Hard puzzle round.
Twelve balls, three weighings
The question
Twelve identical-looking balls, one of which differs in weight, and a balance scale. Find the odd ball and whether it is heavy or light in three weighings.
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
Yes, and twelve is the maximum for three weighings.
Three weighings give three outcomes each, so 27 distinguishable results, and there are 24 possible answers (12 balls times heavy or light). It fits, barely, which tells you every weighing must be near-maximally informative.
Weigh 1-4 against 5-8. If balanced, the odd ball is among 9-12 and two weighings comfortably resolve four candidates.
If 1-4 is heavier, then one of 1-4 is heavy or one of 5-8 is light. Weigh 1,2,5 against 3,4,6 to split those eight possibilities into groups of three, three and two.
The third weighing resolves whichever group remains. The design principle is to keep the number of surviving hypotheses at or below three before the final weighing.
What it is testing
Information budgeting before you start, rather than trial and error.
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