There are 500 prisoners in a prison. The warden has put red mark on the forehead of some of them. They all can see each other once every day, but can not talk to each other, and can not see their own forehead (there is no mirror). The warden says that whoever identifies correctly that their forehead has a red mark, can leave the prison. Any day a prisoner identifies correctly that he is marked he can leave. All of them are logicians and so can make logical deductions.

1. Suppose that 25 of them have been placed red mark on their forehead, and the warden announces this number to all of them. How many days are needed for (the 25 of) them to identify correctly and how?

2. Suppose only one of them has been placed red mark, and the warden announces that at least one of them has been placed red mark, how any days are needed for the one with the red mark to identify correctly and how?

3. Suppose that 25 of them have been placed red mark on their forehead, and the warden announces that at least one of them has been placed a red mark, then in how many days can they all find out, and how?

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