Expert: Paul Klarreich - 6/18/2008

Question
Hi Paul

I was looking through allexperts and was wondering if you could help me with this problem as I see you have answered a weighing problem in the past and thought you might be the chap to help me understand this. The question is for no reason other than frustrating interest. QUESTION: In the days before metrification and digital balances, there was a merchant who sold items by the pound. He had a traditional set of scales and 4 weights. With this equipment he could accurately weigh anything between 1 and 40 pounds in units of one pound. What were the weights he used?
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I don't know where to start with this.  I came across it in a puzzle book and have tried a number of combinatorics but can't really formulate the logic.  It may even be a non mathematical lateral thinking problem.  Any ideas ? Thanks in anticipation of any response. Cheers

Answer
Questioner:   Steven Simpson
Category:  Advanced Math
Private:  No
 
Subject:  Balance & Measuring Problem
Question:  Hi Paul

I was looking through allexperts and was wondering if you could help me with this problem as I see you have answered a weighing problem in the past and thought you might be the chap to help me understand this. The question is for no reason other than frustrating interest. QUESTION: In the days before metrification and digital balances, there was a merchant who sold items by the pound. He had a traditional set of scales and 4 weights. With this equipment he could accurately weigh anything between 1 and 40 pounds in units of one pound. What were the weights he used?
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I don't know where to start with this.  I came across it in a puzzle book and have tried a number of combinatorics but can't really formulate the logic.  It may even be a non mathematical lateral thinking problem.  Any ideas ? Thanks in anticipation of any response. Cheers
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Hi, Simpson,

[I conclude you are from the U.K., where people like to be addressed by their surnames, as in "Amazing, Holmes!", or "Elementary, Watson."]

If he could only put his weights on one side of his balance, he would use a set that is in powers of 2:

1,2,4,8, will give 1..15, for example.  But he could put them on both sides.  That suggests powers of 3:

1,3 can give you all up to 4:  2 = 3-1, for example.

1,3,9 = all up to 13.  

Examples:  5 = 9 - 3 - 1,  7 = 9 - 3 + 1

Now how about  1,3,9,27?

I think, without checking them all out, that this might do it.