Muddy Children Puzzle

The canonical worked example of Epistemic Logic and Common Knowledge. n children have muddy foreheads; each sees the others’ faces but not their own. A father announces publicly “at least one of you is muddy” and then repeatedly asks “does anyone know whether they are muddy?”. Although the announcement seemingly tells the children nothing they cannot already see, it creates common knowledge of the fact, and after k rounds of simultaneous “no” the k muddy children can all deduce their state. The puzzle shows how a public announcement changes group knowledge and how reasoning about others’ knowledge yields conclusions about one’s own.

In this vault

• Reasoning About Knowledge
• Epistemic Logic
• Common Knowledge
• Distributed Knowledge