# Can you solve the famously difficult green-eyed logic puzzle? - Alex Gendler | Summary and Q&A

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June 16, 2015
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TED-Ed
Can you solve the famously difficult green-eyed logic puzzle? - Alex Gendler

## TL;DR

In an island where 100 prisoners have green eyes but can't communicate, the prisoners find out their eye color through a logical deduction after a visitor's statement.

## Key Insights

• 😃 The prisoners use inductive reasoning to deduce their eye color based on the behavior of others.
• 💁 The concept of common knowledge, where everyone knows that everyone else knows an information, is crucial in solving the puzzle.
• #️⃣ The number of prisoners doesn't affect the process; it only determines the number of nights needed for everyone to leave.

## Transcript

Imagine an island where 100 people, all perfect logicians, are imprisoned by a mad dictator. There's no escape, except for one strange rule. Any prisoner can approach the guards at night and ask to leave. If they have green eyes, they'll be released. If not, they'll be tossed into the volcano. As it happens, all 100 prisoners have green eyes, bu... Read More

### Q: How do the prisoners figure out their eye color without communicating?

The prisoners rely on the logical deduction that if there is at least one person with green eyes, everyone will eventually leave.

### Q: Why didn't the prisoners leave on the first night?

The prisoners needed to wait and see if anyone else would leave on the first night, indicating that they themselves have green eyes.

### Q: How does adding more prisoners affect the process?

Adding more prisoners extends the number of nights they need to wait until everyone has deduced their own eye color and leaves.

### Q: What role does the visitor's statement play?

The visitor's statement doesn't provide new information but creates common knowledge among the prisoners, enabling them to make deductions.

## Summary

In this video, there is an island where 100 perfect logicians are imprisoned by a mad dictator. The only way to escape is if a prisoner has green eyes, but they are not allowed to communicate with each other. The narrator visits the island and makes a statement, which eventually leads to all the prisoners being released. This is achieved through the concept of common knowledge, where each prisoner knows that everyone else knows about the green-eyed people.

### Q: What is the strange rule on the island that allows prisoners to be released?

The rule is that any prisoner can approach the guards at night and ask to leave. If they have green eyes, they will be released. If not, they will be thrown into the volcano.

### Q: How do the prisoners know that they have green eyes?

The prisoners have lived on the island since birth and are unable to learn their own eye color. There are no reflective surfaces and all water is in opaque containers, so they cannot see their eye color directly.

### Q: What is the condition given to the narrator before visiting the prisoners?

The condition given to the narrator is that they can only make one statement and cannot provide any new information to the prisoners.

### Q: What statement does the narrator make to the prisoners?

The narrator tells all the prisoners, "At least one of you has green eyes."

### Q: How does the narrator's statement help free the prisoners?

The narrator's statement introduces the concept of common knowledge to the prisoners. Each prisoner now knows that everyone else heard the statement and also knows that everyone else knows about the green-eyed people.

### Q: How do the prisoners use inductive reasoning to determine their eye color?

By observing that no one has left the island in the first night after the narrator's statement, each prisoner gains new information. They realize that if they themselves had non-green eyes, the others would have left. Therefore, each prisoner deduces that their own eye color must be green.

### Q: Why does the pattern of prisoners leaving continue for multiple nights?

The pattern continues because each prisoner knows that everyone else is keeping track of all the green-eyed people they can see. And each prisoner also knows that everyone else knows this, creating a chain of common knowledge. It takes as many nights as the number of prisoners on the island for each prisoner to gain this level of certainty.

### Q: What is the concept of common knowledge?

Common knowledge, coined by philosopher David Lewis, refers to the idea that everyone knows something, knows that everyone else knows it, and knows that everyone else knows that they know it, and so on. In this case, the prisoners all know about the green-eyed people and know that everyone else knows as well.

### Q: What information was contained in the narrator's statement?

The narrator's statement itself did not provide any new information. It simply stated that at least one person has green eyes. However, the new information came from the fact that the statement was shared simultaneously with everyone, creating common knowledge among the prisoners.

### Q: Why did the narrator not tell the prisoners that at least 99 of them had green eyes?

Telling the prisoners that at least 99 of them had green eyes would have immediately led to their release. However, in the presence of a mad dictator, it was important to ensure a good headstart for the prisoners. Therefore, the narrator chose to give a statement that allowed the prisoners to figure out their own eye color gradually.

## Takeaways

The video showcases a clever solution to the prisoner's dilemma on the island. By introducing the concept of common knowledge through a statement, the prisoners are able to deduce the color of their own eyes in a logical manner. This demonstrates how reasoning and information sharing can lead to collective decision-making and individual freedom.

## Summary & Key Takeaways

• In an island, 100 prisoners have green eyes but can't communicate and don't know their own eye color.

• A visitor is allowed to make one statement to the prisoners but can't provide new information.

• The prisoners deduce their eye color through logical reasoning and eventually leave the island.