You start at point S and move north. When you hit a wall you randomly choose with equiprobability to go either way along the wall. When you hit a corner you choose with equiprobability to following the wall round the corner or to turn back on yourself. If the wall disappears from next to you then you carry on in the direction you are moving.
There are two exits $Z_1$ and $Z_2$. What is the ratio of $Z_1:Z_2$
So obviously $Z_1$ is going to be larger as $P(Z_1)>0.75$ just from the fact half the first hit against the wall go out of $Z_1$ and then half of the 2nd hit.
I started to pen and paper this and the probabilities begin the following;
$P(Z_1)=\frac{1}{2}+\frac{1}{4}+\frac{1}{32}+\frac{1}{64}+...$
This pattern continues infinitely so denominator powers are $2^1,2^2,2^5,2^6,2^9,2^{10}...$
$P(Z_2)=\frac{1}{16}+\frac{1}{32}+\frac{1}{32}+\frac{1}{256}+\frac{1}{512}+...$
This pattern continues infinitely so denominator powers are $2^4,2^5,2^5,2^8,2^9,2^{9}...$
Now I think there is another series that needs adding to both the $P(Z_1)$ and $P(Z_2)$ to complete the comparison then you can cancel and solve the ratio as tend to infinity.
Can someone confirm this is the correct way to go about this?
Here is another method.
There are four walls you can hit, and three corners. Let $w_1$ be the probability of eventually exiting by $Z_1$ when you hit the top wall, and $c_1$ the probability of eventually exiting by $Z_1$ when you hit the top right corner. Then, as you say, $$w_1=\frac12(1+c_1)$$ There are seven variables, and you can form an equation like that for each one.
Perhaps you can solve the complete set. How much do you know about simultaneous equations?