The plane is marked with an infinite checkerboard grid: all vertical lines x = k and all horizontal lines y = k
for every integer k (a square grid of unit side length).
A random line segment of length D is dropped on it: one endpoint is placed uniformly at random on the plane
(equivalently, since the grid repeats, uniformly at random in one unit square), and the segment's direction is an
angle chosen uniformly from [0, 2π), independent of the position. Ignore probability-zero events such as the segment
passing exactly through a grid corner or an endpoint lying exactly on a line.
Let f(D) be the probability that the segment crosses exactly one grid line (counting vertical and horizontal lines
together).
Which length D > 0 maximizes f(D), and what is the maximal probability?
Answer format: two exact values, in this order: D, then the maximal probability (closed forms with radicals and π are allowed).