Comment by cornel_io

jmount has an explanation elsewhere in this thread linking to https://win-vector.com/2021/12/26/how-to-read-sourav-chatter... which does a great job of explaining the intuition, but in a nutshell the normalization factor of 3 comes from the fact that if you select two random points between 0 and 1, the mean distance between will be 1/3 (which is pretty easy to write down and solve, boils down the the fact that a pyramid of height 1 that's 1x1 at the base has volume = 1/3).