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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-2/20h/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 2 / 20H / ii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

h

be the expected hitting time from any neighbour of the origin back to the origin. After the first step, the return-time calculation from part (i) gives

2^{n} = 1 + h,

(1)

h = 2^{n} - 1

. Translation by the target vertex is a graph automorphism of the hypercube that exchanges the origin and that adjacent vertex, so the reverse expected hitting time is the same. Hence

E_{0} T_{(0, \dots, 0, 1)} = 2^{n} - 1.

(2)

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