9.4. In the castle, there are 9 identical square rooms, forming a $3 \times 3$ square. Nine people, consisting of liars and knights (liars always lie, knights always tell the truth), each occupied one of these rooms. Each of these 9 people said: "At least one of the neighboring rooms to mine is occupied by a liar." What is the maximum number of knights that could be among these 9 people? Rooms are considered neighboring if they share a common wall.
Answer: 6 knights.

Solution: Note that each knight must have at least one neighbor who is a liar. We will show that there must be at least 3 liars (thus showing that there are no more than 6 knights). Suppose there are no more than 2 liars, then there will be a "vertical row" of rooms where only knights live. But then each of these knights must have a neighbor who is a liar (and these neighbors are different). Therefore, there must be at least 3 liars.

The diagram below shows how 6 knights and 3 liars could be accommodated.

| $\mathrm{P}$ | L | $\mathrm{P}$ |
| :--- | :--- | :--- |
| $\mathrm{P}$ | $\mathrm{P}$ | L |
| L | $\mathrm{P}$ | $\mathrm{P}$ |

Comment: A correct answer without justification - 0 points.

Example of accommodating 3 liars and 6 knights - 2 points.

Proven that there are no more than 6 knights - 5 points.

Note 1: In the example, the liars should not be neighbors.

Note 2: There are other examples.