It got to me...Arghhh..

My cousin came to me with a maths problem that her teacher had given her. It goes like this:

A king is looking for a wise man. He held a contest in search for one wise one. The contest consisted of solving this problem. “The product of the ages of the king’s daughters is 36 and the sum is equal to the number of gates in the royal palace. Determine the age of the princesses.” The most intelligent contestant approached the king after thinking through the problem and said, “Your Highness, I am missing one piece of information.” The king promptly replied “The oldest of my daughters has blue eyes.” The contestant immediately gave the ages of the princesses. What was his answer?

This is a sec one problem. Tell me if I'm stupid. I could not find out the answer. What about you?

My thought process below:
The oldest of my daughters -- meaning there are at least 3 daughters.
Based on possible permutations of products, the following can be derived:
1 x 2 x 3 x 6 Sum = 12
2 x 3 x 6 (2 x 2 x 3 x 3) Sum = 11
1 x 2 x 18 (or 1 x 2 x 2 x 9) Sum=21
1 x 3 x 12 (or 1 x 3 x 3 x 4) Sum= 16
1 x 4 x 9 (or 1 x 2 x 2 x 3 x 3) Sum=14

Assuming that there are no twins, the oldest daughter can be 6, 9, 12 or 18.

My hair was a mish-mesh of bird's nest after trying to pull them out from the frustration i suffered from not being able to solve the problem.

I concocted all possible explanations to eliminate some answers. Ridiculous by any standards, but that's what happens when you're trying all possible means to squeeze out an answer. I believe i've superseded my present level of absurdity with my explanations below! LOL :)

Non sequitur is my first name for the day. Ha. Solid, flawed arguments.
1) Blue eyes. Is it connected to blue moon? If it's blue moon, maybe the answer is 12 or 18 since blue moon only happens once a year or less, (or never?!!?). Probability of blue moon will thus be 1/12 or 1/18 (or 0/infinity?!?), hence deriving the numbers 12 or 18.

2) Gates have to come in pairs since gates in the medieval periods always have two doors. Assuming two doors = two gates, the summation of ages must be an even number. My most ridiculous explanation yet! Yes, yes i know. Gates and doors are different. One gate can have two doors and the summation of gates can still be an odd number.

2) My sis offered that blue is a primary colour. Primary colour has an equivalent status as prime number. So look for a prime number. But alas, 6, 9, 12 and 18 are not prime numbers!!

Any wise one out there who knows the answer? My cousin's teacher gave the clue that "the oldest of the daughters has blue eyes" is the crux of the matter.

Blue eyes, blue eyes, why is this the key to the answer?

Oh man, I can't sleep tonight without knowing the answer! I NEED TO KNOW! :P