2. On an island, there live liars and knights. Knights always tell the truth, and liars always lie. Each islander knows whether every other islander is a knight or a liar. One day, 28 islanders met. Two of them said: "Exactly two of us are liars," then four of the others said: "Exactly four of us are liars," then eight of the remaining said: "Exactly eight of us are liars," finally, all the remaining 14 said: "Exactly 14 of us are liars." How many liars were among those who met? Provide all possible options and prove that there are no others.
Answer: 14 or 28.

Solution. Let the last statement be true. Then all who said it are knights, and there must be exactly 14 liars. Therefore, all who said the other statements are liars, and such a situation is possible. Let the last statement be false. Then those who said it are liars and there are no fewer than 14 liars. Therefore, the first three statements are false and, thus, there are liars in the first three groups as well. Therefore, all those who met are liars.