Question 11
English source version: 30 persons, each of whom is either a Knight or a Liar, sit around the table (the places around the table are numbered $1,2,\ldots,30$ in consecutive order). Knights always tell the truth, while Liars always lie. Each person has exactly one friend among the others. The friend of a Knight is a Liar, and the friend of a Liar is a Knight. Each person was asked “Is it true that your friend is a neighbor of you?” Fifteen persons sitting in odd-numbered places answered “Yes”. Find the number of persons in even-numbered places who also answered “Yes”. (S. Agakhanov)
Russian source version, translated: 30 people, Knights and Liars, sit around a round table (Knights always tell the truth and Liars always lie). Each has exactly one friend at the same table; a Knight's friend is a Liar and vice versa (friendship is always mutual). To the question “Is your friend sitting next to you?”, people in alternate seats answered “Yes”. How many of the others could also have answered “Yes”? List all possibilities and prove there are no others. (S. Agakhanov)
Russian source version, translated: 30 people, Knights and Liars, sit around a round table (Knights always tell the truth and Liars always lie). Each has exactly one friend at the same table; a Knight's friend is a Liar and vice versa (friendship is always mutual). To the question “Is your friend sitting next to you?”, people in alternate seats answered “Yes”. How many of the others could also have answered “Yes”? List all possibilities and prove there are no others. (S. Agakhanov)