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?

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 2005-10-19 13:37:14