Three Children, Two Friends and One Mathematical Puzzle

Two close friends, Robert and Thomas, met again after a gap of several years.
Robert Said: I am now married and have three children.
Thomas Said: That's great! How old they are?
Robert: Thomas! Guess it yourself with some clues provided by me. The product of the ages of my children is 36.
Thomas: Hmm... Not so helpful clue. Can you please give one more?
Robert: Yeah! Can you see the number on the house across the street?
Thomas: Yes! I can.
Robert: The sum of their ages equal that number.
Thomas: Sorry! I still could not determine their ages.
Robert: My oldest child has red hair.
Thomas: OH.. Oldest one? Finally I got it. I know age of each of your children.


What were the ages of Robert's children and how did Thomas know?

Discussion and probable answer

This is a very good logical problem. To do it, first write down all the real possibilities that the number on that building might have been. Assuming integer ages one get get the following which equal 36 when multiplied:

Age of 1st Age of 2nd Age of 3rd Sum(House No.)
1 1 36 38
1 2 18 21
1 3 12 16
1 4 9 14
1 6 6 13
2 2 9 13
2 3 6 11
3 3 4 10

The biggest clue is that the Thomas DID NOT KNOW after having been told the sum equaled the number on the house. Why didn't he know? The only reason would be that the number was 13, in which case there are two possible answers. For any other number, the answer is unique and the Thomas would have known after the second clue. So he asked for a third clue. The clue that the oldest had red hair is really just saying that there is an "oldest", meaning that the older two are not twins. Hence, the answer is that the redhead is 9 years old, and the younger two are both 2 years old.

Source of The Puzzle: This is a popular puzzle and is a modified form of a puzzle published in Science Reporter Magazine.