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Two old friends, Jack and Bill, meet after a long time. Jack: Hey, how are you man? Bill: Not bad, got married and I have three kids now. Jack: That’s awesome. How old are they? Bill: The product of their ages is 72 and the sum of their ages is the same as your birth date. Jack: Cool… But I still don’t know. Bill: My eldest kid just started taking piano lessons. Jack: Oh now I get it.

How old are Bill’s kids?

I am not able to figure out how Jack get it when Bill says "My eldest kid just started taking piano lessons"?

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Possibly similar problem:… – Gerry Myerson Oct 22 '12 at 12:42
Bill: "The product of their ages is 72 and the sum of their ages is the same as your birth date." Jack: "After all these yaers, you still haven't developed enough social skills to hold a normal conversation. Hang in there, Bill!" – Michael Joyce Oct 22 '12 at 12:55
up vote 5 down vote accepted

Since the product of the ages is $72$, the ages must be one of the following combinations:

$$\begin{align*} &2,2,18\\ &2,3,12\\ &2,4,9\\ &2,6,6\\ &3,3,8\\ &3,4,6\\ \end{align*}$$

The sums are $22,17,15,14,14$, and $13$ respectively. Since the sum and product of the ages didn’t give Jack enough information, the sum must have been $14$, the only possibility that admits more than one solution. The ages must therefore have been either $2,6$, and $6$ or $3,3$, and $8$. If they were $2,6$, and $6$, the two oldest would have been twins, and Bill (probably) wouldn’t have referred to his eldest child: it’s true that one of the six-year olds would technically have been his eldest child, but he’d probably have thought of them as being the same age. Jack inferred that the eldest child wasn’t a twin and concluded that Bill’s children were aged $3,3$, and $8$ years.

Added: Oops! As noted in the comments, $1$ is a possible age. That adds the sets $$\{1,1,72\},\{1,2,36\},\{1,3,24\},\{1,4,18\},\{1,6,12\},\{1,8,9\}$$ to the collection, with sums $74,39,28,23,19$, and $18$; fortunately, these add no further ambiguities.

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You seem to exclude the possibility that one of the ages could be $1$. That adds $1,1,72$ (!), $1,2,36$, $1,3,24$, $1,4,18$, $1,6,12$ and $1,8,9$ to the list. The sums are $74, 39, 28, 23, 19, \text{ and } 18$, so your conclusion still holds. Although if Jack was born in '14 and assuming Bill and Jack are roughly comparable in age, it's hard to see how Bill could have such young kids! – Michael Joyce Oct 22 '12 at 12:59
I like your reasoning, but I think you should also include the cases were one of the children is 1 year old: 1,3,24, 1,4,18, 1,6,12, and 1,8,9 (1,2,36 and 1,1,72 are impossible, as they do not sum up to a birthdate). Since the sums of those are 28, 23, 19 and 18, respectively, no further ambiguity is introduced. – Martin Oct 22 '12 at 13:01
@Michael: That was indeed a silly oversight. Thanks (and also to Martin). – Brian M. Scott Oct 22 '12 at 13:06

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