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?
(In reply to
re(8): Simplification by Leonidas)
Let us take one of the 6set weightings which work. So (A, B, C, D, E, F) = (1,4,5,2,3,6).
Now let place A on the unit circle at an angle k from the positive xaxis, anticlockwise. So A is positioned at (Cos[k],Sin[k]).
Continuing (anticlockwise) we place B, C, D, E, F at intervals of PI/3, so F is positioned at (Cos[k+5 Pi/3], Sin[k+5 Pi/3]).
Since k is arbitrary, we can let the yaxis be the one we want to balance over, so the moment in this direction is 0.
Applying the weightings we obtain for the xmoment
mx = Cos[k]+4Cos[k+pi/3]+5 Cos[k+2 pi/3]+3 Cos[k+4 pi/3]+6 Cos[k+5 pi/3].
Expanding
mx = Cos[k]+(2 Cos[k]  2 sqrt(3) Sin[k]) + (5 Cos[k]/2 5 sqrt(3)/2 Sin[k])+ (3 sqrt(3)/2 Sin[k] 3/2 Cos[k]) + (3 Cos[k] + 3 sqrt(3) Sin[k]) = 0.
QED

Posted by goFish
on 20051101 20:12:46 