The Centrifuge Problem

Twelve slots, five vials, and how to balance them.

A centrifuge spins samples fast enough that an uneven load will wreck it. The rotor turns thousands of times a minute, and every rotation of an unbalanced load drags the axle sideways. Load it badly and the motor tears itself off its mount.

So the weight has to be spread evenly around the axis. This rotor has twelve slots, arranged like a clock face.

Two vials are easy: put them opposite each other, at 12 and 6. Three are easy as well — space them a third of the way around, at 4, 8 and 12. In both cases you can see the symmetry without doing any arithmetic.

Five is the interesting one. Five does not divide twelve, and there is no way to space five vials evenly around a twelve-slot circle.

Reason about it for a minute and decide for yourself. Is it possible to load five vials into a twelve-slot centrifuge and have it balance?