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?
As the comment type says, "just some thoughts", with a possible partial solution:
I'm working on the assumption that "balanced", in the true sense, means a center of gravity at the center of rotation. Under those assumptions:
1) We know that the total weight on the centrifuge is 465g. If an arbitrary slice is made BETWEEN samples (through the center of the centrifuge), it is impossible to have the same weight on each side of the slice. This alone tells me it is not possible for 30 slots.
2) Here is a way to get very close to balanced:
- Place the 1g weight anywhere on the circle
- Place the 2g weight opposite the 1g weight
- Place the 3g weight NEXT to the 2g weight
- Place the 4g weight opposite the 3g weight
- Place the 5g weight NEXT to the 4g weight, and so on
- The sequence should look like 1,4,5,8,9,12,13,16,17,20,21,24,25,28,29,2,3,6,7...
In this arrangement, any slice made between samples gives 232g on one side of the slice, and 233g on the other. Any slice made THROUGH the samples yields balanced halves).