Twelve Balls, Three Weighings

Question

Twelve visually identical balls; exactly one is counterfeit, either heavier or lighter, you don't know which. Using a balance scale only three times, identify the counterfeit AND whether it's heavy or light. Describe a complete strategy.

Accepted Answer

Label balls 1–12. Every ball starts with two live hypotheses (H = heavy, L = light): 24 cases; three 3-outcome weighings give 27 distinguishable results, just enough, so no weighing may waste information. Weighing 1: 1,2,3,4 vs 5,6,7,8. Case A, balance. Counterfeit ∈ 9,10,11,12 (8 cases). Weighing 2: 9,10,11 vs 1,2,3 (known good). - Balance → it's 12; weighing 3: 12 vs 1 tells heavy/light. - Left heavy → one of 9,10,11 is heavy; weighing 3: 9 vs 10 (heavier one is it; balance → 11 heavy). - Left light → one of 9,10,11 is light; symmetric. Case B, left heavy. Live cases: 1,2,3,4 ∈ H or 5,6,7,8 ∈ L (8 cases). Weighing 2: 1,2,5 vs 3,4,6 (rotate suspects; 7,8 sit out). - Balance → counterfeit ∈ 7,8 light; weighing 3: 7 vs 8, lighter is it (cannot balance). - Left heavy → consistent cases: 1H, 2H, or 6L; weighing 3: 1 vs 2 (heavier is it; balance → 6 light). - Right heavy → consistent cases: 3H, 4H, or 5L; weighing 3: 3 vs 4 (heavier is it; balance → 5 light). Case C, right heavy. Mirror of Case B. Every branch terminates in exactly 3 weighings with ball + direction identified. ∎ What's really being tested: (1) starting from the information-theoretic bound to know the target, (2) design

Twelve Balls, Three Weighings

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