A man has three daughters, and someone asks him their ages. He provides the following clues:
- The product of their ages is 36.
- The sum of their ages is equal to the number of the house next door.
- The oldest daughter has blue eyes.
Objective: Based on these clues, find the ages of the three daughters.
Solution:
To solve this puzzle, we need to break it down using logical deduction and the clues provided.
Step-by-Step Approach:
Step 1: Consider the Possible Combinations for the Product of Ages
Since the product of the daughters' ages is 36, we need to list all possible combinations of three numbers that multiply to give 36. These combinations are:
- 1, 1, 36
- 1, 2, 18
- 1, 3, 12
- 1, 4, 9
- 1, 6, 6
- 2, 2, 9
- 2, 3, 6
- 3, 3, 4
Step 2: Use the Clue About the Sum of Ages
The second clue states that the sum of their ages is equal to the number of the house next door. Since the sum of ages is critical, let’s calculate the sum for each of the combinations:
- 1 + 1 + 36 = 38
- 1 + 2 + 18 = 21
- 1 + 3 + 12 = 16
- 1 + 4 + 9 = 14
- 1 + 6 + 6 = 13
- 2 + 2 + 9 = 13
- 2 + 3 + 6 = 11
- 3 + 3 + 4 = 10
Step 3: Analyze the Second Clue
Since the second clue gives the sum as the number of the house, we now understand that the person asking the question doesn’t know the answer yet. This must mean that the sum of the daughters' ages is ambiguous—in other words, the sum could match more than one of the combinations.
Looking at the sums, we see that two combinations share the same sum of 13:
- 1 + 6 + 6 = 13
- 2 + 2 + 9 = 13
This must be the house number since it’s the only ambiguous case.
Step 4: Use the Final Clue About the Oldest Daughter
The third clue states that the oldest daughter has blue eyes, which tells us there is a distinct oldest daughter. Therefore, the combination 1 + 6 + 6 cannot be correct because there isn’t an “oldest” daughter (the two oldest are both 6 years old).
Thus, the correct combination must be 2 + 2 + 9.
So, the correct answer is
The ages of the three daughters are 2, 2, and 9.
