"The Green-Eyes Riddle" is a famous puzzle of public knowledge and logical reasoning.
Puzzle Setup
On an island, there lives a group of fanatical cultists with perfect logical reasoning abilities.
- There are 100 blue-eyed people and 100 green-eyed people on the island.
- Their religious rule states: if a person knows they have green eyes, they must leave the island by midnight that day.
- There are no mirrors on the island, and they cannot speak to discuss eye colors. Each person can see the color of others' eyes but cannot directly know their own color.
- One day, an outsider arrives on the island and publicly announces: "I see green-eyed people here."
Question: What will happen next?
Core Conclusion
On the 100th day at midnight, all 100 green-eyed people will leave the island simultaneously.
Logical Deduction (Induction)
To understand what happened over these 100 days, let's first analyze smaller cases:
1. If there is only 1 green-eyed person on the island ($n=1$)
- He sees that all others have blue eyes.
- The outsider says, "There are green-eyed people."
- Since he sees no other green eyes, the only candidate must be himself.
- Result: He leaves at midnight on the first day.
2. If there are 2 green-eyed people on the island ($n=2$)
- Green-eyed A sees green-eyed B. A thinks, "If B is the only green-eyed person, he would leave on the first day."
- The first midnight passes, and B does not leave.
- A realizes, "Since B did not leave, it means B also saw a green eye (which is me)."
- Result: A and B leave simultaneously at midnight on the second day.
3. If there are $n$ green-eyed people on the island
- By induction, everyone waits for others to leave. If no one leaves for the first $n-1$ days, it indicates there are at least $n$ green-eyed people on the island.
- Result: On the $n$th day, everyone realizes the truth and leaves simultaneously.
Did the outsider say something trivial?
This is the essence of the riddle. Many people ask:
"Everyone can already see that there are green-eyed people on the island (because there are 100), why does the outsider's statement change the outcome?"
The answer is: The outsider transformed 'shared knowledge' into 'common knowledge'. "Shared knowledge" is what everyone knows, while "common knowledge" is what everyone knows that everyone knows.
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If we compare knowledge to light: Shared knowledge: Everyone has a lamp illuminating their own path, but you don't know if others' lamps are lit or if they can see the path. Common knowledge: Everyone gathers under a large lamp post in the square, where you not only see the path clearly but also see everyone in the light, and everyone sees you standing in the light.
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In logic and game theory, only common knowledge can generate "consistent expectations," which leads to collective action. Shared knowledge: Although everyone knows, due to the existence of "information islands," each person hesitates, worrying that others may not be informed, resulting in no one taking action (just like the blue-eyed islanders before the outsider spoke). Common knowledge: Breaks the mutual suspicion of information, establishing a "starting point for deduction" that allows everyone to synchronize their logical exercises.
Summary
Without the outsider's so-called "trivial statement," the logical chain could not begin. Just like a line of dominoes, the outsider's words are the finger that knocks down the first tile.