There is a monastery with 100 monks in it. One day, a messenger arrives and announces that there is a disease going around and at least one monk in that monastery is infected. The disease is not contagious, but you die from it at sunrise on the day after you know that you have it. You know you have it when you know that you have a blue dot on your forehead. There are no mirrors in the monastery and the monks have no other ways of seeing their own foreheads. They eat lunch together everyday at a very big round table, and so each one of them can see the status of the other 99 monks. They have all taken vows of silence and cannot communicate to each other what they see on each other's foreheads. Two weeks later, some monks die.
How many monks die and why?
Fourteen monks die because after 14 days pass, they all realize that they must have a mark on their forehead. How do they know? Let's start with one monk. If only one monk had the disease, then he would look at the other 99, see that none of them had any marks, remember that at least one monk has the disease, and deduce right then that he must have the disease. So he would die the next day at sunrise. If two monks had the disease, then they would each look at the other 99, see that one of the 99 had the mark, remember that at least one monk has the disease, and feel sorry for the poor guy. However, the next day, when the poor guy is still alive, the second monk realizes that there must be a second poor guy and it must be him. He himself must also have the disease, otherwise the lone monk would have died already. The two infected monks reason in the same way and while they weren't sure after the first day if they themselves were infected or not, they are sure after the second day. So if only two monks had the disease, they would both realize they had it after two days and die after two days. A similar thing happens when each of three infected monks looks out and sees two infected monks. After three days, they each realize that the two infected monks that they see cannot be the only infected monks—otherwise they would have died after two days. You can see how the reasoning continues on for any number of monks. If 14 monks had marks on their forehead, they would realize this after 14 days. On the 14th day, they could no longer think that only 13 monks had the disease, because if only 13 monks had the disease, then they would have realized this and died after 13 days.