A circular centrifuge has 30 slots spaced evenly around its circumference. Thirty samples need to be spun in the centrifuge, their masses being 1g, 2g, 3g, . . . 29g, 30g. How can all the samples be placed in the centrifuge at once while keeping it balanced properly?
For what other values of N is it possible to balance an N slot centrifuge with samples weighing 1g, 2g, 3g, . . . (N-1)g, Ng?
I think it may come down to arranging the weights such that any selection of half A vs half B will be shown to be equal.
For example, six doesn't work, but here's what I mean. The
arrangement 1 6 3 2 5 4 will always be out of balance 10 vs 11 when
splitting the centrifuge along any of the 3 half divisions.
Hmmm. But each "imbalance" is offset by 120 degrees from every
other. It's too late for me to think too clearly, but
perhaps this arrangement would work after all. 6v5, 4v3, and 2v1
all "imbalance" symmetrically and create a balance. If this is
the case, then it should work for any n=2m, m is odd, where you can
create these symmetrical imbalances.