Two bugs start on adjacent vertices of an icosahedron. Every "move" they each randomly walk along one of the five edges available to them. What is the expected number of moves for them to meet (on either an edge or a vertex)?

Assume that if they meet on an edge, that counts as a full move, even though they have each only traversed half of the edge.

If the answer is $\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers, what is $a+b$ ?

The answer is 601.

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Define the following states:

Note:Except for state 4, the number of the state represents the Hamming distance between the bugs.And let,

$E_n =$ Expected number of moves to meet from state $n$ .

For each possible state where they don't meet ( $n=1,2,3$ ) we can set up equations like this:

$E_n = 1 + \sum_{m=0}^{4} P(n \rightarrow m)E_m$

And clearly, $E_0 = E_4 = 0$

The $1$ accounts for the move going from $n$ to $m$ . And the sum account for all the possible outcomes after the move.

This gives the following set of linear equations:

Solving, $E_1 = \frac{550}{51}$

$550+51 = \boxed{601}$