puzzle 67 curious village

Puzzle 67: The Three Daughters (from Professor Layton and the Curious Village) Question: A man is trying to figure out the ages of his friend's three daughters. He is given the following clues: The product of their ages is 36. The sum of their ages is equal to the house number across the street. The eldest daughter has red hair. What are the ages of the three daughters? Step-by-Step Solution: List all possible age combinations with a product of 36: We need to find all sets of three positive integers (a, b, c) where ( a imes b imes c = 36 ). Here are the possibilities (including permutations): - (1, 1, 36) Sum = 38 - (1, 2, 18) Sum = 21 - (1, 3, 12) Sum = 16 - (1, 4, 9) Sum = 14 - (1, 6, 6) Sum = 13 - (2, 2, 9) Sum = 13 - (2, 3, 6) Sum = 11 - (3, 3, 4) Sum = 10 Analyze the sums: The second clue says the sum equals the house number across the street. The man doesn't know the house number, but he still can't determine the ages, which implies the sum must be non-unique. Looking at the sums: - 13 appears twice (for 1, 6, 6 and 2, 2, 9). - All other sums are unique. This means if the house number were any other sum (e.g., 10, 11, 14, etc.), the man would know the answer immediately. The ambiguity implies the sum must be 13. Use the third clue ("eldest daughter has red hair"): This suggests there is a single eldest daughter (no twins). Between the two possible sets with sum 13: - (1, 6, 6) has two eldest daughters (twins aged 6). - (2, 2, 9) has a single eldest daughter (9). Thus, the correct ages are 2, 2, and 9. Answer: The daughters are 2, 2, and 9 years old.