6.5. Around a round table, 12 people sat down, some of them are knights, and the rest are liars (knights always tell the truth, while liars always lie). Then each of them said: "Among my neighbors, there is a liar." What is the maximum number of people sitting at the table who can say: "Among my neighbors, there is a knight"?
# Answer: 8.

Solution. Note that two liars cannot sit next to each other (otherwise, each of them would be telling the truth). Therefore, no liar can say the second phrase.

On the other hand, 3 knights also cannot sit next to each other (otherwise, the middle one would have lied by saying that he has a neighbor who is a liar). Therefore, among any three sitting next to each other, there is a liar, meaning that no more than two of them can say the second phrase. By dividing the people sitting into four groups of three sitting next to each other, we get that no more than \(4 \cdot 2 = 8\) people could have said the second phrase.

Exactly 8 (knights) of those sitting at the table could have said the required phrase if the people at the table are seated in the following order: LRR LRR LRR LRR.

Remark. For a knight to say the second phrase, he must sit next to another knight. From this, it is not difficult to see that the optimal example is unique (up to rotation of the table).

Comment. Noted that 3 knights cannot sit next to each other - 1 point.

Noted that 2 liars cannot sit next to each other - 1 point.

Shown that the number of people who said the required phrase is no more than \(8 - 3\) points.

Provided an example showing that exactly 8 of those sitting at the table could have said the required phrase - 2 points.