Blue-Eyed Island Induction and Common Knowledge Epistemic Logic

Scenario

100 people with blue eyes live on an isolated island. Everyone is a perfect logician. Rule: Anyone who discovers with mathematical certainty that their own eyes are blue must leave the island on a ferry departing precisely at midnight of that day. There are no mirrors or reflective surfaces, and direct communication regarding eye color is forbidden. Each person can see everyone else's eye color, but has no initial information about their own. Thus, each islander sees 99 blue-eyed people. On Day 1 at noon, a truthful visitor publicly announces to the entire island: 'At least one person on this island has blue eyes.'

Your Task

1. Using mathematical induction, determine what occurs on each midnight following the announcement.
2. State the exact day on which the blue-eyed islanders leave.
3. Resolve the apparent paradox: Since every islander already saw 99 blue-eyed people, what new information did the visitor convey?

Constraints

• Common knowledge of rationality is strictly maintained.
• Departures occur only at midnight.

Examples If $k = 1$, the lone blue-eyed person sees 0 blue eyes and leaves on Night 1. If $k = 2$, both see 1 blue eye, observe no one leaves Night 1, and both leave on Night 2.