You are a biochemist, working with a 12-slot centrifuge. This is a gadget that has 12 equally spaced slots around a central axis, in which you can place chemical samples you want separated. The slots on the centrifuge are numbered 1 through 12 clockwise (and clock-like).
12 1
11 2
10 3
°
9 4
8 5
7 6
When the machine is turned on, the samples whirl around the central axis and do their thing. To ensure that the centrifuge runs smoothly, the samples must be distributed in the 12 slots such that it is balanced evenly.
Can you use the centrifuge to run 5 samples?
For example, if you have four samples, they may be placed in slots 1, 4, 7, and 10 to balance the centrifuge.
I believe we CAN run 5 samples!
If we think of the slots on the centrifuge being numbered as the face of a clock (with 12 at the top), we can use slots 12, 1, 5, 6 and 9.
Draw yourself a diagram of a clock face with radius 1 centered at the origin of the x-y plane and consider these 5 points on it. Obviously they will have equal weighting in the y-direction, as 9 is on the x-axis (so has no weighting), and the other points balance out in pairs (12 with 6, 1 with 5). Now in the x-direction, the points at 12 and 6 have no weighting, and the point at 9 balances the two points at 1 and 5. This is because the points at 1 and 5 are at a 60 degree angle from the x-axis, and hence using trigonometry their distance d in the x-direction is 1/2.
[cos 60 = d/1
d = 1/2]
So each of these two points is 1/2 unit to the right of the origin, and this balances perfectly with the point at 9 which is 1 unit to the left.
Interestingly, this means that we could also run a set of 7 samples, simply by using the opposite slots of the centrifuge (ie 2,3,4,7,8,10 and 11).
I wonder if there are any other patterns that would work (not counting rotational copies of the same pattern of course)
Thanks DJ - another fun problem!
-John
Proposed solution
