6. (3 points) On a $2 \times 4$ 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: 6.

Solution: On the corner cells, liars are obviously standing: they simply do not have three neighbors.

A situation where all people on the board are liars is impossible, as it would mean that four of these liars are telling the truth.

Therefore, there is at least one knight. Then all his neighbors are liars, and we get the following situation (accurate up to symmetry):

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

The cell marked with a question mark can only be a knight, since all his neighbors are liars. Don't forget to check that the left and upper neighbors of this knight have only two liar neighbors, meaning they are indeed lying.

Thus, we end up with 6 liars.