The Lighthouse Keeper's Bottles

3 weighings. Divide 12 into groups of 4, then 4 into 2+2, then test the final pair. Dividing into thirds maximises information from each weighing. General rule: n weighings can find one lighter item among up to 3^n candidates.

A balance scale gives 3 outcomes per weighing. With 3 weighings, you can distinguish 3^3 = 27 different scenarios. Since there are only 12 candidates, 3 weighings are sufficient. The strategy mirrors the 9-ball solution but extended to 12. Step 1: Split 12 bottles into three equal groups: Group A (bottles 1-4), Group B (bottles 5-8), Group C (bottles 9-12). Step 2: Weighing 1: Place Group A on the left pan and Group B on the right pan. Step 3: If left pan rises (lighter): the counterfeit is in Group A. If right pan rises: it is in Group B. If balanced: it is in Group C. Step 4: You now have a group of 4 suspect bottles. Split this group: weigh bottle X1 vs bottle X2 (from the suspect group). Step 5: Weighing 2: If one side rises, that bottle is lighter — found in 2 weighings. If balanced, the counterfeit is X3 or X4. Step 6: Weighing 3: Weigh X3 vs X4. Whichever side rises contains the lighter counterfeit bottle. Done in exactly 3 weighings. Key Insight: The key is dividing into thirds at each stage, not halves. Halving wastes the 'balanced' outcome — it gives you no useful information. Dividing into three groups extracts maximum information from every weighing, since all three outcomes (left light, right light, balanced) each eliminate two-thirds of the possibilities.