[Examples and counterexamples. Constructions] Evaluation + example $\quad]$

On an island, there are 100 knights and 100 liars. Each of them has at least one friend. One day, exactly 100 people said: "All my friends are knights," and exactly 100 people said: "All my friends are liars." What is the smallest possible number of pairs of friends, one of whom is a knight and the other a liar?
Consider a resident $A$ of the island, claiming: "All my friends are liars". If $A$ is a knight, then his friend is a liar. If $A$ is a liar, then his friend is a knight. In either case, $A$ is part of a pair of friends of "different" types. Since 100 people made such a statement, the number of such pairs cannot be less than $100: 2 = 50$.

Example: Suppose 50 knights are friends with each other, 50 liars are friends with each other, and there are 50 pairs of friends of the type "knight - liar", and no other friends for the residents in these pairs. Then each of the first two groups can say: "All my friends are knights", and each of the third group can say: "All my friends are liars".

## Answer

50 pairs.