HMMT 十一月 2017 · 冲刺赛 · 第 6 题

HMMT November 2017 — Guts Round — Problem 6

[ 3 ] Lunasa, Merlin, and Lyrica each have a distinct hat. Every day, two of these three people, selected randomly, switch their hats. What is the probability that, after 2017 days, every person has their own hat back? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2017, November 11, 2017 — GUTS ROUND Organization Team Team ID#

解析

[ 3 ] Lunasa, Merlin, and Lyrica each have a distinct hat. Every day, two of these three people, selected randomly, switch their hats. What is the probability that, after 2017 days, every person has their own hat back? Proposed by: Yuan Yao Answer: 0 Imagine that the three hats are the vertices of an equilateral triangle. Then each day the exchange is equivalent to reflecting the triangle along one of its three symmetry axes, which changes the orientation of the triangle (from clockwise to counterclockwise or vice versa). Thus, an even number of such exchanges must be performed if the orientation is to be preserved. Since the triangle is reflected 2017 times, it is impossible for the final triangle to have the same orientation as the original triangle, so the desired probability is 0 .