The Ages of the Three Daughters
The puzzle
A census taker asks a woman the ages of her three daughters. She says the product of their ages is 36, and the sum of their ages is the number on the house next door.
The census taker walks next door, looks at the number, comes back and says that is not enough information. She replies: "My eldest daughter plays the piano." The census taker thanks her and leaves, satisfied.
How old are the daughters?
Stuck?
One at a time, smallest first. Open only as many as you need — the whole point of the puzzle is the part you work out yourself.
Hint 1
List every set of three whole numbers whose product is 36, and the sum of each.
Hint 2
The census taker saw the sum and still could not tell. What does that rule out, and what does it leave?
Show the answer Hide the answer
Two, two and nine.
How to get there
The sets of three whole numbers multiplying to 36, with their sums: (1,1,36) sums to 38, (1,2,18) to 21, (1,3,12) to 16, (1,4,9) to 14, (1,6,6) to 13, (2,2,9) to 13, (2,3,6) to 11, (3,3,4) to 10.
The census taker knows the sum, because he read it off the house. If that sum were any of 38, 21, 16, 14, 11 or 10, it would pick out exactly one set and he would have had his answer. He says he cannot tell - so the sum must be 13, the one value shared by two different sets: (1,6,6) and (2,2,9).
The mother then mentions her eldest daughter. In (1,6,6) the two oldest are twins, so there is no single eldest. In (2,2,9) there is. The answer is 2, 2 and 9.
The technique worth keeping
The decisive clue is a statement of ignorance: the census taker not knowing is itself information, and it eliminates every sum that would have been conclusive. Puzzles that hinge on what somebody failed to deduce are common, and the method is always the same - work out what they would have known, and keep only the cases where they would not.