A census taker knocks on a man's door and asks about the children who live there.
"I have three daughters," the father replies. "The product of their ages is 72, and the sum of their ages is the number of this house."
The census taker looks at the house number and says, "I still can't determine their ages."
Now the census taker knows exactly how old each girl is. What are their ages?
Solution:
First, enumerate all triples of positive integers whose product is 72:
(1,1,72), (1,2,36), (1,3,24), (1,4,18), (1,6,12), (1,8,9), (2,2,18), (2,3,12), (2,4,9), (2,6,6), (3,3,8), (3,4,6).
Compute their sums. Among these, only two triples share the same sum:
(2,6,6) and (3,3,8) both sum to 14.
Because the census taker knew the house number yet still couldn’t decide, that number must be 14, leaving those two possibilities.
The father then mentions an oldest daughter. The triple (2,6,6) has no single oldest child (the two six-year-olds tie), whereas (3,3,8) does. Therefore the daughters’ ages are 3, 3 and 8.
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