Having read the title, I thought "hey, all you need is 2 measures".
So the final constraint (not mentioned in the title) is to find out whether the anomalous ball is heavier or lighter than the rest.
That's where the third measure comes in.
So the final constraint (not mentioned in the title) is to find out whether the anomalous ball is heavier or lighter than the rest.
That's where the third measure comes in.
Each experiment gives has 3 possible outcomes: left, right, and balanced. So in 2 experiments you can a maximum of 9 = 3*3 different outcomes. That's not enough to pick one ball out of 12.
If you have 3 experiments, you have 27 outcomes. You need 24 to tell which ball, + heavier/lighter, and 3 outcomes are impossible.
Edit: impossible or redundant, depending of your strategy.