4. At a round table, 30 people are sitting, each of whom is either a knight, who always tells the truth, or a liar, who always lies. Each person was asked: "How many knights are among your neighbors?" (Two people are called neighbors of each other if there is no one else sitting between them.) 10 people answered "one," 10 - "two," and 10 - "none." What is the maximum number of knights that can be at the table?
Answer: 22 knights.

Solution. From the condition of the problem, it follows that the number of knights who have two knight-neighbors does not exceed 10; the same can be said about the number of knights who have one knight-neighbor. Therefore, the number of pairs of knight-neighbors does not exceed $(2 \cdot 10 + 1 \cdot 10) : 2 = 15$. Since there are 30 pairs of neighbors at the table, at least $30 - 15 = 15$ pairs include people who are not knights; the number of such people is no less than $\left[\frac{15}{2}\right] + 1 = 8$.

Let's provide an example showing that there can be exactly $30 - 8 = 22$ knights at the table. Suppose those sitting are numbered from 1 to 30 clockwise, and the knights are all those whose numbers are different from $1, 3, 5, 7, 11, 15, 19,$ and 23. Then all the necessary conditions will be met if No. 1 answers "two," and Nos. $3, 5, 7, 11, 15, 19,$ and 23 answer "none."

Criteria. Only the answer - 0 points. Proven that the number of knights is no more than 22 (estimation) - 3 points. Provided an example with 22 knights - 3 points. Complete solution - 7 points.