Problem #13 HARD
The Outbreak of Blue Eyes
Meta Logic Lateral Thinking
Problem Statement
An island contains 100 people. Some have blue eyes, and the rest have brown eyes. They all live under a strict religious law: if anyone discovers they have blue eyes, they must leave the island at noon the next day.
No one is allowed to speak of eye color, and there are no mirrors or reflective surfaces. Everyone knows everyone else's eye color but not their own.
One day, a visitor arrives and announces to everyone: "At least one of you has blue eyes."
Assuming everyone is perfectly logical, what happens next?
Answer & Quick Explanation
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If there are N blue-eyed people, they will all leave the island together on the N-th day after the announcement.
Detailed Editorial Solution
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This riddle uses mathematical induction and common knowledge.
Let's analyze the problem for different numbers of blue-eyed people (N):
- **Case N = 1**: If there is only 1 blue-eyed person, they look around, see only brown-eyed people, and realize they must be the one. They leave on the 1st day.
- **Case N = 2**: If there are 2 blue-eyed people (A and B), A sees B's blue eyes and B sees A's blue eyes. On Day 1, no one leaves (since A thinks B might be the only one, and B thinks A might be). On Day 2, when A sees B didn't leave, A realizes B must have seen someone else's blue eyes (A's own). Thus, both realize they have blue eyes and leave together on Day 2.
- **Induction**: If there are N blue-eyed people, no one leaves for the first N-1 days. On the N-th day, they all realize that if there were only N-1 blue-eyed people, they would have left on day N-1. Since they didn't, all N blue-eyed people realize they themselves must have blue eyes, and they all leave at noon on the N-th day.
If there are, say, 15 blue-eyed people, they will all leave on the 15th day.