Oh, how the years go by: The Weekly Puzzle by Dilip D’Souza

It’s a college reunion and one-time classmates are meeting and comparing notes. Can you guess the ages of three children, and figure out where the ice-cream fits in?

At their college reunion, Alka and Bishamber met for the first time since they graduated. They compared notes about their families. This conversation ensued:

Alka: “I have three kids, Bish.”

Bishamber asked: “Three! Wow! How old are they?”

Alka: “Hey, I know you like puzzles, so why don’t you work it out? The product of their ages is a perfect square less than 150.”

Bishamber: “Doesn’t say much, tell me more.”

Alka: “OK, the sum of their ages is a prime number.”

Bishamber: “Ah! What a great clue, madam! But I still don’t know their ages.”

Alka: “Well, one kid’s age is the product of the other two ages.”

Bishamber: “Moving right along! But I need more.”

Alka: “How about this: the sum of their ages is your daughter’s age.”

Bishamber: “Fine, but which daughter? The one who became a teenager two months ago or the one who won’t be a teen any longer next week?”

Alka: “The one who won’t be a teen after next week.”

Bishamber (groaning): “Damn, I still don’t know!”

Alka: “All right, my eldest can’t stand chocolate ice-cream.”

Bishamber: “Got it!”

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Question: What are Alka’s kids’ ages?

Scroll down for the answer.

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Answer:

The children are aged 3, 4 and 12.

How do you solve such a puzzle?

Alka’s first clue tells us the product of the three kids’ ages is 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 or 144.

Given her next clue, you should write out all the possible age triples for each of these possible products, with their sums. For example, if the product was the perfect square 16, the kids could be 1, 1, and 16 (sum 18); 1, 2, 8 (11); 1, 4, 4 (9); 2, 2, 4 (8).

There are 59 such. Mark the ones that have a prime sum (eg, the 1, 2, 8 triple above).

Of those, pick out the ones in which the product of two of the ages is the third (ie, the third is the square root of the product of all three). There are nine such. Their sums are 3, 5, 11, 13, 17 (twice), 19 (twice) and 23.

Alka now says the sum of the ages is Bishamber’s daughter’s age. Bish’s response to this tells us he has two daughters, aged 13 and 19.

Alka says she means the 19-year-old. Bish groans because there are two triples that add up to 19: (1, 9, 9) and (3, 4, 12). But now Alka refers to her “eldest” — which rules out 1, 9 and 9.

Thus her kids are 3, 4, and 12 years old.