Problem 6.6. On an island, there live knights who always tell the truth, and liars who always lie. One day, 65 inhabitants of the island gathered for a meeting. Each of them, in turn, made the statement: "Among the previously made statements, there are exactly 20 fewer true statements than false ones." How many knights were at this meeting?
Answer: 23.

Solution. The first 20 people are liars, because before them, there were no more than 19 statements made, which means there cannot be exactly 20 fewer true statements than false ones.

Next, the 21st person tells the truth (since before him, there were exactly 20 false statements), so he is a knight. Then the 22nd person lies (before him, there were 20 false and 1 true statement), so he is a liar. Next, the 23rd person tells the truth (the number of false statements is again 20 more than the number of true ones), so the 23rd is a knight, and so on.

Among the people starting from the 21st, those with odd numbers are knights, and those with even numbers are liars. Among the numbers from 21 to 65 inclusive, there are exactly 23 odd numbers, so there were exactly 23 knights at the meeting.