100 monks In Normandy, there is monastery of 100 Trappist monks. They are highly logical and can determine the answer to any solvable mathematical problem. The monks also keep strictly to the rule of St Benedict. To discourage vanity, they have no mirrors or any other reflective surfaces. They remain completely silent and all times and indeed do not communicate with each other in any way at all. However, they always take meals together each day at breakfast.

On a routine medical visit, the local doctor informed them one afternoon that at least one of the monks had contracted a rare and fatal disease. All affected monks displayed the characteristic symptom of a red dot on their forehead. He told them that the illness had already passed the contagious stage so that no new monks would be affected. Most bizzarely, the disease works directly on the brain and kills those infected during the night on the day they realise that they are ill.

The Puzzle: Everything is fine until the morning of the 13th day after the announcement, when some of the monks do not turn up for breakfast (for they are dead). How many monks died?

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Think carefully about the conditions. Each monk knows which of the other monks has the disease, but knows nothing about his own state of health. There is no way for the monks to communicate with each other, none the less they somehow are able to determine that they are ill, and consequently die. Note that the solution doesn't require some trick mechanism by means of which the monks can look at themselves. Remember that the monks are highly logical and hence can precisely determine the consequences of the doctor's proclamation.

Turn the problem on its head. Suppose you know how many monks are infected, then try to determine when they will die. Start by considering the case where only one monk has the disease

Were there only one infected monk, he would immediately know from the instant of the doctor's statement (day zero) that he was the sole infected monk. The reason? Because he knows at least one monk is infected and yet he sees no monks with the red dot, and hence deduces that he is infected. Consequently he would die that evening and would be missing from breakfast on the first day after the announcement.

If there were two infected monks, neither would be able to deduce immediately that he was ill. For concreteness, suppose Abelard and Bernard were both ill. On day zero Abelard knows either that both he and Bernard are ill, or that Bernard only is ill. Bernard makes a similar deduction. However on the first day, Abelard sees that Bernard has not died. He deduces that Bernard cannot have been the only ill monk. Bernard makes a similar assumption. They both die that evening and are missing from breakfast the next day.

Continuing the process, one can determine that if n monks were infected, then they will all die on the n-th day after the announcement. Given that the monks all died on the 13th day, we deduce that 13 monks were affected.

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