The Green-Eyed Prisoners Logic Puzzle: Can You Solve It?

A remote island, a dictator, a hundred green-eyed prisoners, and a single way to escape. Despite the strict conditions, the outcome is always the same: all prisoners manage to leave the island, and the dictator falls. The question is: Where is the key? What is the exact answer? Thanks to this TED-Ed video, we can explore the logic behind the puzzle quickly and simply.

The rules of the puzzle

It’s possible that over time our readers have encountered other versions of the same puzzle, but the story goes roughly like this: On a remote island, a power-hungry dictator keeps a hundred people prisoner, all of them perfectly logical. The island is full of dangers, so escaping seems impossible... However, there is one rule: If a prisoner approaches the guards during the night and asks to leave, the guards will allow it, as long as the prisoner in question has green eyes. Otherwise, the prisoner will be thrown into the volcano. A critical detail is that all hundred prisoners have green eyes, but the dictator has made sure they don’t know this. There are no mirrors or other reflective surfaces. Water is distributed in opaque containers, and as if that were not enough, the prisoners are forbidden from communicating with each other, though they can look at each other.

Now, anyone might “run the risk” with the guards, but we mentioned earlier that the prisoners are “perfectly logical”, meaning they won’t try to leave the island without absolute certainty of success. At this point, you appear. After much external pressure, the dictator gives you special permission to visit the island and speak to the prisoners, with two conditions: You can make only one statement, and you cannot give them new information. The first impression is that the dictator has set a trap for you, but in reality, his armor has a crack. Your statement meets his conditions, and still, all the prisoners manage to escape, leaving the island empty. What did you say? Do you have the answer?

Collective knowledge

In this version, your announcement should be: “At least one prisoner has green eyes” (I’m only sharing it because it appears in the video, I don’t like giving away puzzle answers). To better understand the dictator’s mistake, imagine there are two prisoners, “A” and “B”, instead of a hundred. From their perspective, each sees a green-eyed prisoner, but without information about themselves and without being able to communicate, both spend another night on the island.

However, upon seeing that the other prisoner is still there, they acquire new information. “A” concludes that if “B” had seen a prisoner without green eyes, all “B” would have had to do was leave, since your statement would only apply to “B”. On the other hand, “B” experiences exactly the same: If “A” had left the first night, “B”’s eyes would not be green.

Adding a prisoner only extends the chain of reasoning by one more night, but on the hundredth day, the dictator’s fate is sealed, and all the “logical prisoners” escape. The puzzle explores the concept of what is called “collective knowledge”: The new information the prisoners needed was not in your statement, but in sharing it with everyone at the same time.