Conversation

My National Review Online "Diary" column for January 2003 included the following brainteaser.  

A woman is walking down the street and meets her neighbor.  The woman says, "I can’t remember the ages of your 3 children."  The neighbor replies, "The product of their ages is 36."  The woman thinks a minute and says, "I still don’t know the ages of your 3 children."  The neighbor replies, "The sum of their ages is your address."  The woman thinks a minute and says, "I still don’t know their ages."  The neighbor replies, "The oldest has red hair." 

What are the ages of the three children?

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Solution.  We have to find three whole numbers that, when multiplied together, make 36,  There are just eight possibilities:  1-1-36, 1-2-18, 1-3-12, 1-4-9, 1-6-6, 2-2-9, 2-3-6, 3-3-4.  If you add up the numbers in each triplet, you get:  38, 21, 16, 14, 13, 13, 11, 10.

Now, when the second woman was told that the ages added up to her address, she still didn't know the answers.  Her address must therefore be 13.  If it had been one of the other numbers — 38, 21, 16, 14, 11, or 10 — she would have been able to pin down the triplet.  Only with 13 can she not do this, because two triplets add up to 13:  1-6-6 and 2-2-9.

However, once the neighbor says: "The oldest...," she knows it must be 2-2-9, since if the kids' ages are 1-6-6, there is no oldest!

Answer:  the children are aged 2, 2, and 9.