Instead of a decision tree, you start from the supposition that each coin out of twelve can have a signature trace through 3 weighings that separates it from the rest. Given the vast amount of weighings possible its not a bad supposition. Then you just have to figure out what 3 weighings will do that (leave the trace) for you.
How do you think Mr. Coles came up with this doozy?
Another alternative solution by Frank Cole Probably in the early 60's, we enjoyed a pre-determined solution which enabled mental calculation of the result; pre-determined in the sense that the 3 set-ups are in writing before any weighings, thus eliminating adjustments at weighings II and III. In addition, once the 3 set-ups are written and the 3 weighings are done, one can orally and immediately announce the number of the odd ball and whether it is light or heavy. The derivation is readily extendable to any number of weighings (>= 2) and the appropriate number (>= 3) of balls.
Left Pan Right Pan Weighing I 4 8 10 11 1 2 5 7 Weighing II 2 4 7 12 3 5 6 11 Weighing III 5 6 10 12 7 8 9 11
Start with a Sum = 0 in the mind. Perform the 3 weighings, observing the tilt, if any.
Weighing I: Add -1 if tilt is left pan down, +1 if tilt is left pan up. Weighing II: Add -3 if tilt is left pan down, +3 if tilt is left pan up. Weighing III: Add -9 if tilt is left pan down, +9 if tilt is left pan up.