8.4. In the castle, there are 25 identical square rooms arranged in a $5 \times 5$ square. In these rooms, 25 people—liars and knights (liars always lie, knights always tell the truth)—have settled, one person per room. Each of these 25 people said: "At least one of the rooms adjacent to mine is occupied by a liar." What is the maximum number of liars that could be among these 25 people? Rooms are considered adjacent if they share a wall.
Answer: 13 liars.

Solution. Note that liars cannot live in adjacent rooms (otherwise, they would be telling the truth). Let's select one corner room, and divide the remaining rooms into 12 pairs of adjacent rooms. Then, in each pair of rooms, there can be no more than one liar. Therefore, the total number of liars cannot exceed $12+1=13$. Consider a chessboard coloring of the rooms in black and white (corner rooms are black). If we place 13 liars in the "black" rooms and 12 knights in the "white" rooms, the condition of the problem will be satisfied (all liars will lie, and all knights will tell the truth).

Comment. A correct answer without justification - 0 points.

An example of placing 13 liars and 12 knights - 2 points.

Only the statement that liars cannot live in adjacent rooms - 1 point.

Proven that there are no more than 13 liars - 5 points.