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 separated. The slots on the centrifuge are numbered 1 through 12 clockwise (and clock-like).

           12  1
        11        2

      10            3
             °
       9            4

         8        5
            7  6

When the machine is turned on, the samples whirl around the central axis and do their thing. To ensure that the centrifuge runs smoothly, the samples must be distributed in the 12 slots such that it is balanced evenly.

Can you use the centrifuge to run 5 samples?

For example, if you have four samples, they may be placed in slots 1, 4, 7, and 10 to balance the centrifuge.

My first thoughts were for the "practical" solutions that Charlie suggests, but I find it quite interesting that Nick found an easy solution by combining both runs of Charlie's second solution (provided there is no overlap).

Examining John's question (whether there are, after accounting for rotations and reflections, more than one way to balance the five samples):

First place three samples in (say) 12, 4, and 8

Since there are three positions between any two samples, either the fourth sample will go next to one of the first three (e.g. position 11 or 1), which leads to John's and Nick's solutions and their reflections/rotations, or the fourth sample goes half-way between two of the first three (e.g. position 6). But if it is placed half-way between two, then it is opposite the third, and the fifth sample, which needs to go opposite the fourth, cannot be placed.

So, except for rotations and reflections, there is only one correct solution.

Posted by TomM on 2003-10-04 21:23:18