What are the ages?

Question

A man taking the census walks up to the apartment of a mathematician and asks him if he has any children and how old they are.

1. The mathematician says: "${\it\mbox{I have three daughters and the product of their ages is 72}}$".
2. The man tells the mathematician that he needs more information, so the mathematician tells him ${\it\mbox{"The sum of their ages is equal to our apartment number"}}$.
3. The man still needs more information so the mathematician tells him ${\it\mbox{"My oldest daughter has her own bed and the other two share bunk beds"}}$.

Answer 1

I cannot think of anything really elegant. Just write down all possible ways of writing 72 as a product of three numbers. Then write the sum of each triplet. The correct answer would be that triplet in which the smallest number occurs exactly twice and no other triplet with this property exists whose sum is same as the sum of elements in this triplet.

Answer 2

By examination, there are $12$ integer triplets that multiply to $72$.

Ten of them have unique sums, so if any of these were the solution, the census taker would not be in ignorance of the triplet, once he knew the sum

Two triplets, $(2,6,6)$ and $(3,3,8)$ both add to $14$. Once the professor implies that he has an oldest daughter, the choice is uniquely defined...

Edited to correct inability to count rows in a spreadsheet....