9.6. In the room, there are 30 people; each is either a knight (who always tells the truth) or a liar (who never tells the truth). Each of them made the statement: "In this room, there are as many liars as there are people with the same eye color as me." How many liars are in the room? Provide all possible answers and prove that there are no others.
Solution: There is at least one liar in the room, since for any knight there is at least one person with the same eye color - the knight himself. Let there be exactly $x$ liars in the room ($x$ is a natural number). Note that if two people have the same eye color, then they either both tell the truth or both lie. Therefore, it is impossible for a liar and a knight to have the same eye color. We will divide all people into groups. We will put all the liars in one group, and divide the knights into groups by eye color. Then in each of the groups of knights there are exactly $x$ people, as follows from the statement of any of the knights in this group. If there are $n$ groups of knights, we get the equation $(n+1) \cdot x=30$. Then the number of liars $x$ is a divisor of 30. Moreover, it is impossible for all liars to have the same eye color - otherwise they would be telling the truth. Therefore, $x>1$. There are no other restrictions: if $x$ is a divisor of 30, different from 1, then all people can be divided into $\frac{30}{x}$ groups of $x$ people each (it is not prohibited for the situation $x=30$, in which case there will be one group). Let the eye color of people from different groups be different, and in each group, except one, the eye color of the participants is the same. And let in this special group all are liars, and there are no other liars. The condition is thus satisfied.

Answer: $2,3,5,6,10,15$ or 30.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| The case when there are no liars at all is not considered | points not reduced |
| It is justified that the number of liars is not equal to 1 | add 1 point if the solution is not complete |
| It is proven that the number of liars is a divisor of 30 and examples are constructed for all possible cases | 6 points |
| It is proven that the number of liars is a divisor of 30, but not all possible examples are constructed | 3 points |
| All possible values of the number of liars are correctly found, but it is not proven that there are no others | 2 points |
| Some possible values (not all) of the number of liars are correctly found | 1 point |
| Answer without justification and/or incorrect answer | 0 points |