The Muddy Children Problem

Many mathematicians grew up on a diet of puzzles like those set by Raymond Smullyan and Martin Gardner. Unfortunately, ingenious and elegant as these puzzles often are, they frequently have solution methods that don’t give rise to generalisable theory.

So I was recently surprised to find that one of my favourite puzzles of this type, the Muddy Children Problem, is actually an important example that appears in courses on mathematical logic, epistemology, computer science and even quantitative finance. If you haven’t met the problem before then have a go at solving it before looking at the various papers and courses on the subject. A fairly detailed elementary treatment can be found here though there are easier to understand informal arguments in existence.

The main academic approaches to the problem are via modal logic and Kripke models.

In less politically correct days it was known as the unfaithful wives problem and Smullyan’s version of this problem involved logicians with coloured hats.

Did I mention that it’s also a drinking game?