The Three Daughters
Two old friends meet in the street after many years.
"I have three daughters," says the first. "Multiply their ages together and you get 36."
"That is not enough," says the second.
"Add their ages together," says the first, "and you get the number on that gate over there."
The second friend looks at the gate, thinks for a while, and says: "That is still not enough."
"Ah, of course," says the first. "My eldest daughter plays the dutar."
"Now I know," says the second — and he is right.
Answer to type: the age of the eldest daughter.
💡 I want a nudge
Write out every triple of whole numbers whose product is 36, and put their sums beside them. Then ask: why would knowing the sum not be enough?
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Answer: 9
Every triple of whole numbers multiplying to 36, with its sum:
- 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
- The second friend can see the gate, so he knows the sum. If the sum had been 14, or 21, or any of the others, it would have appeared exactly once in the list and he would have known the ages at once.
- He says it is still not enough. That sentence is the key to the whole puzzle: the sum he is looking at must be one that appears twice. Only 13 does — so the ages are either 1, 6, 6 or 2, 2, 9.
- Then the father says "my eldest daughter". For 1, 6, 6 there is no single eldest — there are two six-year-olds. So that possibility is out.
- The ages are 2, 2 and 9, and the eldest is 9.
The trick: "I do not know" is a fact, and a very sharp one. It rules out every case in which the person would have known. Learning to squeeze information out of somebody's ignorance — not just out of what they state — is what separates this puzzle from an ordinary equation.
Nobody cracked this one. It is still open for you.