[Ashutosh]

A correction. In the case 998/2, both blued eyed men commit suicide 2 days after the visitor's speech (and not 1 day) since the rule says you die a day after you know your eye colour. In the case 900/100, it is exactly 100 days after which all blue eyed people commit suicide all together. To prove this properly we may use induction. For instance in the case 997/3, say blued eyed men are A, B and C. Then A will argue thus:

"Say I didn't have blue eyes. Then A and B will use the 998/2 argument and I should see them die in 2 days from now."

B and C have similar thoughts. But 2 days later no one dies so A, B and C simultaneously deduce that they've blue eyes. Hence on the third day they kill themselves. You can see now how induction works.

The reason why the brown eyed people don't kill themselves is simple. This argument will not work if the visitor hadn't made his statement. The brown eyed men still don't share the "common knowledge" that there are brown eyed people on the island. To see the meaning of common knowledge more clearly, suppose there were only 2 brown eyed people say A and B on the island. Denote by S, the statement "There is at least one brown eyed person on the island". Then while A knows S and B knows S, A doesn't know that B knows S and B diesn't know that A knows S. This is the crux of matter. What the visitor did was that he made S common knowledge.