The Return of The Da Vinci Puzzle

Oh well, I was not inclined to publish the solution. However, as Spielberg said to Harrison Ford, or as Sly Stallone thought to himself, never say never.

The Original Da Vinci Puzzle is here:
http://randomjunkyramblings.blogspot.com/2008/03/da-vinci-puzzle-with-apologies-to.html
Sure beats calling it The Vulcan Mind Meld, doesn't it? Beats Einstein's Puzzle hands down? I suspect you will find clues in the puzzle once you see the solution.

Anyhow, I got some awesome, and some crazy, solutions. Below are two approaches I thought of. In the first one, we take liberties with the puzzle/ twist the puzzle rules a bit. In the second one, we incorporate optimization ideas to fit the rules of the game.

Approach 1:
1> Cut the sets of balls, so that each set of balls has a prime number numerator in a ratio that makes up its weight.

2> Identify the duplicate set of balls on weighing the balls once by using properties of prime numbers.

Approach 2:
1> There is no unique solution.

2> Avoid using the scale. Solve this on paper or write a program.

3> You would like to minimize the error in the solution. The error here is the possibilities of jars holding the duplicate set of balls. E.g. You could come up with a solution where you weigh set 1,2,3,4,5 and the weight indicates that the duplicate set could be 1,2 or 5,6. You could also come up with a set 3,4,5,6,7 where the duplicate set could be 3,4 or 5,6 or 8,9. You minimize error with the first solution, i.e. the first solution would be *more* correct.

4> As pointed out by a Mensan, you have to try out all the possibilities before you can identify the optimal solution set. Yep, welcome to the weary world of The Traveling Salesman. This is literally a "hard" problem.
Pick any solution string from the optimal set as your solution. At this point, if you haven't lost your cool already, you are welcome to also weigh this solution string on the scale. Once.

5> The same bright Mensan suggested a book for this kind of funky thinking:
Numerical Recipes in C: The Art of Scientific Computing
http://www.amazon.com/Numerical-Recipes-C-Scientific-Computing/dp/0521431085/ref=sr_1_4?ie=UTF8&s=books&qid=1211417740&sr=1-4
I don't have my copy of this book anymore. Its been a while since I was in engineering, however, I might borrow it if you have a copy.

6> Don't shoot me, I'm just the messenger for the crazy ideas running about in my attic. :-D

So, what do you think?