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?
(In reply to You can get very close, but...
As a follow-up, I did verify that this "close" solution does indeed balance completely by simply summing moments about any arbitrary cross-cut (thanks goFish).
It seems that 10 slots will work as well, with the sequence:
And 6 does work with:
With the little algorithm I used, I couldn't get 4, 8, or 12 to work. My supposition is that it can be balanced for 4N+2 (N=1,2,3...), but perhaps there is another arrangement different than mine that will allow others.