7.5. At a round table, 10 people are sitting, each of whom is either a knight, who always tells the truth, or a liar, who always lies. Each of them was given a token. Then each of them passed their token to one of their two neighbors. After that, 5 people said: "I have one token," while the other 5 said: "I have no tokens." What is the maximum number of knights that could have been among these 10 people?
# Answer. 7.

Solution. Evaluation. After passing the chips, each person sitting at the table can have 0, 1, or 2 chips. The total number of chips will be 10. Note that if a person lies, they will state a number of chips that differs from the actual number by 1 or 2. Since the total number of chips according to the answers differs from the actual number by \(10-5=5\), at least 3 people must have lied. Therefore, there are no more than 7 knights at the table.

Example. Suppose the knights sitting at the table pass the chips as follows (arrows indicate the direction of the chip transfer; the number of chips after the transfer is in parentheses):

$$
\leftarrow \mathrm{P}(0)-\mathrm{P}(0) \rightarrow \mathrm{P}(1) \rightarrow \mathrm{P}(1) \rightarrow \mathrm{P}(1) \rightarrow \mathrm{P}(1) \rightarrow \mathrm{P}(1) \rightarrow \mathrm{L}(1) \rightarrow \mathrm{L}(2) \leftrightarrow \mathrm{L}(2) \leftarrow
$$

In this case, all knights tell the truth, while the liars lie, saying they have 0 chips.

Comment. A fully justified solution - 7 points. Correct evaluation - 5 points, a correct example constructed - 2 points, points are cumulative. The problem is not solved or solved incorrectly - 0 points.