On a certain island there are people with assorted eye colors. There are 100 people with blue eyes and 100 people with brown eyes. Since there are no mirrors on this island, no person knows the color of their own eyes.
The people on the island are not allowed to talk or communicate with each other in any way. They are also not aware of the number of blue or brown eyed people on the island. For all they know, they could have red eyes too. But they are allowed to observe other people and keep count of the number of people with a certain eye color. There is a rule that the people on the island have to follow – any person who is sure of their eye color has to leave the island immediately.
One day, a person comes to the island and announces to the people that he sees someone with blue eyes. Everyone knows that the person only speaks the truth. What do you think happens?
If there was only one blue-eyed person on the island, then that person would look around and see that there is no other blue-eyed person. So he realizes that he is the only person with blue eyes on the island and leaves on the day of the announcement.
If there are 2 blue-eyed people, then they look at each other. Each one expects the other to leave on the day of the announcement. However, on the next day, when they realize that neither of them left the island, they would be able to deduce that both of them have blue eyes. They both leave the island on the second day.
Through mathematical induction, this logic can be applied to the 100 blue-eyed people on the island. So on the 100th day, all the 100 blue-eyed people leave the island.