On an island in the middle of the ocean lives several people of a peculiar sort: They are all logicians. One hundred of them have blue eyes, while the rest have brown eyes. The leader of the group, known as the guru, has green eyes. It so happens that no one on this island knows the color of their own eyes.
At midnight every night a ferry arrives at the island, and leaves shortly thereafter with anyone who can tell the ferryman their eye color.
One day, the guru stands up and says, so that everyone hears:
“I can see at least one person with blue eyes.”
What will happen? Will anyone leave the island? Who, and when?
Please not that no one will tell anyone else their eye color, or communicate it directly in some other way. As soon as someone knows their eye color she will leave the island with the next ferry - they all want to leave this weird place, after all. During any given day, all islanders will see each other, and notice any absences.
Answer
It is easier to figure out what would happen if there were fewer people with blue eyes.
We know that there is at least one blue-eyed person - the guru says so. If this were the case, the blue-eyed person would see that everyone else has brown (or green) eyes, and so she would know that she must have blue eyes. She would leave on the first midnight.
If there were two blue-eyed people, they would both see the other, and know that either there is only one blue-eyed person (the other), or there are two, and their own eyes are blue. They can’t distinguish these two cases yet, but, if there were only one with blue eyes, she would leave the first night, as in the previous example. Since this doesn’t happen, they know that they both have blue eyes. They would leave on the second night.
This process can be continued (inductively) with larger groups of blue-eyed people. If there were one hundred blue eyed people they would all leave on the hundredth night.
There are several variations of this puzzle, with different names, including Josephine’s Problem and The Muddy Children Problem. The most popular version might be Blue Eyes by Randall Munroe.