### N Monks problem

There's a monastry of fifty monks that have taken a vow of silence. They go through the same routine every day. They wake up, pray. They go to lunch, their only meal of the day, were they all sit round a big circular table, and eat a simple meal. Then they go back to their rooms and pray. Then they fall asleep.

One day, the head monk of the area comes to visit. He's a bit different - he's allowed to talk and has a life outside the monastry. He tells them that there is currently a plague ravaging the land. People everywhere are dying. The disease manifests itself as red blotches on the forehead. The blotches are the only manifestation of the disease for three months, whereupon the next stage starts, a horribly painful death.

The head monk tells them that at least one of their midst will have the disease, probably more. Anybody with the disease should kill themselves, to save all of the pain and suffering. By killing themselves, they will restrict the movement of the disease, and will go to heaven. Anybody with the disease will show the first symptoms within a month. The chief monk then leaves. He returns two months later, to find that all of the infected monks have killed themselves, and they all did it on the same day.

Bear in mind the following:

* They have no mirrors or any other way of seeing themselves
* The blotches appear only on the forehead and cannot be seen by the monk
* Infected monks feel no different - the only manifestation of the first part of the disease are the blotches.
* The monks cannot talk to each other or in any other way communicate.
* Any monks with the disease will display the blotches within a month of the head monk leaving.
* At least one monk definitely has the disease.
* The monks only see each other once per day, at lunch, when they are all sat round the round table.
* These monks are brighter than the average monk....

How did they know whether or not they were infected and why did they all kill themselves on the same day?