## Demonstrating objects are of equal weight

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### Demonstrating objects are of equal weight

Given N objects drawn from pools of objects with two distinct weights, how many balance-scale tests are required to show that they're actually all the same weight?

### Re: Demonstrating objects are of equal weight

This may or may not be optimal.

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### Re: Demonstrating objects are of equal weight

ekim wrote:Given N objects drawn from pools of objects with two distinct weights, how many balance-scale tests are required to show that they're actually all the same weight?

I can do it with Ceiling(log[sub]2[/log]N) weighings:

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### Re: Demonstrating objects are of equal weight

Deleted because I didn't read the question properly.

This is my signature. There are many like it, but this one is mine.

### Re: Demonstrating objects are of equal weight

Zero. The problem can be solved without using a balance scale at all.

Or one, if weighing on a balance is the only permissible method to compare weights.

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Or one, if weighing on a balance is the only permissible method to compare weights.

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wee free kings

### Re: Demonstrating objects are of equal weight

Qaanol wrote:Zero. The problem can be solved without using a balance scale at all.Spoiler:

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Summum ius, summa iniuria.

### Re: Demonstrating objects are of equal weight

Thesh wrote:Qaanol wrote:Zero. The problem can be solved without using a balance scale at all.Spoiler:Spoiler:

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### Re: Demonstrating objects are of equal weight

Thesh wrote:Qaanol wrote:Zero. The problem can be solved without using a balance scale at all.Spoiler:Spoiler:

The problem specifies exactly two distinct weights, so that objection doesn't work.

The problem states "... how many balance-scale tests are required to show ..."

Normal people infer that what is meant is ".. if you were to use only a balance scale in the normal way, how many weighings are required to show ..."

I can only conclude that Qaanol is not normal .

### Re: Demonstrating objects are of equal weight

Okay, I parsed it as "There are two pools of objects, x and y, pool x weighs a, pool y weighs b, a != b."

Summum ius, summa iniuria.

### Re: Demonstrating objects are of equal weight

jaap wrote:The problem states "... how many balance-scale tests are required to show ..."

Normal people infer that what is meant is ".. if you were to use only a balance scale in the normal way, how many weighings are required to show ..."

I can only conclude that Qaanol is not normal :D.

I am quite entirely comfortable with that conclusion. However to be fair, I did make a slight mistake in one of my answers.

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wee free kings

### Re: Demonstrating objects are of equal weight

ekim wrote:Given N objects drawn from pools of objects with two distinct weights, how many balance-scale tests are required to show that they're actually all the same weight?

This doesn't make complete sense to me...if there are two distinct weights, how can they all be the same weight?

Or do you mean they are SUPPOSED to be two distinct weights, but they're actually not?

Or do you mean it's a given that every object you get will either be of weight A or weight B, but you don't know for any given object whether it has weight A or weight B, and also you don't know how many objects of weight A you have nor how many objects of weight B you have, and also you don't know if A is greater than or less than B, and also you don't know the value of A or B—but you do know that A does not equal B—and the question is to determine in a minimum number of weighings whether you have been given a sample consisting entirely of weight A objects? (Or, equivalently, a sample consisting entirely of weight B objects, since you don't know the values of A or B and would be labeling them arbitrarily.)

I think this last one is complicated enough that it's probably the intention of the puzzle.

In other words, if my guess is correct, you mean that you have N objects, which either (case 1) have all the same weight or (case 2) have exactly two distinct weights, and you're trying to determine which case is correct.

Is that right?

There's no such thing as a funny sig.

### Re: Demonstrating objects are of equal weight

Wildcard wrote:ekim wrote:Given N objects drawn from pools of objects with two distinct weights, how many balance-scale tests are required to show that they're actually all the same weight?

This doesn't make complete sense to me...if there are two distinct weights, how can they all be the same weight?

Or do you mean they are SUPPOSED to be two distinct weights, but they're actually not?

Or do you mean it's a given that every object you get will either be of weight A or weight B, but you don't know for any given object whether it has weight A or weight B, and also you don't know how many objects of weight A you have nor how many objects of weight B you have, and also you don't know if A is greater than or less than B, and also you don't know the value of A or B—but you do know that A does not equal B—and the question is to determine in a minimum number of weighings whether you have been given a sample consisting entirely of weight A objects? (Or, equivalently, a sample consisting entirely of weight B objects, since you don't know the values of A or B and would be labeling them arbitrarily.)

I think this last one is complicated enough that it's probably the intention of the puzzle.

In other words, if my guess is correct, you mean that you have N objects, which either (case 1) have all the same weight or (case 2) have exactly two distinct weights, and you're trying to determine which case is correct.

Is that right?

Yeah. Another way of looking at it is that you started with two large sacks of, say, 100N objects, one sack containing objects of weight A; the other containing objects of weight B, dumped them both out, the objects got mixed up, then you took a fairly small sample of N objects (half of 1% of the heap) and want to know whether you got a mixture of objects from both sacks, or all the objects in your sample came from the same sack.

The pools with 2 distinct weights means that out there in the general population of objects you could be working with, there are two (and only two) different weights. Having drawn some objects from that population, there's no guarantee you got a representative sample...

### Re: Demonstrating objects are of equal weight

In that case

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There's no such thing as a funny sig.

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