2. At a round table, 100 people are sitting. Each of them is either a knight, a liar, or a fool. A knight always tells the truth, a liar always lies. A fool tells the truth if a liar is sitting to their left; lies if a knight is sitting to their left; and can say anything if a fool is sitting to their left. Each of them said: "A liar is sitting to my right." How many liars are at the table? List all possible answers and prove that there are no others. (V. Migrin)
Answer: 0 or 50. Solution: Evaluation. Suppose there is a liar among those sitting. Then to the right of him is a knight or a fool. Any of them in this situation will tell the truth, meaning to the right of him is again a liar, and so on, which means there are exactly 50 liars. Examples. For 0 liars: at the table are only fools, and each can lie that to the right of him sits a liar. For 50 liars: at the table, on even places sit liars, and on odd places sit fools or knights (in any arrangement).