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.
blue eyes will all leave on day 100(with day 1 being the day of the announcement), and thebrown eyes will leave on day 101.