What are the ages of the three sons?

[Ackbach]

Here is this week's POTW:

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Two men meet on the street. They haven't seen each other for many years. They talk about various things, and then after some time one of them says: "Since you're a professor in mathematics, I'd like to give you a problem to solve. You know, today's a very special day for me: all three of my sons celebrate their birthday this very day! So, can you tell me how old each of them is?"

"Sure," answers the mathematician, "but you'll have to tell me something about them."

"OK, I'll give you some hints," replies the father of the three sons, "The product of the ages of my sons is 36."

"That's fine," says the mathematician, "but I'll need more than just this."

"The sum of their ages is equal to the number of windows in that building," says the father pointing at a structure next to them.

The mathematician thinks for some time and replies, "Still, I need an additional hint to solve your puzzle."

"My oldest son has blue eyes," says the other man.

"Oh, this is sufficient!" exclaims the mathematician, and he gives the father the correct answer: the ages of his three sons.

What are the ages of the three sons?

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[Ackbach]

This problem is on page 9 of How to Solve It: Modern Heuristics, by Michalewicz and Fogel.

Kiwi had the right idea in his partial solution, so I will simply post his solution with my completion thrown in:

Spoiler

The sons are 2, 2 and 9.

Since the product of their ages come to 36 their ages are one of the following:

1,1,36 (sum = 38)
1,2,18 (sum = 21)
1,3,12 (sum = 16)
1,4,9 (sum = 14)
1,6,6 (sum = 13)
2,2,9 (sum = 13)
2,3,6 (sum = 11)
3,3,4 (sum = 10)

The building has 13 windows because if it has any other number of windows, then the professor would have sufficient information to answer the problem without the last clue.

The last clue tells us that the age of the oldest son is distinct. So having twins aged 6 is not a possibility. Therefore, the man has twins aged 2 and a 9 year old with blue eyes.