For one of the exercises of my homework I need to answer the following question, but I am not sure how I should apply gamblers ruin theory to solve this problem (it is stated as a hint, not that I must use it). The problem is as follows:
A mouse is in a room in a circular hallway with $N+1$ rooms. In the the other $N$ rooms there is a cheese. He has a chance of $\frac{1}{2}$ to go either way every step. If he enters a room with a cheese he eats the cheese. Prove that all cheeses have an equal chance of being eaten last, namely $\frac{1}{N}$.
I see that eating the $i^{\textrm{th}}$ cheese as last has two possibilities, first mouse walks to $i+1$ and then goes to $i-1$ without passing $i$, and the other way around. But I am at loss what to do next.
Hint: you may try setting up the 1-step analysis (as they do in the analysis of random walks). Here, you denote $p_i(j)$ the probability, starting at $i$, to eat the cheese at $j$ the last. Then, do one step of the random walk and recalculate $p_i(j)$'s...