Logic riddles: The paradox of the blue-eyed islanders

Consider the following riddle: There is an island with 100 inhabitants, 10 with blue eyes and 90 with brown eyes. They can all see the color of other people’s eyes, but not their own. They cannot communicate about it and there are no mirrors, and there is a law that says that if someone discovers that they have blue eyes, then that person must leave the island at 8 am the next day. All the inhabitants of the island are very good at logical reasoning. One day, an outsider arrives on the island, takes one look at its inhabitants and says: “I am pleased to have seen at least one person with blue eyes.”

The conventional solution holds that everyone with blue eyes will leave the island on the 10th day. If there were only one inhabitant with blue eyes, he or she would immediately realize that the stranger is referring to him/her, since the other 99 peole have brown eyes. This person would leave the next morning. If there were two people with blue eyes, seeing that the next morning no one leaves, they would both know that they have blue eyes, for they would think: “If the only one of the other 99 who has blue eyes saw that mine are brown, he/she would immediately know that his/hers are blue and would have left the next morning.” Therefore, they would both leave the second day. If we repeat this reasoning in this fashion, we conclude that on the 10th day, all 10 blue-eyed islanders would leave.

Irrefutable reasoning? Perhaps not. Those who wish to delve deeper into the subject will find abundant documentation on the web (see, for example, the blog of the great Australian mathematician Terry Tao).

The conventional solution to the riddle of the blue-eyed islanders is sometimes refuted with the following reasoning: if there are 90 brown-eyed people and 10 blue-eyed people on the island, all the islanders know there are some blue-eyed people there; therefore, for an outsider to arrive and say that there is at least one person with blue eyes does not add any information, since each and every one of the islanders already knew it. And if the stranger doesn’t bring any new information, how could his statement cause the 10 blue-eyed inhabitants to leave the island?

The problem thus becomes the paradox of the blue-eyed islanders, which I submit to the consideration of our astute readers.

Carlo Frabetti is a writer and mathematician based in Spain who has published more than 50 works of popular science for adults, children and young adults.