$[$ Mathematical logic (miscellaneous).]

Authors: Folklore, Folklore

In a school of witchcraft, there are 13 students. Before the clairvoyance exam, the teacher seated them at a round table and asked them to guess who would receive the clairvoyant diploma. They modestly kept quiet about themselves and their two neighbors, but wrote about everyone else: "None of these ten will get it!" Of course, all those who passed the exam guessed correctly, while all the others were wrong. How many wizards received the diploma?

#
Assume that no one received a diploma. Then the statement of each student is true. In this case, everyone should have received a diploma - a contradiction. Therefore, at least one of the students received a diploma of a seer. He told the truth, so no one except his neighbors received a diploma. If both neighbors also did not receive diplomas, then the statement "None of these ten will get it!" for each of them is true, but they should have been wrong! If both neighbors passed the exam, then both of them were wrong in their statements. Therefore, only one of the neighbors could have passed the exam successfully. Indeed, in this case, his statement is true, and the statement of the second neighbor is false.

## Answer

Two wizards.