Mathematicians and their sons

Algebra Level 3

I got this problem from a book. It was and still is one of my absolute favorite problems of all time.

Two mathematicians meet at a coffee shop.
"You have three sons, correct?" Bob says, "what are their ages?"
"The product of their ages is 36" says Bill, " and the sum of their ages is today's date"
"Well, I know the date" bob says, "but that doesn't tell me how old your children are"
"Oh, I forgot to tell you," Bill says, "my youngest one has red hair"
"Ahh, thank you, now I know how old they are" bob replies.

What is the sum of Bill's children's ages?

Assumption/hint: Assume if two children are of the same age, they are born on the same date.

Image credit: Wikipedia Sloan hadfield

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