6.4. At a round table, 120 people are sitting - knights, who always tell the truth, and liars, who always lie. Each of them made exactly one of the following two statements: 1) "My right neighbor is a knight"; 2) "The person who is one to the right of my right neighbor is a knight." How many liars could there be at the table? (Don't forget to explain your answer.)
Answer: $0,60,120$. Solution. The cases where all sitting at the table are liars, or all sitting at the table are knights, are obviously valid. Suppose there are both knights and liars sitting at the table. Then there is a liar (L) whose right neighbor is a knight (R). L could not have made statement 1). Therefore, he made statement 2), and to the right of R sits a liar (L2). This means R also could not have made statement 1, i.e., he made statement 2), so to the right of L2 is again a knight. Continuing this reasoning, we see that knights and liars alternate around the table.

Comment. Any answer without justification is 0 points. One answer with justification is 0 or 1 point for the answer $120-1$, both of these answers - 2 points. The answer 60 with justification - 5 points.