charges on sphere
This problem with $N$ point charges on a sphere is a famous problem in electrostatics known as the Thomson problem. For large $N$, it is in general an open problem still under active research.
References:
Wikipedia.org
Mathworld.wolfram.com
Mathpages.com
This has been a problem since Thomson proposed the arrangement of electrons and positive charges (nucleus was not known at that time) in rigid electron shells of atom which is what called Plum-pudding model of atom. He suggested that electrons are arranged in a symmetrical pattern with respect to the center of sphere which is applicable only to smaller elements in periodic table (Old-timer wandered a lot after discovering the $e/m$ ratio).
While googling, I found this applet which generates some arbitrary patterns (up to 5000). I think there are many algorithms which can be used to solve these kind of patterns up to some finite value.