# 8. Variant 1.

On the Island of Misfortune, there live truth-tellers, who always tell the truth, and liars, who always lie. One day, 2023 natives, among whom $N$ are liars, stood in a circle, and each said: "Both of my neighbors are liars." How many different values can $N$ take?
Answer: 337.

Solution: Both neighbors of a knight must be liars, and the neighbors of a liar are either two knights or a knight and a liar. Therefore, three liars cannot stand in a row (since in this case, the middle liar would tell the truth). We can divide the entire circle into groups of liars/knights standing in a row. These groups alternate. Each "group of knights" consists of one person, and each "group of liars" consists of one or two people. Let \( x \) be the number of "single" liars, and \( y \) be the number of "pairs of liars." Then the number of knights will be \( x + y \), and \( x + y + 2y + x = 2023, 3y + 2x = 2023, N = x + y \). It is easy to notice that \( y \) must be odd, \( x \geq 2 \) (since \( 2x \) must give the same remainder when divided by 3 as 2023), accordingly, \( 3y \leq 2019, 1 \leq y \leq 673 \). In this range, there are \( 674 / 2 = 337 \) odd numbers, i.e., \( y \) can take 337 different values. For each of these values, the number of knights \( x + y \) is uniquely determined, and thus the number of liars \( x + 2y \) \( (2y + x = 2023 - (x + y)) \).