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(2): Solution by rohit)

I'm pretty sure that my example holds, because you should use the sine instead of the cosine.  As the angle drops to zero, sine goes to zero and cosine goes to one.  And as the angle goes to zero, the distance to axis also goes to zero.  Also, the radius perpendicular to the axis has an angle of ninety and distance to axis of one, which is consistent with sine instead of cosine.

But that's just one example.  Try calculating your wheel at an angle of 13 degrees, or 79.  As soon as you get into the awkward spaces between the weights, your calculator should explode with irrational numbers, meaning infinitely long strings of digits past the decimal.  It should become clear pretty quickly that you'll never get the sum of the irrational numbers on one side to equal the sum of the irrational numbers on the other side---unless you arrange the wheel so that the irrational numbers always perfectly line up with each other by putting identical weights opposite each other.

My objection is predicated on the assumption that the goal is to make the wheel balance perfectly on every axis through the circle's center.  The problem asks that the wheel be "balanced properly."  I'm not sure what that means. 

Rohit, your solution is probably the right one, in that it's the most balanced you can make the wheel.  I just wanted to point out that the wheel isn't perfectly balanced along every axis, and probably can't be within the constraints of the problem.

Posted by Leonidas on 2005-10-20 15:23:10