3. The village of knights and liars on the map has the shape of a $9 \times 9$ grid, with one person living in each cell - a knight or a liar. Knights always tell the truth, while liars always lie. Cells are considered neighbors if they share a side or a corner. Each resident said: “Among my neighbors, there is an odd number of liars.” Is the number of liars in the village even or odd?
Solution. Divide the board into nine $3 \times 3$ squares. We will prove that in each such square there is an odd number of liars. Consider a resident from the central cell. If he is a knight, then among his neighbors there is an odd number of liars, and thus in the entire square there is an odd number of liars. If he is a liar, then among his neighbors there is an even number of liars, plus himself, and in the end, in the square there is an odd number of liars.

Since the sum of nine odd numbers is odd, the total number of liars is also odd.

Evaluation criteria. The idea of dividing into $3 \times 3$ squares - one point.

It is stated that in a $3 \times 3$ square there is an odd number of liars, but it is not proven - three points. Note. It is not necessary to provide an example that this is possible, although it is not difficult. For example, nine liars in the central cells of the $3 \times 3$ squares, and the rest are knights.