Pentagon walk, reaching a non-neighboring corner

Question

Five corners A, B, C, D, E form a ring (edges A-B, B-C, C-D, D-E, E-A). A walker starts at A and each step moves to an adjacent corner, either direction with equal chance. The target is C, which is two steps away from A.

On average, how many steps until the walker first reaches C?

Accepted Answer

Let a, b, d, e be the expected steps to reach C from A, B, D, E, with C itself costing 0. Reflecting the ring across the line through C swaps B with D and A with E, so a = e and b = d. Writing the "1 plus average of neighbors" equations: b = 1 + 	frac12 a + 	frac12 \cdot 0 = 1 + 	frac a2, a = 1 + 	frac12 b + 	frac12 e = 1 + 	frac12 b + 	frac12 a \implies 	frac12 a = 1 + 	frac12 b \implies a = 2 + b. Substitute b = 1 + 	frac a2: a = 2 + 1 + 	frac a2 = 3 + 	frac a2 \implies 	frac a2 = 3 \implies a = 6. Takeaway. Set up one equation per corner and use the ring's reflection symmetry;…

Pentagon walk, reaching a non-neighboring corner

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