puzzle 021 curious village

Puzzle 021: The Three Daughters (from Professor Layton and the Curious Village) Problem: A man asks a woman how old her three daughters are. The woman gives two clues: "The product of their ages is 36." "The sum of their ages is equal to the number of the house across the street." The man replies, "I still cant tell their ages!" The woman then adds a third clue: "My eldest daughter plays the piano." Question: What are the ages of the three daughters? Solution: Step 1: List all possible age combinations with a product of 36. We list all sets of three positive integers (ages) that multiply to 36: Ages (a, b, c) Product (abc) Sum (a + b + c) 1, 1, 36 36 38 1, 2, 18 36 21 1, 3, 12 36 16 1, 4, 9 36 14 1, 6, 6 36 13 2, 2, 9 36 13 2, 3, 6 36 11 3, 3, 4 36 10 Step 2: Analyze the sum clue. The man couldnt determine the ages even after knowing the sum. This implies the sum must correspond to multiple possible age combinations. Looking at the sums: 13 appears twice (for 1, 6, 6 and 2, 2, 9). - All other sums are unique. Thus, the house number must be 13 (otherwise, the man would know the answer immediately). Step 3: Apply the piano clue. The woman mentions her "eldest daughter" (singular), implying there is a unique oldest child. Between the two possible sets for sum=13: 1, 6, 6: No unique eldest (two age-6 daughters). 2, 2, 9: Unique eldest (age 9). Thus, the piano clue eliminates (1, 6, 6). Answer: The daughters are 2, 2, and 9 years old. Key Logic: The ambiguity in the sum points to duplicate sums (only 13 fits). The piano clue breaks the tie by implying a single eldest child. This puzzle combines factorization, logic, and careful interpretation of hints!