Determining the Ages of John’s Three Children

This brain teaser revolves around inferring the ages of three children from partial, gradually revealed information: first a product of their ages, then a sum that ties to an external reference, and finally a structural hint about the "eldest" child. The setup forces the solver to reason about multiple possible age triplets consistent with the initial clues, then track how each new piece of information changes what Mary can deduce. It mimics the style of classic logic puzzles sometimes seen in brainteaser rounds for quantitative or logic-heavy roles, where candidates must reason about what other people know and do not know.

The solution leans on discrete combinatorics, reasoning about integer factorisations, and the symmetry of sum and product. It also requires meta-reasoning about knowledge: why Mary's uncertainty after each clue is itself informative. Interviewers are watching for systematic enumeration rather than guesswork, clear organisation of cases, and awareness of when symmetric information is insufficient. Strong answers explicitly track which candidate sets remain possible at each step and use the final structural clue to eliminate residual ambiguity in a logically airtight way.

What it tests

This problem class is governed by the interplay between the uniqueness (or lack thereof) of solutions to systems of equations involving symmetric functions—here, the product and sum of integers. When you know the product of several positive integers, there may be multiple unordered sets that satisfy this. Adding the sum as a second constraint often narrows the possibilities, but ambiguity remains if two or more sets share the same sum. The key insight is that information about the structure (such as the presence of a unique maximum or repeated values) can resolve ambiguity when symmetric constraints alone cannot. This pattern holds because symmetric functions like sum and product 'lose' information about the order and sometimes about the distinctness of the values, so extra structural clues are required to uniquely identify the set.