the number of hares remaining after m minutes, in terms of p and m, which is optimal up to a multiplicative constant, and prove it.
GRiddles Series B: Puzzle 3
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Hare share
Let p > 2 be a prime and m < p − 1 be a positive integer. There are p spots arranged in a circle. Each spot is either a rabbit hole or a tree stump. Exactly p+12 of these spots are rabbit holes and the rest are tree stumps. At the start, on each tree stump, p hares, labelled 1, 2, . . . , p, are placed. Every minute, each remaining hare labelled i makes a single hop, landing on the spot exactly i positions further clockwise. Any hare that lands in a rabbit hole falls in and disappears. Find an upper bound, the best you can, for the number of hares remaining after m minutes, in terms of p and m.
Example for p = 7: the rabbit holes are shown in blue and the tree stumps in grey.
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