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Three Weights


Date: 12/07/97 at 17:08:59
From: Jeff MacDougall
Subject: Advanced Algebra Honors

I can't figure this out, it is a problem for my Algebra 2 class that I 
and my friends can't figure out:

A boy selling fruits has only three weights, but with them he can 
weigh any whole number of pounds from 1 pound to 13 pounds inclusive. 
What three weights does he have?


Date: 12/17/97 at 12:00:25
From: Doctor Mark
Subject: Re: Advanced Algebra Honors

Hi Jeff,

Problems like this have intrigued people for a long time, and the 
problem you are looking at was originally solved by a man named Claude 
Bachet (1587-1638).  Here's the idea.

First of all, we have to realize that if he puts the fruit in one pan 
of the balance (you have to imagine that this is how the boy is going 
to weigh the fruit), then he can put a particular weight in either the 
pan the fruit is in, or the other pan, or not put the weight in any 
pan at all.

Let the three weights be w, W, and K, where w is the lightest weight, 
W is the middle weight, and K is the heaviest weight.

Now clearly the heaviest weight we can weigh is, by assumption, 13, 
and this must be when the fruit is in one pan, and all three weights 
are in the other pan, so:

   w + W + K = 13

Now the second heaviest weight we can weigh is 12, and this must be
balanced when we have the two heaviest weights, W and K, in the other 
pan. Thus

   W + K = 12.

Hence, w = 1.

We could weigh a fruit of weight 11 by putting w in the pan with the
fruit, and W+K in the other pan. Then there would be 12 pounds in each 
pan.

How could we weigh a fruit of weight 10? If you think about it, this 
must involve putting the fruit in one pan, and w and K in the other 
pan.

That means that w + K = 10. Since w = 1, this means that K is 9, and 
hence that W must be 3.

So the weights are 1, 3, and 9. That's an interesting pattern, don't 
you think?

The original problem that Bachet considered was how many weights, and 
of what values, were necessary to weigh all weights up to 40. What 
answer do you think he got?

(Hint: what number do you get if you double 13 and add 1? And how is 
that related to the weights you needed? What if you do a similar thing 
for 40? Maybe this tells you what the pattern is. You might also think 
about how numbers are written in base 3...)

As a historical note, Fermat's Last Theorem, which was just solved 
4 years ago after defying the best mathematicians for more than 
350 years, was originally scribbled in the margins of Fermat's copy 
of Bachet's translation of Diophantus' Arithmetica (boy, that's a lot 
of possessives, eh?).

-Doctor Mark,  The Math Forum
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