On average, how many edge lengths will the spider walk before getting to fly?

Question

A spider and a fly are diametrically opposite vertices of a web in the shape of a regular hexagon. The fly is stuck and cannot move. On the other hand, the spider can walk freely along the edges of the hexagon. Each time the spider reaches a vertex, it randomly chooses between two adjacent edges with equal probability and proceeds to walk along that edge. On average, how many edge lengths will the spider walk before getting to fly?

Answer 1

9

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Let the hexagon be ABCDEF, with the spider standing on A. By symmetry, the
expected number of steps before the spider reaches the fly on B and F, and on C and E, are the
same. Let the expected number of steps before the spider reaches the fly from A be a, from B be
b, and from C be c.
Note that from A, the spider will take one step, and with equal probability, move to either
B or F. The the number of steps the spider will take is 1 plus one-half times the number of
steps it takes from B, plus one-half times the number of steps it takes from F. By linearity of
expectation, we have
a = 1+1/2b +1/2b.
Writing similar equations gives us
b = 1+1/2a +1/2c,
c = 1+1/2b +1/2*0.
Solving yields (a, b, c) = (9, 8, 5), giving the answer 9