Problem $6.$ Weights and Scales $(7$ points$)$

You have a double pan balance scale on which you can place weighted objects on both sides. For example, if you have two objects, one with weight $5$ and one with weight $7,$ you can measure items of weight $5,7,$ or $12.$ You can also measure items of weight $2,$ by placing $7$ on one side and $5$ on the side which contains the object to be weighed. If the scale balances, you know that the item has weight $2$ since $5+2=7.$

$($a$)[3$ points$]$ Suppose you have six objects, with weights $1,2,4,8,16,32.$ Clearly explain how you can use the balance scale to measure an item of weight $N,$ where $N$ is any whole number between $1$ and $50.$

Solution:

$($b$)[4$ points$]$ Suppose you have four objects with weights $1,1,4,$ and $9.$ I claim that there is a value $x$ such that you can use the balance scale to weigh any item of a whole number weight from $1$ to $x?$ Determine the value of $x$ and provide a brief justification as to why you can weigh each number in the range, but not $x+1.$

Solution: