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Week 1 Homework

 
 
Picture of Alex Gou
Week 1 Homework
by Alex Gou - Monday, March 7, 2022, 7:42 PM
 

Does anyone know how to do this problem?

What is the minimum number of weights which enable us to weight any integer number of grams of gold from 11 to 10001000 on a standard balance with two pans if you are only allowed to use the weights on one side of the pan? Explain your answer using number bases.

Thanks.

 
Picture of John Lensmire
Re: Week 1 Homework
by John Lensmire - Tuesday, March 8, 2022, 10:27 AM
 

Hi Alex,

Let me give both an example of weights and a hint for this problem.

As an example, suppose you have the weights of $5$, $8$, and $13$ grams. Using one of these weights you can get $5$, $8$, or $13$ grams, with two weights you can get $13$, $18$, or $21$ grams, and using all three gives $26$ grams. Thus, in total you can get$$5, 8, 13, 18, 21, \text{ or } 26$$grams. (Note $13$ we can get two ways for later.)

As a hint for the actual problem, try to work your way up. Clearly you need to get $1$ gram, so you need a $1$ gram weight. How can we $2$ grams? Well either we add another $1$ gram weight or a $2$ gram weight. To avoid duplicates, suppose you add a $2$ gram weight. Now we can get $1$, $2$, or $1+2 = 3$ grams, so $4$ grams is the smallest weight we cannot get so far. What if we then add a $4$ gram weight?

Try to continue from here and see if you can find a pattern.

Hope this helps!