Number of Weight stones

Vishalini · 2014-02-20 00:37:08

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??

bobbym · 2014-02-20 01:04:26

Hi;

Notice the powers of 3 and go from there.

Agnishom · 2014-02-20 01:08:37

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.

Vishalini · 2014-02-20 01:18:19

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

Vishalini · 2014-02-20 01:19:59

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

Bob · 2014-02-20 01:30:05

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

Agnishom · 2014-02-20 01:32:15

Sorry, I have no idea. bobbym might know

bobbym · 2014-02-20 01:36:08

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

Agnishom · 2014-02-20 01:39:21

Actually I had not seen his post

anonimnystefy · 2014-02-20 02:21:20

Hi Bob

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

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.

bobbym · 2014-02-20 02:24:57

Hi anonimnystefy.

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

anonimnystefy · 2014-02-20 02:29:45

Hm?

bobbym · 2014-02-20 02:33:22

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?

anonimnystefy · 2014-02-20 02:43:24

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

bobbym · 2014-02-20 02:48:45

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

anonimnystefy · 2014-02-20 02:52:25

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

Bob · 2014-02-20 03:00:47

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

bobbym · 2014-02-20 03:01:05

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

Bob · 2014-02-20 03:04:41

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

Bob

bobbym · 2014-02-20 03:12:51

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.

Bob · 2014-02-20 03:29:45

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

bobbym · 2014-02-20 03:36:18

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.

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

Number of Weight stones

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

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