PUMaC 2025 · 组合（A 组） · 第 8 题

PUMaC 2025 — Combinatorics (Division A) — Problem 8

Trevor the traveller starts at the origin of R . He then takes a series of 8 steps that change his 2 position by a vector in the set { ( a, b ) ∈ Z : | a | , | b | ≤ 1 and ( a, b ) ̸ = (0 , 0) } . He takes each of the 8 possible steps exactly once, and takes the steps in uniformly random order, so that he returns 2 to the origin. The winding number of each point P in R , except those exactly on the path, is defined as the number of times any ray starting at P going out to infinity crosses a left-pointing portion of Trevor’s path minus the number of times the ray crosses a right-pointing portion of 2 the path, where left and right are defined relative to the ray. Let A be the area in R weighted 2 by winding number. An example is shown below. Find the expected value of A . y winding = 1 winding = 0 6 winding = − 1 3 1 3 A = 1 · + 0 · + ( − 1) · = 0 4 4 4 2 5 1 7 right left P x 3 8 4 Weighted areas for steps, in order, (1 , 1) , ( − 1 , 0) , ( − 1 , − 1) , (1 , 0) , ( − 1 , 1) , (0 , 1) , (1 , − 1) , (0 , − 1) 2

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