4. At the School of Sorcery and Wizardry, there are 13 students. Before the Divination exam, the teacher seated them at a round table and asked them to guess who would receive the Seer's diploma. Everyone 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?
Solution. Suppose 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 they both 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.