**9**

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**