A Logic Puzzle

The Puzzle

Alice has three friends who Beth has never met and knows nothing about. She challenges Beth to deduce their ages.

Beth asks for a clue. Alice kindly gives her two clues:

“The product of their ages is 36. The sum of their ages is my favourite number in the whole world.”

After thinking carefully, Beth tells Alice that the challenge is unfair: Alice has not given enough information.

Alice feels unhelpful, so she tells Beth some random details to waste time:

“The eldest of the three friends is called Carlos and it was his birthday yesterday. The other two had their birthdays a few weeks ago. Maybe you can come along with me next year and ask them yourself?”

Beth laughs at Alice and tells her that that won’t be necessary. She has just worked out their ages.

How old are Alice’s three friends? What is Alice’s favourite number?


I strongly recommend trying the above problem for yourself before asking for help or reading further.

The puzzle is my unsubtle re-telling of a puzzle I came across on Wikipedia, originally called the Ages of Three Children puzzle. I found the original Wikipedian wording of the riddle unsatisfactory, mostly because the puzzle talks about ages but part of the solution demands that the numbers be treated as integers, not reals. I also realised that, since Beth works out the answer, the puzzle works even when the clues are a little less obvious than the house number.

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