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?

(In reply to

re: Solution by Hugo)

You made a mistake in when n = 2. It would balance as:

7 - 10 - 11 - 8 - 9 - 12

Thus, the order for any n would be:

(6n+1) - (6n+4) - (6n+5) - (6n+2) - (6n+3) - (6n+6)

As for how to place them in the centrifuge, assuming the centrifuge had a factor of 6 slots, you would place them in that order starting from any given centrifuge slot, spacing them with the same number of empty slots between each centrifuge tube if there are more than 6 slots.

As for the problem in question, where there are 30 tubes, one could simply use this pattern numerous times with different values of n, as the rohit suggested. So, you would use the above protocol with n=1, then repeat it until n=4 (starting from a different empty centrifuge slot and again placing them equally apart from eachother, ignoring the previously balanced centrifuge tubes).

I hope this clarifies Rohit's solution.