Can you solve the famously difficult green-eyed logic puzzle? - Alex Gendler

Summary

On an island where 100 logical prisoners, all with green eyes, are trapped, a dictator allows escape if a prisoner knows their eye color. However, the prisoners don't know their own eye color and can't communicate. When a visitor announces, 'At least one of you has green eyes,' it sparks a chain reaction of logical deductions. By the 100th morning, all prisoners leave, as they each realize they have green eyes. This puzzle highlights inductive reasoning and the concept of common knowledge, where shared awareness among individuals leads to eventual certainty.

Takeaways

• 🟢 The prisoners are all logicians and cannot learn their own eye color due to restrictions imposed by the dictator.
• 🟢 The prisoners are told they can leave if they know they have green eyes, but risk death if they are wrong.
• 🟢 All prisoners have green eyes, but they don’t know it themselves, creating a dilemma where no one is sure of their own eye color.
• 🟢 The speaker can only make one statement and cannot give new information. They say, 'At least one of you has green eyes.'
• 🟢 This statement initiates a chain of inductive reasoning among the prisoners, each observing the others during daily head counts.
• 🟢 The number of prisoners plays a critical role in the number of days it takes for them to leave; they all escape on the 100th day.
• 🟢 The reasoning process works because each prisoner waits and sees if others leave, revealing new information with each passing day.
• 🟢 The situation can be simplified with fewer prisoners: with two prisoners, both would leave on the second day.
• 🟢 The concept of 'common knowledge' is central to this puzzle; every prisoner knows that others are making the same deductions.
• 🟢 The dictator is outsmarted because the statement reveals nothing new individually but collectively triggers the prisoners’ escape through logical deduction.

Q & A

What is the only way for the prisoners to escape from the island?

A prisoner can approach the guards at night and ask to leave if they are certain they have green eyes. If they do, they'll be released; if not, they'll be thrown into the volcano.

What restriction is placed on the prisoners' knowledge about their own eye color?

The prisoners cannot learn their own eye color because there are no reflective surfaces, and they are not allowed to communicate with each other.

What is the significance of the statement 'At least one of you has green eyes'?

The statement is crucial because it provides common knowledge to all prisoners. Now, each prisoner knows that there is at least one green-eyed person and that everyone else knows this too.

How does inductive reasoning help the prisoners figure out their own eye color?

Each prisoner waits to see if others will leave. If they don't, it means the others are unsure and are waiting for more information. After a certain number of nights, each prisoner deduces that they must have green eyes because the others haven't left.

Why do all the prisoners leave on the 100th night?

Because each prisoner sees 99 other green-eyed people. On the 99th night, they realize that if they didn't have green eyes, the others would have left already. The fact that no one has left by that point means they all must have green eyes.

What would happen if there were only two prisoners on the island?

If there were only two prisoners, each would see the other’s green eyes. After the first night, they would both stay, but after seeing each other still there in the morning, they would realize they both have green eyes and leave on the second night.

What is the concept of 'common knowledge' as described in the script?

Common knowledge means that not only do all the prisoners know a piece of information, but they also know that everyone else knows it, and that everyone knows that everyone else knows it, and so on.

Why doesn't the dictator’s rule change anything initially?

The dictator believes that because the visitors didn't tell the prisoners any new information (since they already see each other’s eyes), the situation hasn’t changed. However, the statement creates common knowledge, which does change the prisoners' reasoning process.

How would the situation change if the statement was 'At least 99 of you have green eyes'?

If the statement was 'At least 99 of you have green eyes,' the prisoners would know on the first night that everyone has green eyes, so they would all leave the following night, reducing the wait time to just one day.

Why is it important that the prisoners can't communicate with each other?

If the prisoners could communicate, they could confirm their eye colors much faster, bypassing the need for this inductive reasoning process. The lack of communication forces them to rely on indirect reasoning based on what they see and what others do.