Single-Cross 2

August 2023 : Puzzle

Consider 3-space (i.e. R3) partitioned into a grid of unit cubes with faces defined by the planes of all points with at least one integer coordinate. For a fixed positive real number D, a random line segment of length D (chosen uniformly in location and orientation) is placed in this cubic lattice.

What length D maximizes the probability that the endpoints of the segment lie in orthogonally adjacent unit cubes (that is, the segment crosses exactly one integer-coordinate plane), and what is this maximal probability? Give your answer as a comma-separated pair of values to 10 significant places (e.g. “1.234567891,0.2468135792”).