Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2014-02-20 00:37:08

Vishalini
Member
Registered: 2014-02-20
Posts: 6

Number of Weight stones

Qn:The answer for "What are the 4 minimum weight stones required to measure any weight from 1kg to 40 kg?" is "1kg,3kg,9kg,27kg". But if I was asked to find the weights between 1kg to 100kg (or) between 1kg to 200kg (or) something, how to find it?? Is there any formula to find the weights?? What is the logic behind that?? dunno

Offline

#2 2014-02-20 01:04:26

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Number of Weight stones

Hi;

Notice the powers of 3 and go from there.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#3 2014-02-20 01:08:37

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Number of Weight stones

Finding a general solution to this problem can be tough but there is another set like this:

With just six weights and a balance scale, you can weigh any unit number of kgs from 1 to 364. What could be the six weights?

1 kg, 3kg,9kg, 27kg, 81 kg, and 243kg.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#4 2014-02-20 01:18:19

Vishalini
Member
Registered: 2014-02-20
Posts: 6

Re: Number of Weight stones

bobbym wrote:

Hi;

Notice the powers of 3 and go from there.

Hello Sir,
I have given the answers as 1,3,9,27 and so u have easily said them as powers of 3. For such a question "Find the weights required to balance all the weights between 1& 100, How will u find?? and explain your logic

Offline

#5 2014-02-20 01:19:59

Vishalini
Member
Registered: 2014-02-20
Posts: 6

Re: Number of Weight stones

Agnishom wrote:

Finding a general solution to this problem can be tough but there is another set like this:

With just six weights and a balance scale, you can weigh any unit number of kgs from 1 to 364. What could be the six weights?

1 kg, 3kg,9kg, 27kg, 81 kg, and 243kg.

Your answer is correct. But, by what logic, u found these weights?? Pls explain

Offline

#6 2014-02-20 01:30:05

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: Number of Weight stones

hi Vishalini

Welcome to the forum.

You may be familiar with binary arithmetic.

eg.  7 = 111in binary = one 4 + one 2 + one 1

If not: http://www.mathsisfun.com/binary-number-system.html

If the problem only allowed weights on one side of the balance, then you couldn't do better than having weights of 1,2,4,8,16 .....

When weights are allowed on either side, this, in effect, allows us to make a weight by subtraction as well as addition

eg.  7 = 9 - 3 + 1

and base three provides the optimal solution, because we can count one up from a base number or one down.

http://hotmath.com/hotmath_help/topics/ … bases.html

eg. 100 = 81 + 27 - 9 + (no 3 needed) + 1

I don't think you will find you can do all numbers up to 100 with less weights.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#7 2014-02-20 01:32:15

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Number of Weight stones

Sorry, I have no idea. bobbym might know


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#8 2014-02-20 01:36:08

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Number of Weight stones

I believe you would use base arithmetic. In this case base 3 ( case base, that rhymes ) as Bob has already done.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#9 2014-02-20 01:39:21

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Number of Weight stones

Actually I had not seen his post


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

#10 2014-02-20 02:21:20

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Number of Weight stones

Hi Bob

Yey, you are correct,  it cannot be done with less (for the simple reason of 3^4 being smaller than 100 smile).

Hi Vishalini

If you want to be able to measure up all weight between 1 and n, you will need no less than floor(log_3(n))+1 weights.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#11 2014-02-20 02:24:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Number of Weight stones

Hi anonimnystefy.

But are there other basis that can also do the problem that are the same length?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#12 2014-02-20 02:29:45

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Number of Weight stones

Hm?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#13 2014-02-20 02:33:22

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Number of Weight stones

Supposing I said that a,b,c and d could be used as a basis for numbers from 1 to 80. How can you check that I am right or not?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#14 2014-02-20 02:43:24

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Number of Weight stones

I didn't say anything about 80. I said it about 81.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#15 2014-02-20 02:48:45

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Number of Weight stones

Supposing it was for 100, how can we determine if a,b,c,d and e can make it?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#16 2014-02-20 02:52:25

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Number of Weight stones

I can just say 1,3,9,27,81...


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Offline

#17 2014-02-20 03:00:47

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: Number of Weight stones

In my head I've just proved that base 3 weights are optimal and that no other set of weights will do the same job.  Shall I write out a formal proof?

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#18 2014-02-20 03:01:05

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Number of Weight stones

Yes, I know but what about another basis that may be the same size but different?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#19 2014-02-20 03:04:41

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: Number of Weight stones

I think no other set will do.  Would you like a proof?

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#20 2014-02-20 03:12:51

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Number of Weight stones

Hi Bob;

Check this out.

http://www.mathsisfun.com/puzzles/measu … ution.html

For 80 he uses 2 , 6, 18, 54. True they are multiples but they are different.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#21 2014-02-20 03:29:45

Bob
Administrator
Registered: 2010-06-20
Posts: 10,052

Re: Number of Weight stones

hi bobbym,

That's a clever idea; but it relies on the 'balance' sometimes not balancing.  I was assuming the requirement was for a balance between the two scale pans.

Looking back to the OP that is permissible so it would appear there are more solutions.


Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

Offline

#22 2014-02-20 03:36:18

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Number of Weight stones

Hi Bob;

I agree with the 1,3,9, 27 answer but I was just wondering in case someone gets the brilliant idea of changing the question a little bit. If someone came in here and asked for 5 stones and asked how many there are I would be stuck, having no easy way to determine the answer.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

Board footer

Powered by FluxBB