On an island, there live only knights, who always tell the truth, and liars, who always lie. One day, they all sat in a circle, and each one said: "Among my two neighbors, there is a liar!" Then they sat in a circle in a different order, and each one said: "Among my two neighbors, there is no knight!" Could there have been 2017 people on the island? (L. Samoilov)
Problem 992
Official solution
Answer. It could not. Solution. In the first round, both neighbors of each liar were knights. By matching each liar with their right neighbor in this round, we can be convinced that there are no fewer knights than liars on the island. In the second round, both neighbors of each knight were liars. By matching each knight with their right neighbor in this round, we can be convinced that there are no more knights than liars on the island. It turns out that there are an equal number of knights and liars on the island. But then the number of inhabitants on the island is even, while the number 2017 is odd.