7.4. On an island, there live knights and liars. Knights always tell the truth, liars always lie (the islanders know who is who). A tourist who arrived on the island met a group of 10 islanders and started asking each one in turn: "Who is more in your group: knights, liars, or are they equal in number?" Five of them said the same thing: "There are more liars." What did the other five people say?
Answer: "Equally". Solution. Suppose first that there are more than five liars in the company. Then the first five told the truth, and thus they are knights. So, there are at least five knights, which means our assumption is incorrect. Now suppose that there are more than five knights in the company. Then among the first five, there must be a knight, and he should have told the truth, but in fact, he said there were more liars, which again leads to a contradiction. Therefore, there are 5 liars and 5 knights in the company. Since the first five lied, they are liars, and the rest are knights, and they said: "Equally".

## 8th grade