8. (50 points) Each of the 12 knights sitting around a round table has thought of a number, and all the numbers are different. Each knight claims that the number they thought of is greater than the numbers thought of by their neighbors to the right and left. What is the maximum number of these claims that can be true?
Answer: 6.

Solution. Let's renumber the knights in order with numbers from 1 to 12. In the pairs $(1,2),(3,4)$, $\ldots,(11,12)$, at least one of the knights is lying (specifically, the one who guessed the smaller number). Therefore, there can be no more than 6 true statements. Now let's provide an example where exactly 6 statements are true. Suppose the knights with odd numbers guessed the numbers from 1 to 6, and those with even numbers guessed the numbers from 7 to 12 (in any order). Then all the knights with even numbers are telling the truth.