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HMMT Combinatorics 2019/6: Sequence of eight reflections preserving center

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February 17, 2019
HMMTcombinatorics

Problem Statement

A point PP lies at the center of square ABCDABCD. A sequence of points {Pn}\{P_n\} is determined by P0=PP_0 = P, and given point PiP_i, point Pi+1P_{i+1} is obtained by reflecting PiP_i over one of the four lines ABAB, BCBC, CDCD, DADA, chosen uniformly at random and independently for each ii. What is the probability that P8=PP_8 = P?
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