8.2. Around a table sit 7 natives: knights, liars, and bores (there is at least one representative of each tribe at the table). A knight always tells the truth, a liar always lies. A bore lies if at least one knight sits next to them, and in other cases says whatever they want.

How many bores could there be at the table if everyone answered the question "Who are your neighbors?" with "Both of my neighbors are bores." Provide all possible options and explain why there are no others.
Answer: 2, 3, or 4.

## Solution:

One bore could not sit at the table alone, because there must be a knight at the table who tells the truth, which means there are at least two bores.

Since representatives of all tribes are sitting at the table, there are no more than five bores.

Let's try to seat 5 bores. The remaining two seats are occupied by a knight and a liar.

The knight and the liar cannot sit next to each other, because otherwise the knight would lie. Therefore, they must sit at least one bore apart. However, the liar cannot be surrounded by bores, because in that case he would tell the truth. Therefore, 5 bores could not sit at the table.

Next, we will provide examples of seating 2, 3, and 4 bores at the table:
![](https://cdn.mathpix.com/cropped/2024_05_06_114931989ed3501ca821g-1.jpg?height=398&width=1746&top_left_y=2000&top_left_x=154)

## Criteria:

7 points - correct answer and a fully justified solution;

4 points - only two correct answers with justification;

3 points - only one correct answer with justification; only the answer - 0 points.