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
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