Problem 5.2. During a physical education class, 25 students from 5B class lined up. Each of the students is either an excellent student who always tells the truth, or a troublemaker who always lies.

Excellent student Vlad stood in the 13th place. Everyone except Vlad stated: "There are exactly 6 troublemakers between me and Vlad." How many troublemakers are there in the line?
Answer: 12.

Solution. Note that students in places $7-12$ are troublemakers, since there are fewer than 6 people between each of them and Vlad. Therefore, the student with number 6 is an excellent student. The same can be said about the student in the 5th place, then about the 4th, the 3rd, the 2nd, and the 1st.

Thus, the first six places are occupied by excellent students, the next six places are occupied by troublemakers, and Vlad is already at place 13. Similar reasoning can be applied to students in places $14-25$. Therefore, there are exactly $6+6=12$ troublemakers in the line.