On anisland remote, lives one tribe which follows a iron rule: Nobody has to know the color of their eyes. There are 1000 inhabitants of the island: 100 have the blue eyes And 900 The brown eyes. However, nobody knows what color your eyes are. Anyone who finds out must start immediately, climbing the ferry that stops on the coast every morning to take away those who have broken the law.
To avoid discovering it, there are no mirrors, the water is kept in opaque containers and talking about the color of the eyes is strictly prohibited. In addition, the islanders are extremely intelligent And they are all capable of thinking logically.
One day, A traveler arrives on the island. During the banquet in his honor, he pronounces a phrase that will change his life on the island:
How beautiful! I’m not the only one here to have blue eyes!
This affirmation On two feet it may seem useless: The traveler simply noticed that there is at least one person with blue eyes. On two feet, it does not seem that this statement can have consequences, there is nothing new in the words of the traveler!
And instead … starts an unstoppable chain of events: at the 100th day After the banquet, All islanders with blue eyes leave the island at the same time. But why? We find out in this article, in which we see how some logical problems can be solved with a principle called mathematical inductionwhich consists in demonstrating a basic case and then generalizing the solution.
So let’s start from the case in which there is only one person with blue eyes.
The solution if there is only one person with blue eyes: he goes away after one day
Let’s start with the simplest case: there is one only person With blue eyes, which we call Andrea. After the traveler’s phrase, Andrea looks around and does not see anyone with blue eyes. So he understands that He must be the only one with blue eyes. Since he now knows with certainty the color of his eyes, he broke the law and the next morning he takes the ferry to leave the island. All the others will be able to continue living on the island because they will know they do not have blue eyes, but they will not know with certainty of what color their eyes are.
If there is One person with blue eyes, the island will leave the island day after The banquet.
If there are two islanders with blue eyes, they go away after two days
Now let’s imagine there are two islanders with blue eyes: Andrea And Beatrice. Both see one person with blue eyes (the other), but they don’t know they have them too. They make this reasoning:
If I don’t have blue eyes, then the only person with blue eyes is the other, who will start tomorrow morning.
The next morning, however, Andrea and Beatrice see that Nobody left the island. This means that the other person also saw someone with blue eyes and hesitated for the same reason. And the other person with blue eyes can only be themselves, because they clearly see that all the others have brown eyes. At this point, They understand both of having blue eyes And, the second morning, they go up on the ferry together.
If there are two islanders with blue eyes, they will leave the island two days After the banquet.
The inductive reasoning: what happens for 100 islanders with blue eyes
Now let’s imagine that people with blue eyes had been three third person with blue eyes – in addition to Andrea and Beatrice there is Chiara – and we try to schematize the reasoning, remembering that the inhabitants of the island are very intelligent and are capable of mathematical reasoning:
- All three know that, if there had been only one person with blue eyes, he would leave after a day;
- They also know that, if there were only two people with blue eyes, they would go to the second day;
- Andrea, Beatrice and Chiara see each two people with blue eyes and think like this:
If in the morning of the second day after the banquet nobody leaves the island, it means that there is a third person with blue eyes and that that person is just me, because I don’t see any other person with blue eyes.
When the morning of the second day And Nobody leaves the islandall three understand that there are not only two, but three people with blue eyes! And that they are the three of them. Abandon Therefore the island together with the morning of third day.
This reasoning can be extended to any number of islanders with blue eyes: if there are on the island 100 people with blue eyeswill all wait until the 99th day. When no one leaves the island on the morning of the 99th day, all 100 will understand that they have blue eyes. So, they will all leave together with 100th day after the banquet.
The induction principle works just like this: the initial case is demonstrated (the one in which there is only one island with blue eyes) and then it is shown that if it applies to N (in this case 99), then it also applies to N+1 (in this case 100) and this guarantees us that the reasoning can be extended to any natural number. If there were 300 people with blue eyes, for example, they would start after 300 days.
The solution of the indovinel comes thanks to common knowledge
Anyone who reads this riddle could do the following objection:
But if there is a man with blue eyes, the other islanders knew it, already because they had seen him! And all the reasoning would be worth still. The fact that the traveler said aloud does not change anything!
Instead, in reality, everything changes! Why? Because the traveler’s phrase introduced a new one crucial information: not so much the fact that there are island -olds with blue eyes, but the fact that everyone knows it in same instant. The affirmation of the traveler has transformed individual knowledge into common knowledge. Before his sentence, every person with blue eyes knew that there were others, but nobody knew that others also knew it, because on the island it is strictly forbidden to discuss the color of the eye.
After the traveler’s declaration, Everyone knows that everyone knows of the existence of the blue eyes. This leads all the islanders to start the reasoning at the same time and start with the count of the days together. Only in this way can everyone be sure that others are also doing the same reasoning at the same time and then get to leave the island together.
