What is the smallest number of integer weights required to exactly balance every integer between $1$ and $40$? 
What is the smallest number of integer weights required to exactly balance every integer between $1$ and $40$.

I do not really understand what this problem is asking for me to do. Any suggestions are appreciated. 

The question simply asks, using which set of nos you can make every no.s between 1 to 40.
Ans: 1, 3, 9 and 27.
Using this set you can make any no from 1 to 40 with different combo.
Mark the series 3^0, 3^1, 3^2, 3^3.
