Maths teaser

[Guest pawnhead]

Here's a teaser from my old school days.

My maths teacher posed this problem to the class, and gave us five minutes.

I came up with the highest number, and the teacher scoffed at my result.

After expliaining my result, one student agreed with me, but the teacher said that I was wrong, and he would devote no more class time to the problem.

To this day, I'm still frustrated at this teacher as I'm certain that I was the only one with the correct solution.

Anyway, thirty years later, I'm after some vindication, so here it is:-

Given a set of equal arm scales, like the Justice scales, and the following weights :- 1kg, 3kg, 9kg and 27kg, how many different weights can be measued?

[Aeschylus]

I believ you can weigh any weight between 1kg-40kg, though I haven't checked.

[Cap'n Refsmmat]

How would you do 29?

[noz92]

40, unless decimals and fractions are counted into the weight.

[Callipygous]

yeah, you can do every number from 1-40. what answer did you give? what did the teacher say it was?

[Treadstone]

do you only have one of each wieght? if so you couldnt weigh out 2kg and so the 1-40 idea is wrong.

 

nathan

[Aeschylus]

How would you do 29?

 

27kg + 3kg on one side 1kg + the object you wnat to weigh.

 

do you only have one of each wieght? if so you couldnt weigh out 2kg and so the 1-40 idea is wrong.

 

nathan

 

For 2kg, 3kg on one side and 1kg plus the object you want to weigh on the other.

[Guest pawnhead]

The answer I gave was forty, and it's obvious that whoever formulated the problem had this answer in mind, because the numbers given follow a formula:-

Take the sum of the previous numbers, double it, and add one.

These are the only four weights that enable you to weigh everything from one to forty kilograms without missing anything, or doubling up on combinations, however;-

I didn't realise this explanation at the time, and the teacher said that I was wrong because the weights should only be used on one side of the scales.

 

Now to the second part:-

I first posted this on my usual forum http://forums.overclockers.com.au/showthread.php?s=&threadid=302198&perpage=15&pagenumber=1

And it opened up a debate on zero weight and negative weights.

I believe that, as in mathematics, zero holds the position of a number, but it is nothing, and has no value as a weight, it’s just nothing. You can’t measure what’s not there, except the absence of something that can be measured.

I realise that nothing has been found that opposes gravity, but a helium balloon, although it has positive mass, has a negative weight, if measured under atmospheric pressure, opening up the possibility of another forty weights.

Somebody pointed out that a one kilo weight is a one kilo weight, whether it’s positive or negative, but a one kilo chicken is not the same as a negative one kilo helium balloon, and they can both be measured by a one kilo weight and scales.

I’m a bit confused, so I signed up to this forum.

Can anyone help clear this up in my mind?

[Aeschylus]

It's rubbish: even helium balloons have 'postive weight' as weight is defined as mg and 'zero' is ther trivial solution and should be ignored.

[haggy]

I think the point of the problem is the concept of the "base 3" number system.

1 = 3^0

3 = 3^1

9 = 3^2

27 = 3^3

 

Any number in base 10 will be able to be represented in base 3.

E.g. 29 = 1*3^3 + 2*3^0

 

But 2*3^n = 1*3^(n+1) - 1*3^n, corresponding to putting the two weights on the opposite side of the scale.

[YT2095]

It's rubbish: even helium balloons have 'postive weight' as weight is defined as mg and 'zero' is ther trivial solution and should be ignored.

the guy DID state "if measured under atmospheric pressure", in which case he`s right, as hellium will exhibit a negative weight due to displacement of a heavier gas combo (air).

[Aeschylus]

the guy DID state "if measured under atmospheric pressure", in which case he`s right, as hellium will exhibit a negative weight due to displacement of a heavier gas combo (air).

 

Weight is not the net force on the balloon it it is the force due to the gravity of the Earth, so even if the helium balloon moves upwards it's weight is still in the same direction.

[YT2095]

I agree entirely, the rubber in the ballon and the helium is measurable, sure.

but it`s the principal behind the lift (or negative weight as he calls it) that`s the important factor in his illustration

 

his example at face value makes sense, going deeper into the WHYS or details is just sidetracking from the maths issue, it`s not a Physics based thread

[Primarygun]

Do air pressures act towards the sky ?

[YT2095]

to a point yes, and then eventualy the helium ballon will reach a pressure equilibrium and go no further upwards

[aommaster]

I beleive ancient people used to use something like this, and they were able to measure from 1-40

[Guest pawnhead]

Weight is not the net force on the balloon it it is the force due to the gravity of the Earth, so even if the helium balloon moves upwards it's weight is still in the same direction.

Giving a negative result on the scales which are measuring a downward force, and thus supporting my argument that a further forty weights can be measured

[Aeschylus]

Giving a negative result on the scales which are measuring a downward force, and thus supporting my argument that a further forty weights can be measured

 

Yep, but you're not measuring weight tho', which is what the original question asked.

[Guest pawnhead]

Yep, but you're not measuring weight tho', which is what the original question asked.

OK so what ARE you measuring, "net force"? Isn’t that what weight is all about?

If a kilo is in orbit, in constant freefall, or in a cave, in the centre of the Earth, then it’s weightless, but it’s still a kilo, in mass.

If a kilo is at high tide level, with an atmosphere, then it gives us what is our impression of what a kilo weighs.

If a kilo is under water then it gives us a distorted impression.

Spring scales would have to be recalibrated according to the medium that they are operating in, and the gravity that they are subjected to, but equal arm scales would operate in any medium, regardless of gravity. (If subjected to acceleration, basically the same force as gravity) So:-

How do you measure the weight of a helium balloon, using equal arm scales? (Assuming that you want to breathe at the same time) Are you saying that it must be subjected to a vacuum? (But equal arm scales operate regardless of the medium) Are you saying that you must apply an upward force on the opposite side, and not a downward force on the same side? (But the result is the same) Or are you saying that a kilo is a kilo regardless of it’s vector? (But a chicken is not the same weight as a helium balloon)

I like YT2095’s retort “going deeper into the WHYS or details is just sidetracking from the maths issue”.

But it’s the “WHYS” and the “details” that’s making this interesting.

And confusing.

[paganinio]

WEIGHT approximately equals MASS*9.8, a balloon's weight is a VECTOR with positive NUMBER

 

If you don't regard it as a vector, then it's a positive number