Problem 1438

1. Define the sets and relationships:
Let the boys be $B_1, B_2, B_3$ and the girls be $G_1, G_2, G_3$. According to the problem, each boy knows exactly two girls and each girl knows exactly two boys. We can represent these relationships as follows:
- $B_1$ knows $G_1$ and $G_2$
- $B_2$ knows $G_1$ and $G_3$
- $B_3$ knows $G_2$ and $G_3$

2. Construct a bipartite graph:
We can represent this problem using a bipartite graph where one set of vertices represents the boys and the other set represents the girls. An edge between a boy and a girl indicates that they know each other. The graph can be visualized as follows:
- $B_1$ is connected to $G_1$ and $G_2$
- $B_2$ is connected to $G_1$ and $G_3$
- $B_3$ is connected to $G_2$ and $G_3$

3. Check for a perfect matching:
A perfect matching in a bipartite graph is a matching that covers every vertex exactly once. In this case, we need to find a perfect matching where each boy is paired with a girl he knows, and each girl is paired with a boy she knows.

4. Find the perfect matching:
We can use the given relationships to find a perfect matching:
- Pair $B_1$ with $G_1$
- Pair $B_2$ with $G_3$
- Pair $B_3$ with $G_2$

Let's verify that these pairs satisfy the conditions:
- $B_1$ knows $G_1$
- $B_2$ knows $G_3$
- $B_3$ knows $G_2$

Each boy is paired with a girl he knows, and each girl is paired with a boy she knows. Therefore, this is a valid perfect matching.

$\blacksquare$