6. (3 points) On a $3 \times 3$ chessboard, there are knights, who always tell the truth, and liars, who always lie. Each of them said: "Among my neighbors, there are exactly three liars." How many liars are on the board?

Neighbors are considered to be people on cells that share a common side.
Answer: 5.

Solution: The people in the corner cells are obviously liars: they simply do not have three neighbors.

If the person in the center is a knight, then all the people in the side cells are also liars. But in this case, the knight has 4 liar neighbors, not three. This leads to a contradiction.

If the person in the center is a liar, then the people in the side cells each have three liar neighbors, so they are knights. Now we check that the person in the center is indeed a liar: they do not have three liar neighbors.