The Universe of Discourse: A puzzle about balancing test tubes in a centrifuge

A centrifuge with n circular slots must be loaded with k test tubes symmetrically to avoid explosion. For n=12, arrangements of 2, 3, 4, and 6 tubes work, while 1 and 11 don't. Surprisingly, 7 tubes can be balanced using a 3+2+2 configuration. The puzzle explores solvable combinations of n and k. Multiples of k always work, but other solutions exist. The general solution is unknown, though factors like gcd(n,k) > 1 guarantee solvability. Prime numbers of slots are particularly challenging, requiring all slots to be filled.