4. On an island, there live knights who always tell the truth, and liars who always lie. One day, 9 natives, among whom were both knights and liars, stood in a circle, and each said: "The two people standing opposite me are liars." Opposite a native stand the fourth native from him in a clockwise direction and the fourth from him in a counterclockwise direction. How many of the 9 natives are knights? Provide all answers and prove that there are no others.

Answer: 3 or 4 knights.
Solution. Let's divide all the natives into three groups (see the figure, natives from the same group have the same numbers).

In each such triplet, there cannot be three liars (since then one of the liars would be telling the truth) and there cannot be three knights (since then one of the knights would be lying). Therefore, there are no fewer than three and no more than six knights. Consider any knight. Rotate our circle so that he ends up in the first triplet in a separate vertex. Then this triplet

![](https://cdn.mathpix.com/cropped/2024_05_06_33d5614ef1cdadfd041cg-2.jpg?height=482&width=434&top_left_y=1158&top_left_x=1416)

$\odot$ is filled in the only possible way (see the figure). Therefore, there are no more than five knights in total. For there to be exactly five knights, the other two triplets are also filled in the only possible way (see the figure). But then we get a contradiction

![](https://cdn.mathpix.com/cropped/2024_05_06_33d5614ef1cdadfd041cg-2.jpg?height=262&width=240&top_left_y=1696&top_left_x=974)

(3)

![](https://cdn.mathpix.com/cropped/2024_05_06_33d5614ef1cdadfd041cg-2.jpg?height=105&width=217&top_left_y=2049&top_left_x=1068)

![](https://cdn.mathpix.com/cropped/2024_05_06_33d5614ef1cdadfd041cg-2.jpg?height=102&width=312&top_left_y=1768&top_left_x=1470)

![](https://cdn.mathpix.com/cropped/2024_05_06_33d5614ef1cdadfd041cg-2.jpg?height=83&width=88&top_left_y=1877&top_left_x=1304)

2

I

(I)

$\odot$

$\odot$ for the highlighted triplet. Therefore, there are no more than four knights in total. Examples for 3 and 4 knights (see the figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_33d5614ef1cdadfd041cg-3.jpg?height=214&width=162&top_left_y=184&top_left_x=248)

(P)

![](https://cdn.mathpix.com/cropped/2024_05_06_33d5614ef1cdadfd041cg-3.jpg?height=86&width=51&top_left_y=437&top_left_x=340)

(I)
(I)

(I)

(I)

(

## Criteria.

Incorrect solution - 0 points.

An example for 3 knights is provided - 1 point.

An example for 4 knights is provided - 1 point.

Proved that there cannot be fewer than 3 knights - 1 point.

Proved that there cannot be more than 6 knights - 1 point.

Proved that there cannot be 6 knights - 1 point.

Proved that there cannot be 5 knights - 1 point.

Correct solution - 7 points.