Census Taker: How old are your three daughters?
Mrs. Smith: The product of their ages is 36, and the sum of their ages is the address on our door here.
Census Taker: (after some figuring) I’m afraid I can’t determine their ages from that …
Mrs. Smith: My eldest daughter has red hair.
Census Taker: Oh, thanks, now I know.
How old are the three girls?
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There must be at least two sets of three numbers whose product is 36 but that have the same sum. Trial and error shows these are (1, 6, 6) and (2, 2, 9), both of which add to 13. Mrs. Smith’s second statement is a “red hairing” — the eldest daughter’s hair color is unimportant, but the fact that Mrs. Smith has an eldest daughter implies that (1, 6, 6) can’t be the right answer. Thus her daughter’s ages are 2, 2, and 9.
From Frank Morgan, The Math Chat Book, 2000.
Some readers object that even twins are not precisely the same age, so the (1, 6, 6) group does contain an “eldest” daughter and can’t be ruled out on that basis. I think that’s a valid criticism.
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