Assuming that the people know that they are either blue or brown eyed, but not which, then we can solve this by induction.
  • If there was 1 blue-eyed and 199 brown eyed people, the blue eyed person would leave on the day of the announcement, and everyone else would leave the next day.
    • This is because the blue-eyed person immediately sees that everyone else has brown eyes so he must have blue, so he leaves as soon as he can.
    • The rest of the brown-eyes don't know immediately whether they had brown or blue eyes, but after they see that the blue-eyed person left, they realize they must have brown, so they can also leave. This happens on the second day after the announcement.
  • If there were 2 blues and 198 browns, the blue eyes would leave on the second day and the brown eyes on the third day.
    • On day 1, the two blue eyes know that there's one other blue but they don't know their own color.
    • However, they know that if they weren't blue, the one blue-eye would leave on day 1.
    • However, since no one leaves on day 1, they realize they must also have blue, so they both leave on day 2.
    • Seeing this on day 3, the browns realize they cannot have blue eyes and they all leave as well.
  • Using the same logic, we can predict that the blue eyes will all leave on day 100 (with day 1 being the day of the announcement), and the brown eyes will leave on day 101.
This is an interesting puzzle because it seems to show that a statement containing no new information (everyone knows at least one person has blue eyes) can lead to a change in outcomes.
However, the statement does contain new information in one hypothetical state of the world (the state of 1 blue 199 brown.) So, information about hypothetical, non-actualized states of the world can actually impact outcomes in the real world. Fascinating.