logic
Blue-eyed islanders
They leave on the nth night, and only because they all heard the guru.
Everyone can see the others’ eye colors but not their own. A guru says: “I see someone with yellow eyes.” If you deduce your own eye color, you leave that night.
Night —
The guru’s public statement creates common knowledge.
Each night that no one leaves gives everyone new information.
On the night that matches the real number of yellow-eyed people, they all deduce their eye color and leave.
Muddy children are the same count. Aumann's agreement theorem uses the same public step.