4.1. In a row, 64 people are standing - each one is either a knight, who always tells the truth, or a liar, who always lies. One of the standing knights said that he stands next to a knight and a liar, and all the other 63 people repeated his phrase. Indicate how many of them were knights.
Answer: 42

Solution. The people at the ends must be liars, as they do not have a second neighbor, and by saying this phrase, those standing there lied. All others can be liars, in which case there are no knights at all, but this option contradicts the condition that at least one knight is present. If there is a knight, then next to him stand a knight and a liar: KKL or LKK. A liar cannot tell the truth, so if there is a knight on one side, there must be a knight on the other side (if the liar is not at the end). We get ...KKLKK... or ...KLKK... A knight cannot stand between two liars, so knights always stand in pairs. We have ...KKLKK... in both cases. Thus, the following arrangement is obtained: LKKLKKLKK...LKKL - if we do not count the last liar, the remaining 63 people can be divided into 21 triplets LKK, each containing 42 knights.