2. On an island, there live liars and knights. Knights always tell the truth, and liars always lie. Each inhabitant of the island knows whether each of the others is a knight or a liar. One day, 19 islanders met. Three of them said: "Exactly three of us are liars," then six of the others said: "Exactly six of us are liars," finally, nine of the remaining said: "Exactly nine of us are liars." How many liars were among those who met? Provide all possible options and prove that there are no others.
Answer: 9, 18 or 19.

Solution. Let the third statement be true. Then all who said it are knights, and there must be exactly nine liars. Therefore, all who said the other statements are liars. There are nine of them, so the one who remained silent is a knight, and such a situation is possible. Let the third statement be false. Then those who said it are liars, and there are no fewer than nine liars. Therefore, in the first two groups, there are also liars. Consequently, there are no fewer than 18 liars. The last one can be anyone.