This is a variation of the blue eyes puzzle.
Suppose the logicians want to get off the island. They are selfish and think only of themselves. They may help others as well, but solely for their own benefit. However, if a logician realises that he has brown eyes, he will try to ensure that the maximum number of people get off the island.
On the first day, the Guru makes her statement. On every subsequent day, 1 logician is secretly picked at random, and is asked to make a statement. The statement comes via an intercom, so no one knows who has given the statement or if it is true. The statement can not refer to individuals (by name, place, etc.) but must refer only to the following three sets - blue-eyed, brown-eyed, all. It must be logical and can use maths.
Will anyone ever make a useful statement? Will anyone ever leave before the 100th day?