Mean Distance between Two Random Points in Unit Cube

Theorem
The mean distance $R$ between $2$ points chosen at random from the interior of a unit cube is given by:

The value $R$ is known as the Robbins constant.

Proof
From Mean Distance between Two Random Points in Cuboid:

The result follows by setting $a = b = c = \dfrac 1 2$.

Hence we have:

and:

So:
 * $r - r_1 = r - r_2 = r - r_3 = \dfrac {\sqrt 3 - \sqrt 2} 2$

Thus: