Pea Wormsworth must be new here...
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12 Identical Balls
I found a solution like this. But I really like the way you worded it. After this I tried to find a solution where the rearrangement of the balls could be done consistently. It can. use 4 balls on each side of the scale and 4 balls not on the scale. Number each ball as having a position (1-12). Then after each measure, move say ball in slot #1 to slot #6, and ball in position #2 to position #4, etc. Each ball takes a new position based on this and re-measure. Then repeat the movement and take the 3rd measure. The result is a unique set of measurements that reveal which ball is unique and it's weight (heavy or light). I am saying that not only are the positions of the balls for each measure known, but also the re-arrangement of the balls between measures is also the same. Then I tried to apply a formula for the rearrangement process. So instead of a lookup table, the rearrangement was based on the position number. And I think there is also a solution for that. I don't have the equation any more, but I think it was like: (A*n1+B)%C = n2. Where n1 is the starting position, A, B and C are integers, "%" is modulus and n2 is the new position. And I think it worked. But I lost the formula, so I cannot prove it now. The end result would also be a unique set of measurements that results in a unique set that can find the solution from a lookup table. Finally, I thought that since each weight results in one of 3 outcomes, that the sum of the weightings is a number in base 3. So, the final step in simplification would be to either modify the rearrangement equation or perform a final equation on the resulting number to make the solution be a number which reveals the solution in order of the starting positions of the balls. This would remove the lookup table required on the measurements made to identify the final ball and its mass. But alas, I never found the answer to this last problem before I lost interest and lost all my notes. The reason for doing all this is that I like "golf", in a programming sense. I think taking a puzzle and reducing it down to minimal code to solve it is very interesting. I suspect that if a computer program was written to provide a structure for the problem at hand, that the resulting solution code would be about 2 or 3 lines of recursive code. Which would be far more efficient than all the if/then type conditions and lookup table required for most of the obvious solutions people are so proud of online. I have searched a little to see if anyone has done it, but I haven't seen it yet. I am passing this to you, because I think someone with math knowledge and an understanding of the problem should be able to either find the solution I described or prove that it is not possible. If you ever hear of someone with that solution you described, encourage them to complete the job and simplify their solution down to its most reduced form.context