# 322 logic/centrifuge.p

## Description

This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.

# 322 logic/centrifuge.p

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?

logic/centrifuge.s

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.

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