A difficult puzzle about blue-eyed prisoners who are stuck on an island

2024 Author: Malcolm Clapton | [email protected]. Last modified: 2023-12-17 03:44

The tyrant keeps prisoners on the island. A brave girl comes to them and makes a bold statement. Discuss what will happen after.

A despotic dictator has 100 people imprisoned on the island. It is impossible to escape from there, but there is one rule. At night, any prisoner can ask the guards for release. If the prisoner has blue eyes, he will be released. If not, they will feed the sharks.

In fact, all 100 prisoners are blue-eyed. But they have been living on the island since birth, and the dictator made sure that no one knew the color of his eyes. There are no mirrors on the island, the prisoners cannot see their reflection anywhere. All water containers are opaque.

The prisoners cannot communicate with each other in any way. They are forbidden to talk, exchange gestures, write messages in the sand, or otherwise communicate. But every morning they see each other at roll call.

The islanders are logical in all their actions, so none of them will dare to ask for release if they are not absolutely sure of success.

One day a dictator falls in love with a girl who always tells the truth. He succumbs to the persuasion of the chosen one, allows her to visit the island and talk to the prisoners. But she sets the following conditions: she can make only one statement and must not give new information to the prisoners.

The girl knows about the situation on the island and wants to help the prisoners to free themselves, but fears to incur the wrath of the dictator. After much deliberation, she informs the crowd of prisoners who were taken to the roll call: "At least one of you has blue eyes."

After the conversion, the dictator's beloved leaves the island. He is not angry with her. It seems to him that the information she gave to the prisoners is not dangerous and the statement made will not change anything. Life on the island seems to go on as usual.

However, 100 days after the girl's visit, the island turns out to be empty: all the prisoners demanded release and left it forever. Consider how it happened. We remind you: all the inhabitants of the island have excellent logic.

The number of islanders in this case does not matter. To simplify the task, we will leave only two prisoners - conditional Andrey and Masha. Each of them sees a prisoner with blue eyes, but knows that this blue-eyed one may be the only one.

On the first night, they both wait. In the morning they see that their companion in misfortune is still here, and this gives them a hint. Andrei guesses that if his eyes were not blue, then Masha would have freed herself on the first night, realizing that she was the only blue-eyed prisoner. In the same way, Masha thinks about Andrey. They both understand the following: "If the other waits, my eyes can only be blue." The next morning they both leave the island.

Now let's consider the situation when there are three prisoners: Andrey, Masha and Boris. Each of them sees two captives with blue eyes, but is not sure how many blue-eyed ones see the others - two or only one. On the first night, the prisoners wait, but the morning does not yet bring clarity.

Boris reasons like this: “If my eyes are not blue, Andrei and Masha are only watching each other. It means that next night they will leave the island together. But on the third morning, Boris sees that they have not gone anywhere, and concludes that the prisoners are watching him. Andrey and Masha think in the same way, so on the third night they all leave the island.

This is called inductive logic. You can increase the number of prisoners, but the reasoning will remain true and will not depend on the number of islanders. That is, if there were four prisoners, they would leave the island on the fourth night, five on the fifth, one hundred on the hundredth.

The key to this puzzle is the concept of shared knowledge. This is the knowledge that each member of the group possesses, and each member of the group knows that all other members of the group know, and everyone knows that everyone knows that everyone knows, and so on ad infinitum.

Thus, it becomes clear that the new information was given to the islanders not by the girl's statement itself, but by the fact that they all heard it at the same time. Now all the prisoners not only know that at least one of them has blue eyes, but that everyone is watching all the blue-eyed, and that they all know this, and so on.

The only thing that each individual prisoner does not know is whether he belongs to the blue-eyed, which is being watched by the rest. He will only know this when as many nights have passed as there are prisoners on the island. Of course, the girl could save the prisoners from 98 nights on the island, saying that at least 99 of them have blue eyes. But with an unpredictable dictator, jokes are bad, and it's better not to risk it.

The puzzle is based on the TedEd video.

Show solution Hide solution