The random walk of two drunks The problem is such: two drunks start at either end of an alleyway of length n. Apart from at the ends, they each move one step forwards or one step backwards randomly. At the ends of the alley they move towards the centre with probability $1$. What is the expected number of steps before they meet?
In the problem I'm looking to solve $n = 99$, though it could be any odd number.
We are looking for the most elegant solution to this problem. It has been solved via Monte-Carlo and Markov Chains in our group, but I want to believe there is a simpler solution not involving $2400 \times 2400$ matrices. I am trying to take inspiration from the question below, but the boundary condition complicates matters.
Exact probability of collision of two independent random walkers after N steps
Links to papers or other questions dealing with this particular problem would be gladly accepted.

Here are some easy remarks on the question (but not a solution). First, if $n$ is odd the two particles never meet, for parity reasons, hence one can assume that $n$ is even. 
Say that the particle starting from $0$ hits $n$ at time $T_n$. The particles cannot cross each other without meeting hence, at time $T_n$, they already met. Thus, the meeting time $M_n$ is almost surely finite and $$E(M_n)\leqslant E(T_n)=n^2. 
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Note that $M_n$ is also at most the hitting time $T'_n$ of $0$ by the particle starting from $n$, hence $$E(M_n)\leqslant E(\min(T_n,T'_n)),$$ where $T'_n$ is an independent copy of $T_n$. 
Conversely, the particles cannot meet before at least one of them hits $n/2$ hence $$E(M_n)\geqslant E(\min(T_{n/2},T'_{n/2})).
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This suggests that there might exist some finite positive $\mu$ such that $E(M_n)\sim\mu n^2$ when $n\to\infty$. If this is so, one knows that $\frac14\theta\leqslant\mu\leqslant\theta$ where $\theta$ in $(0,1)$ is defined by the equivalent $E(\min(T_n,T'_n))\sim\theta n^2$. 
Note finally that $\min(T_n,T'_n)$ is the first hitting time of the boundary of the square $[-n,n]^2$ by a random walk on $\mathbb Z^2$ starting from $(0,0)$ with steps $(\pm1,\pm1)$.
