In a distant monastery there are more than 50 monks all of them great experts in logic. They are all day in their cell praying and meet once a day for dinner at a round table where faces can be seen. When they finish dinner, each one returns to his cell in absolute silence. The monks have taken a vow of silence, they cannot gesture or communicate in any way and there are no mirrors in the monastery or any way of being reflected.
One day the prior father arrives and before beginning dinner he tells you: one or more of you have been pointed out by an angel who has made a red mark on your forehead. Those who have the mark must go on a pilgrimage as soon as they know it. Finally, the prior father left without indicating who the elect were. After 7 days, all the monks with the red mark realized that they were marked and only they went on a pilgrimage.
Can you discover how many were the chosen monks how they realized it?
There will be 7 monks who will go on pilgrimage.
To reach this conclusion we will make the following reasoning: If only one monk were marked, the first day, during dinner, he would see that no one is marked, then if the prior father said that one or more were marked he would deduce that he was the only one elected and would leave the first day.
If they were 2 monks marked, the first day, during dinner, each of them would see another monk marked so he could not know if he himself is or not, so neither of them could leave. On the second day, when he sees that the marked monk continues there he deduces that that other monk also sees another with the mark since if he had not left the first day applying the previous deduction. Since he only sees a marked monk, he deduces that he himself has the other mark and the two leave on the second day.
If the marked monks were 3, the first day each would see two other monks with a mark. Each of them would apply the above reasoning and deduce that if only the other two monks had each mark they would see a single marked monk, which would take two days to realize that they have the mark and therefore would march on the second day . But since there are three marked monks, on the third day, they will be seen at dinner which means that the other two marked monks also see two marked monks and so they have not been able to leave. Therefore he deduces that there is a third marked monk who is himself and can all march on the third day.
In the same way we could extrapolate the rest of the cases until reaching the 7 days proposed by the statement and since the number of monks that march coincides with the number of days elapsed we deduce that there are seven marked monks.