In order for the conversation to proceed as it did, the product of the ages of Bob’s children needs to be able to be factored in 2 different ways, one of which has the largest 2 factors being the same as each other, and which have the same sum of factors. For example, with 72, Andy would have been unsure whether Bob’s children were aged (8,3,3) or (6,6,2) even knowing his address was 14, until having his final question answered.

The next largest number that satisfies these constraints is 288. Some submitters interpreted “next largest” to mean the largest number smaller than 72, not the smallest number larger than 72. Since this was a bit vague, we accepted 36 as correct as well.

One of the most popular incorrect answers was 225, with the thought that Bob’s children could have been (25,3,3) or (15,15,1). However, 25 is too old to be a child!

Congratulations to all of the correct responders, especially Alice Scarpa, this month’s randomly-chosen winner of a Jane Street t-shirt!