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I have a balancing problem which I am not sure if I understand completely correct.

Centrifuge has 36 holes. If in a hole only one test tube can be placed and all the test tubes are with equal weight.

What are the number of test tubes which can be used to balance the Centrifuge?

The answer that I have is:

The Centrifuge can work with even number of test tubes. (2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34 and 36)

And cannot work with odd number of test tubes (1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35)

The reasoning behind that is the following: If a test tube has equal weight as all the others in order to balance it each and every test tube should be placed in front of another.

Is this a correct approach?

Math Riddle
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    See this Numberphile video. – Jens May 15 '20 at 13:19
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    Why should 3 not work? Place the children at place 1,13,25 (equally spaced). Then there is no resulting torque, and the thing will be in balance – daw May 15 '20 at 13:27
  • Yes, this video is really helpful.

    I will try later to answer for each number balance is possible and will post it here to check if I am correct.

    I reword the question so it is obvious that it is about Centrifuge.

     – Math Riddle May 15 '20 at 13:33
  • I think any number of tubes that can be arranged to have rotational symmetry should do the trick. Like daw said, 3 should do it as well. – Galen May 15 '20 at 14:30
  • Surely, from a physics point of view, a cientrifueg that is weighted should be able to spin fast enough so it doesn't matter whether it's balanced or not because the speed itself should creste a centrifugal force that in turn gives a gyroscopic effect. that will automatically balance the centrifuge – William Pennanti May 15 '20 at 15:10
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    @WilliamPennanti So I guess you've never seen a washing machine go through a spin cycle with an unbalanced load. – Jaap Scherphuis May 15 '20 at 16:40
  • @WilliamPennanti Physically, you can get wobbling that can damage centrifuges over time by not balancing. Ultracentrifuges running at 100000 g have even exploded to tremendous effect. Balancing centrifuges is standard procedure in molecular biology. – Galen May 15 '20 at 16:55

2 Answers2

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My answer is

Every set except 1 and 35

Even numbers
It is obviously possible to place one pair of tubes on opposite sides.
Then you can add another pair of tubes at any free opposite places.
The centre of gravity remains in the centre of the wheel.
And so on, this covers $2$ to $36$ even numbers.

Odd numbers
Three tubes can be placed symmetrically like this where marked $3$:

enter image description here

As before, you can add another pair at any free opposite places, eg marked $5$.
Again, the centre of gravity remains in the centre of the wheel.
And so on, place them in 15 pairs of places – all except the original set of $3$.
Which leaves 3 holes while cannot be filled.
This gives $3$, $5$ ... $33$ odd numbers.

So all combinations are possible except $1$ and $35$.
You can't balance a single tube, or a single hole.

Weather Vane
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  • My "odd" example of $5$ looks unbalanced, but is not. Rot13(Vg qbrf abg znggre jurer lbh cynpr nal cnve, nf ybat nf gurl ner bccbfvgr). – Weather Vane May 15 '20 at 19:02
  • Thanks, I tried it with the formula and by placing tickmarks on the holes to try to balance. – Math Riddle May 15 '20 at 19:28
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Thank you all. Weather Vane, your drawing explains it really good and I can visualize it in my head.

I tried to do the math behind the conjunction: You can balance $k$ identical test tubes, $1 ≤ k ≤ n$, in an $n$-hole centrifuge if and only if both $k$ and $n-k$ can be expressed as a sum of prime divisors of n.

So the first thing that we need to to is to find the divisors of $n$. If >!$n$ is $36$ its divisors are: $$\text{1 2 3 4 6 9 12 18 and 36}$$

The prime divisors are: $$\text{2 and 3}$$

So now for all numbers both $n$ and $n - k$ should be made by summing only $2$ and $3$.

It is obvious that for 1 and 35 we cannot do it because for:

$$n=1 \text{ cannot be made of prime divisors of 36}$$ $$=> n - k = 35 \text{ sum of prime 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 2}$$ $$\dots$$ $$n=35 \text{ sum of prime 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 2}$$ $$n=1 \text{ cannot be made of prime divisors of 36}$$

for all the others we can make the numbers by summing 2 and 3.

$$n = 2 \text{ sum of prime 2}$$ $$ => n - k = 34 \text{ sum of prime 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 2 + 2}$$ $$\dots$$

$$n = 3 \text{ sum of prime 3}$$ $$ => n - k = 33 \text{ sum of prime 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3}$$

$$\dots$$

$$n = 4 \text{ sum of prime 2 + 2}$$ $$ => n - k = 32 \text{ sum of prime 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 2}$$

etc for all the other numbers.

Math Riddle
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