How would I go about selecting $3$ random points $p$, $q$, and $r$ on the perimeter of a square of side a such that they are as far apart from each other as possible (i.e. the sum of the euclidean distances from $p$ to $q$, $q$ to $r$, and $p$ to $r$ is maximized)?
Is there a general method to do this such that I can extend it to (for example) finding $5$ such points on the surface of a cube of side $a$? $7$ on the surface of a sphere of radius $a$? Not a textbook problem, was just thinking for fun and got lost.
Can try solving the square problem with a simulation on a computer, but looking for mathematical insight. Thank you.