The expected number of steps a bug walks through all edges

Question

(It's from one Quant related interview question)

There is a tetrahedron(A, B, C, D) and a bug at one of the vertices(like A) is performing a random walk on the edges (all edges with the same length), at any vertex where it can select any of the 3 edges to walk on with same probability(1/3), What is the expected number of steps it needs till it walks through all different edges(six different edges)?

It's easy to work out the expected number of steps it walks from one to two different edges, E(0->1) = 1, E(1->2) = 3/2, of course, it's easy to call the Markov chains expected time to absorption, but it's complex if you continue to calculate the transition probability recursively.

Answer 1

Hint: the analogous problem for vertices can be modelled through a Markov chain on $8$ states, representing the visited vertices. These states can be lumped into $4$ states $L_0,L_1,L_2,L_3$, leading to a random walk on $L_0 L_1 L_2 L_3$ moving toward the right. We go from $L_0$ to $L_1$ with probability $1$, from $L_1$ to $L_2$ with probability $\frac{2}{3}$, from $L_2$ to $L_3$ with probability $\frac{1}{3}$. It follows that the expected number of steps for visiting the whole set of vertices is $1+\frac{3}{2}+3=\frac{11}{2}$. Can you do the same (or something similar, based on the sub-problems

    /|\       |\       |\     |\     \
   / | \      | \      | \    | \     \
   \ | /    \ | /    \ |      | /     /
    \|/      \|/      \|      |/     /

) with the edges?