The Blind Muddy Child

Seven children are playing outside; some number of them have mud on their foreheads. Every child is a perfect logician, and it is common knowledge that each child can see every other child's forehead, except Blind (one specific child), who can see nobody but can hear everything. A child never sees their own forehead. Their father says: "At least one of you has mud on your forehead." He then repeats, in rounds, the question "Does any of you know whether you have mud on your own forehead?" In each round every child answers "yes" or "no" aloud, simultaneously, and all answers are heard by all. If everybody answers "no", the next round begins. In fact Blind and two other children are muddy (three muddy children in total). In which round does some child first answer "yes"?

• graph theory
• inclusion exclusion
• induction
• information

Source: Variant of the classic muddy-children puzzle (Littlewood; popularised in epistemic logic by Halpern and Moses); the blind-child twist is written for AxiomIQ. Statement written for AxiomIQ.