This thread is about one of my favorite logic puzzles, The Blue Eyed Islanders. It has been retold by many people, with different variations and wordings, but it’s not a puzzle about language, and instead just about logic.
Here is an exemplary presentation by xkcd: Blue Eyes - A Logic Puzzle
A group of people with assorted eye colors live on an island. They are all perfect logicians – if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.
On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.
The Guru is allowed to speak once (let’s say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:
“I can see someone who has blue eyes.”
Who leaves the island, and on what night?
xkcd’s discussion of the solution (spolier): Blue Eyes - The Hardest Logic Puzzle in the World - Solution
The Fields medalist Terence Tao has a few blog posts about it
Relevant Wikipedia article