【Analysis】First, according to the problem requirements, rotations that are the same are considered the same method, so we only need to consider the case where one person is in the first position, and then sort according to the conditions given in the problem.

【Solution】Solution: For convenience, let's use the numbers $1, 2, 3, 4, 5, 6$ to represent the 6 people. Then, 1's friends are 2 and 6, meaning that the ones adjacent to 1 can only be $3, 4, 5$.

Since rotations that are the same are considered the same method, we can only consider the sorting methods starting with 1. Since it is a circle, the second and the last positions can only be chosen from 3, 4, 5.

The possible seating arrangements with 1 as the base are:
（1）If the second position is 3, then the third position can be 5 or 6,
(1) If the third position is 5, then the fourth position can only be 2, leaving 4 and 6. Since the last position can only be $3, 4, 5$, the fifth position is 6, and the sixth position is 4, i.e., $1,3,5,2,6,4$;
(2) If the third position is 6, leaving $2, 4, 5$, if the fourth position is 2, then 4 and 5 are left, which are adjacent, which does not meet the requirements, and 6 and 5 are adjacent, so the fourth position must be 4, then the fifth position is 2, and the sixth position is 5, i.e., $1,3,5,2,6,4$;
（2）If the second position is 4, we can similarly deduce two arrangements, i.e., 1, 4, 6, 2, 5, 3 and 1, 4, 2, 6, 3, 5,
（3）If the second position is 5, we can similarly deduce two arrangements, i.e., 1, 5, 2, 4, 6, 3, and 1, 5, 3, 6, 2, 4.

In total, there are 6 arrangements.
Therefore, the answer is: 6.
8. (10 points) King Arthur summoned 6 knights to the palace. Each knight has exactly 2 friends. They sat around a round table (the names and seats of the knights are shown in the figure), and it turned out that in this arrangement, any two adjacent knights are friends. King Arthur wants to rearrange the seats, so how many different ways can King Arthur arrange the seats such that no knight sits next to his friends (rotations that result in the same arrangement are considered the same method)?

