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, . . . (N1)g, Ng?
You can start by arranging the numbers 1 to 6. Once you have arranged 1 to 6 the others can be arranged in a similar manner because all others can be expressed as 1 to 6 (+ 6x).
For example 7  12 can be expressed as 1+6,2+6 and so on.
1 to 6 can be arranged in the following sequence:
1  4  5  2  3  6. This sequence is balanced because the centrifuge is balanced across all diagonals. For example the 1  2 diagonal has 4,5 at one side and 3,6 at the other. The forces perpendicular to the diagonal are 4sin60 + 5 sin60 and 3sin60 + 6 sin60. Which being equal the centrifuge is balanced. This holds true for all diagonals.
Therefore it is possible to balance any multiple of 6 (including 30).
Therefore it is possible to balance 6n.

Posted by rohit
on 20051019 13:37:14 