A number of monks live in a monastery and they do not communicate in any way. Monks gather just once a day in the prayer hall; there are no mirrors and monks never see their own faces. A disease is ravaging the land; its first visible sign is a mark on the forehead of the victim. The disease causes no discomfort initially but it is eventually so painful that anyone who has acquired it will immediately commit suicide. A visitor one day announces: 'At least one monk in this place has a mark on the forehead". It is common knowledge that all monks are rational.

1. Assume there are only two monks in total and both have the disease. After the announcement is made, will they commit suicide simultaneously? If yes, in how many days?

2.If there were three monks in total and all of them committed suicide simultaneously? How many monks had the disease and in how many days they commuted suicide?

3. Suppose you are one of the four monks living the monastery; on the third day after the accountant all the monks (including you) show up for the praying, but the next day only you show up for the praying, how many monks with a mark did you see? Can you say that you are safe from the disease?

4.If after exactly one week has passed all monks simultaneously commit suicide, how many monks were in the monastery? How many had the disease?

there are a lot of variations to this. unfortunately, this is a bad variation since suicide depends on their own pain threshold, not on knowing if they have the disease. There's one about prisoners with hats, another about an island.

however, assuming they commit suicide as soon as they realize they have the mark, and they trust the visitor not to lie, i'll do the first one.

i see the other person is marked. two possibilities: 1) i have the mark, 2) i don't have the mark.
Suppose 2). if i dont have the mark, the other person knows he has the mark since at least one of the two has it, and he will die immediately.
However, he doesn't commit suicide. That must mean he's not sure if he has the mark. The only way he's not sure is because I also have the mark. So I die after i realize he's not dying. Since I waited before dying, he knows he has the mark as well, and dies afterwards.

I'll let you do the rest.