Ages of Three Children
Date: 9/4/96 at 20:40:5 From: Jason Subject: Ages of Three Children During a recent census, a man told the census taker that he had three children. When asked their ages, he replied, "The product of their ages is 72. The sum of their ages is the same as my house number." The census taker ran to the door and looked at the house number. "I still can't tell," she complained. The man replied, "Oh that's right, I forgot to tell you that the oldest one likes chocolate pudding." The census taker promptly wrote down the ages of the 3 children. How old are they?
Date: 9/5/96 at 13:47:14 From: Doctor Tom Subject: Re: Ages of Three Children Hi Jason, After I know that the ages multiply to 72, here is a complete list of the possibilities: Ages: Sum of ages: 1 1 72 74 1 2 36 39 1 3 24 28 1 4 18 23 1 6 12 19 1 8 9 18 2 2 18 22 2 3 12 17 2 4 9 15 2 6 6 14 ** 3 3 8 14 ** 3 4 6 13 Note that every combination of possible ages which has a product of 72 has its own unique sum of ages - except for 2, 6, 6 and 3, 3, 8, both of which share the sum of 14. Since the census taker can't figure out the ages after looking at the house number, the house number must be 14, because then the ages could be either 2, 6, 6 or 3, 3, 8. Now, the next clue is that the _oldest_ child likes chocolate pudding. This means that there is _one_ oldest child. Well, there is no oldest child of the ages are 2, 6, 6, so the ages of the children must be 3, 3, and 8 years old. -Doctors Tom and Chuck, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 11/22/2008 at 17:22:11 From: John Subject: The census taker logic problem The fact the you have an oldest child does not mean there cannot be two children that same age. Therefore logically the problem cannot be solved. I have twin daughters and one is 3 minutes older than the other.
Date: 11/22/2008 at 23:40:43 From: Doctor Peterson Subject: Re: The census taker logic problem Hi, John. I've made the same comment more than once, because I'm in that position myself. Dr. Rick (another Math Doctor) is my twin brother, five minutes older than I am, and I've been aware all my life that he is the oldest! It's not quite as bad as you say, though; in real life, though we shouldn't just assume anything, we can look at the facts and figure out what someone else is thinking, rightly or wrongly. In this case, once you see that the two remaining possibilities are distinguished by the presence of twins, you can see what is intended even if you know it isn't quite right. This is not a useless skill in a world where logic isn't always recognized! My usual comment when I point this out has been, "It's a cute puzzle, but you have to take it with a grain of salt." I've seen several versions of the puzzle. Here is an interesting variation I dealt with earlier this year: Question: A census taker knocks on the door and a woman answers the door. She informs the census taker that she lives in the house with her three sons. The census taker asks for the ages of the boys and the women informs him that the product of their ages equals 36 and the sum of their ages equals the address of the house next door. He finds this rather odd, but walks to the house next door, realizes that he needs more information, and returns. He asks for another hint, and the women tells him that only her oldest son was born in a leap year. How old are the three sons? Explain your answer and show all work! My answer: The main idea is that knowing the product to be 36 is not enough, even WITH the knowledge of the sum of the ages. If you list possible products that would give 36, you can find a couple sums for which this would be true -- and one for which the added knowledge that the oldest son is the only one born in a certain year would clear up the uncertainty. This version of the puzzle fixes the interesting "error" in the usual formulation in which you are supposed to deduce that the oldest simply EXISTS, and therefore is not a twin. I have a twin brother and a younger brother, and the former is definitely the OLDEST son; his mere existence as such is not enough. The fact that he was born in the same year as I would do the trick. I suppose there is still one small loophole: twins could be born in different years, but still be the same age now. But at least this version shows that someone has noticed the problem! If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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