This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
You are a biochemist, working with a 12-slot centrifuge. This is a gadget
that has 12 equally spaced slots around a central axis, in which you can
place chemical samples you want centrifuged. When the machine is turned on,
the samples whirl around the central axis and do their thing.
To ensure that the samples are evenly mixed, they must be distributed in the
12 slots such that the centrifuge is balanced evenly. For example, if you
wanted to mix 4 samples, you could place them in slots 12, 3, 6 and 9
(assuming the slots are numbered from 1 to 12 like a clock).
Problem: Can you use the centrifuge to mix 5 samples?
The superposition of any two solutions is yet another solution, so given
that the factors > 1 of 12 (2, 3, 4, 6, 12) are all solutions, the
only thing to check about, for example, the proposed solution 2+3 is
that not all ways of combining 2 & 3 would have centrifuge tubes
from one subsolution occupying the slot for one of the tubes in
another solution. For the case 2+3, there is no problem: Place 3
tubes, one in every 4th position, then place the 4th and 5th
diametrically opposed (each will end up in a slot adjacent to one of
the first 3 tubes). The obvious generalization is, what are the
numbers of tubes that cannot be balanced? Observing that there are
solutions for 2,3,4,5,6 tubes and that if X has a solution, 12-X has
also one (obtained by swapping tubes and holes), it is obvious that
1 and 11 are the only cases without solutions.
Here is how this problem is often solved in practice: A dummy tube
is added to produce a total number of tubes that is easy to balance.
For example, if you had to centrifuge just one sample, you'd add a
second tube opposite it for balance.