Let theta be the angle of the point's initial vector. After traveling a distance r, the point has moved r*cos(theta) horizontally and r*sin(theta) vertically, and thus has struck r*(sin(theta)+cos(theta))+O(1) walls. Hence the average distance between walls will be 1/(sin(theta)+cos(theta)). We now

average this over all angles theta
2/pi * intg from theta=0 to pi/2 (1/(sin(theta)+cos(theta))) dtheta

which (in a computation which is left as an exercise) reduces to

2*sqrt(2)*ln(1+sqrt(2))/pi = 0.793515.

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