Question

A census taker approaches a house and asks the woman, "How many children do you have, and what are their ages?"

Woman: "I have three children. The product of their ages is 36, and the sum of their ages is equal to the house number of the house next door."

The census taker checks the house number, then says, "I need more information."

Woman: "My oldest child is sleeping upstairs."

What are the ages of the three children?

• A. 9, 2, 2
• B. 6, 6, 1
• C. 12, 3, 1
• D. 6, 3, 2

Hint:

Find product-36 triples whose sums tie (this is the ambiguous house number), then use "a single oldest child" to rule out the twin option.

Answer 1

The Correct Option is A

Step 1: List all triples whose ages multiply to 36.
Since the product of the three children's ages is 36, list every way to write 36 as a product of three positive whole numbers, along with each triple's sum.
\[ \begin{array}{c|c} \text{Ages} & \text{Sum} \\ \hline 1,1,36 & 38 \\ 1,2,18 & 21 \\ 1,3,12 & 16 \\ 1,4,9 & 14 \\ 1,6,6 & 13 \\ 2,2,9 & 13 \\ 2,3,6 & 11 \\ 3,3,4 & 10 \end{array} \]

Step 2: Use the "sum equals house number" clue.
The census taker knows the exact house number next door (the sum), yet still says "I need more information." This can only happen if two different triples share the exact same sum, so that knowing the sum alone cannot tell them apart.
Scanning the sums above, only 1, 6, 6 and 2, 2, 9 both add up to 13. Every other triple has a sum that no other triple matches, so the house number must be 13, and the two candidate triples are (1, 6, 6) and (2, 2, 9).

Step 3: Use the "oldest child" clue.
The woman then says her oldest child is sleeping upstairs, implying there is a single, clearly identified oldest child.
In the triple (1, 6, 6), two children are both aged 6, so there is no single oldest child, this triple is ruled out.
In the triple (2, 2, 9), the 9 year old is uniquely the oldest, which fits the clue perfectly.

Step 4: Final Answer:
The children's ages are 9, 2 and 2 years, matching option (A).
\[ \boxed{9, \; 2, \; 2} \]