Puzzle – 57| Priests in Temple

There are 20 priests in a temple. One day, Lord Shiva appears before them and tells them that some of them have sinned, and that a black spot would appear on the forehead of all the priests who have sinned. The priests are not allowed to look into a mirror or communicate with each other. When any priest finds out that there is a spot on his forehead, he should leave the temple on that day itself. At least 1 priest has sinned. How can a priest find out whether he has a spot on his forehead. What would be the pattern of the priests leaving the temple?

Solution:
Scenario 1 (Only 1 priest has sinned): On the first day itself, the priest who has sinned would see that no other priest has the spot on his forehead and would know that he is the one who has sinned(because Lord Shiva said that atleast one of them had sinned). Therefore, he would leave on the first day. Note that, in this case, the priest left because he did NOT see the spot on anyone’s forehead.

Scenario 2 (2 priests have sinned): In this case, one of the priests (Priest A) who has sinned would see only 1 priest with the spot (Priest B). However, unlike the first case, this priest (Priest B) does not leave on the first day because he can see 1 priest with the spot (Priest A). Now, priest A knows that since priest B did not leave on the first day, he must be seeing another priest with the spot. The same is true for priest B. Therefore, they both realize on the second day that both of them have the black spot and they both leave on the second day.

Scenario 3 (3 priests have sinned): Following the same approach from scenario 2, one of the priests (Priest A) who has sinned would now see two priests with the spot (Priests B & C). However, unlike the second case, priests B and C do not leave on the second day because they can each see 2 priests with the spot (Priest A). Now, priest A knows that since priests B & C did not leave on the second day, they must be seeing two more priests with the spot. The same is true for priests B & C. Therefore, all three realize on the third day that all three of them have the black spot and they all leave on the third day.

In a similar manner, by extension, we can determine that if n priests have sinned, all n of them would leave on the nth day together.

Reference: http://txti.es/hosted-sites-monk

This puzzle is contributed by Feroz Baig.