``Two Russian mathematicians meet on a plane. 
“If I remember correctly, you have three sons,” says Ivan. “What are their ages
today?” 
“The product of their ages is thirty-six,” says Igor, “and the sum of their
ages is exactly today’s date.” 
“I’m sorry, Igor,” Ivan says after a minute, “but that doesn’t tell me the ages
of your boys.” 
“Oh, I forgot to tell you, my youngest son has red hair.” 
“Ah, now it’s clear,” Ivan says. “I now know exactly how old your three sons
are.” 
How did Ivan figure out the ages?

“The product of their ages is thirty-six,”

The only way we can get 3 whole numbers, smallest to largest, such that when
they are all three multiplied together we get 36.

1*1*36 = 36 
1*2*18 = 36 
1*3*12 = 36 
 1*4*9 = 36 
 1*6*6 = 36 
 2*2*9 = 36 
 2*3*6 = 36
 3*3*4 = 36

“and the sum of their ages is exactly today’s date.” 
   
Now we'll add all those possibilities:

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

This rules out 1,1,36 since 1+1+36 = 38 which is too
big to be today's date, since no month has more than
31 days.  So the possibilities are now

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

"my youngest son has red hair"

This tells us that there IS a youngest son, so that
rules out 2,2,9 and 3,3,4 in which there is no
youngest son because the youngest two are twins, and
thus are the same age.

Ages are 1,2,18, product = 36, sum = 21, date = 21st
Ages are 1,3,12, product = 36, sum = 16, date = 16th 
Ages are 1,4,9, product = 36, sum = 14, date = 14th 
Ages are 1,6,6, product = 36, sum = 13, date = 13th 
Ages are 2,3,6, product = 36, sum = 11, date = 11th  
 
It could be any of those 5 possibilities.  Are you sure
you didn't leave something out?  

Edwin