What are the ages of the three sons?

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SUMMARY

The ages of the three sons can be deduced from the conversation between the mail carriers. The product of their ages equals the age of mail carrier A, while the sum of their ages corresponds to the number of windows in a building. The clue about the middle son being red-haired indicates that there are no twins, leading to unique ages. The only set of ages that satisfies these conditions is 2, 3, and 7, which multiply to 42 and sum to 12, allowing for a unique solution.

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Homework Statement



Two mail carriers meet on the route and have a conversation
A: You have three sons. How old ar ethey?
B: if you multiply their ages, the result will equal your age
A: That's not enough to tell me the answer!
B: the sum of these three numbers equals the number of windows in that building.
A: But it's still not enouhg!
B: My middle son is red-haired
A: Ah, now it's clear!"
How old are the sons?

Homework Equations



None

The Attempt at a Solution



I tried just plugging a bunch of three setted numbers but i just don't know what the hell I'm tryin to go for. I can deduce from the red-haired part that there are no twins at all in the three kids so the numbers have to be unique but besides that I've gotten nowhere!
 
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Assuming people don't live more than 100 years, the age of mail carrier A is less than or equal to 100 so the product of mail carrier B's son's ages is less than or equal to 100.

Also, since most building are rectangular in nature, the number of windows will normally be the product of two numbers (what do these numbers represent?).
 
The importance of "My middle son is red-haired" is that there IS one "middle son". Look at the sets of 3 numbers that satisfy the conditions and see if there is only one that does not have 3 distinct numbers.
 

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