You can use these numbers to figure out how many weights of each color you need to put on the hemisphere. It can also help you to determine which position to put the three fulcrum controls on the platform.
In this particular lever, the middle fulcrum position will require an equal amount of weight on both sides; the right (as in not left) position will need half the weight on the hemisphere of the orange ball; the left one will require twice as much weight on the counterweight as on the orange ball on the other side. Considering what weights you have, which position do you think you should use?
Using our numbers, we can rule out which position(s) won't work with the available weights. The middle position would require 11 units of weight (the same as the orangish-colored ball). Although, we have to remember that there are four brown weights on the bottom of the counterweight. So, we would need to put 7 more units of weight on the hemisphere. With two 1's, two 4's, and two 16's, I can't think of any possible way to make seven. The right (as in not left) position requires 5.5 units, one half of the orange ball; when you take away the four at the base, you're dealing with 1 and a half units, which wouldn't work too well. By using logical reasoning, we can say the left position is correct. The orangish ball's weight doubled is 22; 22 minus 4 for the bottom hemisphere is 18. I think it would be possible to make 18 out of the brown, crystal, and/or black weight(s).
Could I have a shorter explanation?
What is the movie about that played after I solved the puzzle?