Solution
The Blind Muddy Child
Show the problem again
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"?
Worked solution
The answer is 7, the number of children. Model a world as the set of muddy children (non-empty). Consider first the worlds in which Blind is clean and k of the six sighted children are muddy. These behave exactly like the ordinary muddy-children puzzle: by induction, all k muddy sighted children answer "yes" together in round k and nobody else does. In particular the worst of these worlds, with all six sighted children muddy, is settled in round 6. Now look at the worlds in which Blind is muddy. After r silent rounds, the surviving worlds are all worlds with a muddy Blind, together with the Blind-clean worlds that have more than r muddy children. A sighted child sees Blind's mud and every other forehead, so their two candidate worlds (own forehead clean or muddy) both have Blind muddy and both survive every round, so they never know. Blind, on the other hand, sees nothing, so Blind's candidates are every world still alive. Blind cannot know until no Blind-clean world survives, and that happens only after round 6, when the last such world (six muddy sighted children) has been ruled out by their silence. So Blind answers "yes" in round 7 and is the only one to do so. In the stated world (Blind plus two others muddy), the first "yes" therefore comes in round 7, however many others are muddy. This was also confirmed by simulating the public announcements over all worlds for 5, 6 and 7 children: with a muddy Blind, Blind alone answers first, in round n.
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.