10.4. In the castle, there are 16 identical square rooms forming a $4 \times 4$ square. Sixteen people, who are either liars or knights (liars always lie, knights always tell the truth), have settled in these rooms, one person per room. Each of these 16 people said: "At least one of the rooms adjacent to mine is occupied by a liar." What is the maximum number of knights that could be among these 16 people? Rooms are considered adjacent if they share a wall.
Answer: 12 knights.

Solution: Note that for each knight, at least one of their neighbors must be a liar. We will show that there must be no fewer than 4 liars (thus showing that there are no more than 12 knights). Suppose there are no more than 3 liars, then there will be a "vertical row" of rooms where only knights live. However, each of these knights must have a neighbor who is a liar (and these neighbors are different). Therefore, there must be at least 4 liars.

The diagram below shows how 12 knights and 4 liars could be accommodated.

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

Comment: A correct answer without justification - 0 points.

Example of accommodating 4 liars and 12 knights - 2 points.

Proved that there are no more than 12 knights - 5 points.

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

Note 2: The estimate can be obtained by noting that in three rooms - a corner room and the two adjacent to it - at least one liar must reside.