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 should point out that the pairings shown on my solution are the diagonals of the circle. The actual arrangement is arrived at by inserting the first 15 weights in 15 consecutive slots clockwise, and the remaining 15 weights in consecutive slots counter-clockwise. The weights 1 and 16 are adjacent.  For 8 weights,

 

                       2        3

                1                       4

                   5                 8

                        6       7

Gee! It's tough drawing a circle in a smooth way, but convince yourself that in a real centrifuge, the weights as shown will balance perfectly.               

Posted by Mike Yadasko on 2005-10-20 04:52:37