I don't do this to make my students hate me, though I am satisfied with the possibility, as I've found that the more they hate me, the more they end up learning. It's a seductive problem, which illustrates the value of deductive reasoning. It is quite simple, though a great number of students (some of them much brighter than I) tear their hair out over it. I first encountered it thanks to a logician named James Garson (at the University of Houston) and I solved it in about ten minutes. That has more to do with the fact that I got over the math anxiety which is common among even the brightest people.
If you're here, it's likely because you're one of my victims; and (given that you're a university student) the laws of probability suggest that you're at least as bright as I am. The only advantage I've got on most of you is the fact that I'm an old man who has a head start on reading boring books. That aside, this is the question, and below is the answer I gave to Dr. Garson, back in the summer of 2015.
We should read the problem in good faith and take the text at face value. The first thing we notice is that there are three clues, and no clue is extraneous, vacuous, or trivial. All three clues are necessary for the solution.
Clue No. 1: There are three numbers, x_1, x_2, x_3, such that (x_1)(x_2)(x_3) = 36
Clue No. 2: There is some positive integer, x_4, such that x_1 + x_2 + x_3 = x_4
Clue No. 3: The eldest son has red hair.
We know x_4 is a positive integer because the post office doesn't number addresses with irrational or negative values.
The last clue is interesting, and clearly designed to provide us with the means of solving the problem, while simultaneously providing us irrelevant information in an attempt to confuse us. We don't care what color hair the boy has. His father said that "the oldest one" was unique. That means there will be one, and only one, eldest child. There are no triplets, and there are no elder twins.
For various number theoretic reasons, we can safely assume that the ages of all the boys will be positive integers also.
The first thing we can do is to define our system of equations.
The next thing to do is to find all the prime factors of 36.
If we were trying to find the ages of four children, then the first clue would alone be sufficient for a solution. In fact, there are three children, and so now we have to work backwards, and attempt to find possible solutions for the second equation in our system.
There is one, and only one, repeating possible solution, suggesting that the building's number is thirteen.
If we take the text at face value, as logicians should, we can now intuit our answer, based upon the necessity of the final clue. Note that one of our solutions admits two elder twins, followed by a younger brother. This contradicts Rule No. 3. Thus we know that the eldest boy is nine, and he is followed by younger twin brothers, each aged two.
Checking our solution against the defined system, we find everything in order.
