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?

My first thought was a sum of X and Y components.

But maybe one can imagine a graphical vector solution, where each vector force has magnitude or length equal to 1, 2, 3, ... centimeters ; with a change in angle from the last vector of 2 pi / n.  In other words, construct an n-gon where sides are of lengths 1, 2, ..., n; all angles are equal.  The problem then becomes finding an ordered set of values for lengths that lead to a closed n-gon.

The math will probably be the same, but maybe there's a geometry trick that makes it easier.   Like considering the n triangles formed by drawing spokes from some central point to the vertices of the n-gon.  Do the perpendicular bisectors of the n-gon's sides all meet at one point?

Posted by Larry on 2005-10-18 23:57:32