2. Six people are standing in a circle, each of whom is either a knight - who always tells the truth, or a liar - who always lies. Each of them said one of two phrases: "There is a liar next to me" or "There is a liar opposite me." What is the minimum number of liars among them? Provide an example and prove that there cannot be fewer liars.


Answer: 2.

Solution. Let's number all the people standing clockwise (this way, people with numbers 1 and 4, 2 and 5, 3 and 6 will stand opposite each other).

Zero liars is obviously impossible (then there would be only knights and no one could say any of the phrases).

If there is one liar, let's say his number is 1, then the knight with number 3 has two knight neighbors and a knight standing opposite him. This means he cannot say either of these two phrases.

Example with two liars. Liars have numbers 1 and 2 and say "There is a liar opposite me," which is also said by knights 4 and 5. The remaining knights (3 and 6) say "There is a liar next to me."