In a little town in West Michigan lives a math professor, who hears one day that the barber has three children. So, on the next visit to the barber, the professor casually inquires, "I have heard you have three children, is that right?" "Yes!" says the barber. "Well, how old are they?" "You are the math professor, aren't you? I tell you, if you multiply the ages of the three, you'll end up with 36." "All right!" the professor answers and walks home. The next day the professor comes back to the barber shop and says: "With the information you have given me, it is impossible to figure out how old your kids are." Then the barber says: "Very good, I see you are a good mathematician. If you add the ages of the three, the sum will be the number of my house." So, the professor walks out, looks at the house number and returns home. Still the professor can't find the solution. The next day, the professor tells the barber that there still must be some information that's missing. "Yes, you are very clever!" says the barber. "The next information I'm giving you is the last word I'm saying about the age of my children. Now you will have enough information. Don't come back again and ask for more. The youngest has blonde hair." The professor goes home and figures out the answer.

Question

What are the ages of the barber's children, and how did the professor figure it out?

Ive narrowed down the answers to
2, 3, and 6
9, 2, and 3

Firstly, because according to BobPursley, a kid whos 1 cannot have blonde hair, meaning noones one. Than the brothers and sisters cant be the same age, and that narrows it down to those 2 answers. Now how can I find the final answer?

Answer 1

How can it be 9,2, and 3? It has to multiply to 36.

Answer 2

Sorry, :)
Its late here.

So that leaves...

2, 3, and 6?

I just wanna know how we can prove the blonde kid cant be 1.

Answer 3

The other alternative set is 1,3,12; an alternative to 2,3,6. Hair is usually reserved for two year olds. I doubt if you can Prove it.

Answer 4

So we can just going to assume its 2, 3, and, 6?

Answer 5

If its the youngest child who's hair color is given, the solution is {1,6,6}

it has nothing to do with the actual hair color, but with mentioning that there exists a unique youngest (or oldest) child to remove the ambiguity in the two age sets with identical sums (i.e. {1,6,6} and {2,2,9) )

all other age sets have a unique sum