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.

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

What are the whole number factors of 36? A kid with blonde hair is not 1. Brothers/sisters cannot be the same age.

The only possible answers I found were 1, 1, 36

1, 2, 18
1, 3, 12
1, 4, 9
1, 6, 6
2, 2, 9
2, 3, 6
2, 4, 4
3, 3, 4
9, 2, 3

2, 3, 6

9, 2, 3

So, Im still trying to narrow it down.

If the sum of the ages were unique, then the mathematician would have been able to solve it after the second piece of information...

To find the ages of the barber's children, we need to analyze the information given step by step.

1. The barber tells the professor that if he multiplies the ages of the three children, the result is 36.
This means that the ages of the three children could be any combination of three positive integers whose product is 36. Let's list all the possible combinations:
1, 1, 36
1, 2, 18
1, 3, 12
1, 4, 9
1, 6, 6
2, 2, 9
2, 3, 6
3, 3, 4

2. The professor then tells the barber that after considering the product of the ages, he still can't determine their ages.
This means that the product of the ages alone is not enough information to uniquely determine the ages of the children.

3. The barber adds that the sum of the ages of his three children is equal to the number of his house.
This additional information allows us to deduce the possible combinations of ages.

- If the sum of the ages is a unique number, then the professor would know the ages based on that number. However, this is not the case because the professor cannot determine the ages after considering the information.
- This implies that the sum of the ages does not uniquely determine the ages.

4. Lastly, the barber mentions that the youngest child has blonde hair.
This is the crucial missing piece of information required by the professor. Now we can deduce the ages of the children.

From the given combinations, only one combination has a youngest child: 1, 6, 6.
Therefore, the ages of the barber's children are 1, 6, and 6.

The professor figured out the answer by considering all the possibilities provided by the product of the ages, excluding combinations where the sum of the ages is a unique number. Once the barber mentioned that the youngest child has blonde hair, the professor could determine the ages by eliminating all other combinations and identifying the unique combination with a youngest child.