The answer

After I know that the ages multiply to 72, here is a complete list of the possibilities: Ages: Sum of ages: 1 1 72 74 1 2 36 39 1 3 24 28 1 4 18 23 1 6 12 19 1 8 9 18 2 2 18 22 2 3 12 17 2 4 9 15 2 6 6 14 ** 3 3 8 14 ** 3 4 6 13 Note that every combination of possible ages which has a product of 72 has its own unique sum of ages - except for 2, 6, 6 and 3, 3, 8, both of which share the sum of 14. Since the census taker can't figure out the ages after looking at the house number, the house number must be 14, because then the ages could be either 2, 6, 6 or 3, 3, 8. Now, the next clue is that the _oldest_ child is eating ice cream. This means that there is _one_ oldest child. Well, there is no oldest child of the ages are 2, 6, 6, so the ages of the children must be 3, 3, and 8 years old.