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## Question

During my 20th wedding anniversary party, John asked me the age of my three children. I told John that given the sum and product of their ages, Smith missed the problem tonight and David missed it at the party two years ago. My youngest child always insists I should tell people their age in the form of sum and product.

Can you determine the age of my three children? You may assume that they were born after my marriage.

## 2 answers to this question

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Is there a non-brute force way to solve this one? I have some guidelines, but I don't know that it cuts down the answers enough for me to start deducing anything.

Let oldest's age be X

Let middle's age be Y

Let youngest's age be Z

(oldest and middle terms are used loosely since they could actually be the same age)

6 < x + y + z < 60

Six since two years ago he had three children, and 60 because he had the children within the bounds of his marriage which is concluding its 20th year

x >= y > z

Since there is a "youngest" one.

This could tighten down the previous constraint to

6 < x + y + z < 59 since the youngest, I guess you could assume, is at least a whole year younger than the middle child.

The sum of their ages cannot be unique within the bounds.

The sum of two less than their current ages cannot be unique within the bounds.

Same goes for the product.

Other than that, I don't know how to solve it without writing a software program.

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It has been a while since posted, but not that much activity.... anyways, here is my solution.

Two years ago, there was ambiguity in the sum and product of the ages (2,7,12) and (3,4,14) -- and that is why David couldn't guess it.

During the party, the ambiguity in the sum and product between (4,8,15) and (5,6,16) -- Smith missed to guess it.

There fore, the age of my children must be 5, 6 and 16.

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