Task 2. On an island, there live knights who always tell the truth, and liars who always lie. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: "My left neighbor is a liar" or "My right neighbor is a liar." What is the smallest number of knights that can be at the table?
# Answer: 10.

Solution. Next to each liar, there must be at least one knight (otherwise, if liars sit on both sides of a liar, the statement made by the liar would definitely be true). Therefore, among any three consecutive residents, there is at least one knight. If we divide all the people sitting at the table into groups of three consecutive residents, we get that there is at least one knight in each group, so there are at least 10 knights in total.

Let's provide an example where there are exactly 10 knights. Suppose the residents sit as follows: knight, two liars, knight, two liars, and so on. Each knight says any of the given phrases—it will be true in any case. A liar says, "My left neighbor is a liar" if the person on the left is a knight, and vice versa, he says, "My right neighbor is a liar" if the person on the right is a knight. It is clear that all the conditions of the problem are satisfied.

## Criteria

## The following criteria are cumulative:

5 p. It is proven that there are at least 10 knights.

2 6. It is proven that there can be exactly 10 knights.