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 re(4): Sums of irrationals by goFish)

I had never heard that sums of irrationals can be rational.  Can you give an example, other than something trivial?  Or do you mean that we can't prove that a given sum of irrationals is irrational?

In either case, keep in mind that for the circle in this problem to truly balance at every angle, the sum of all those sines with coefficients must not only resolve to something rational, but resolve to zero at every angle.  I don't know how to prove that it can't happen, but it seems pretty obvious.

Posted by Leonidas on 2005-10-31 16:32:45