The Blue-Eyed Islanders

On an island, 100 people have blue eyes and 100 have brown eyes. Everyone can see everyone else's eyes but not their own, no one may communicate about eye color, and anyone who deduces their own eye color must leave the island that midnight. All are perfect logicians, and all of this is common knowledge. One day a visitor announces to everyone: "At least one of you has blue eyes." Counting that day as day 1, on which midnight do the blue-eyed islanders leave?

• graph theory
• inclusion exclusion
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Source: Common-knowledge puzzle in wide circulation for over a century; a version appears in Littlewood's 'Mathematical Miscellany' (1953). Statement written for AxiomIQ.