In a party, each person knew exactly $ 22$ other persons. For each two persons $ X$ and $ Y$, if $ X$ and $ Y$ knew each other, there is no other person who knew both of them, and if $ X$ and $ Y$ did not know each other, there are exactly $ 6$ persons who knew both of them. Assume that $ X$ knew $ Y$ iff $ Y$ knew $ X$. How many people did attend the party?
[i]Yudi Satria, Jakarta[/i]
1. Let \( n \) be the total number of people at the party. Each person knows exactly 22 other people. Consider an arbitrary person \( A \). Let \( F(A) \) be the set of friends of \( A \) and \( S(A) \) be the set of strangers of \( A \). Then, we have:
   \[
   |F(A)| = 22 \quad \text{and} \quad |S(A)| = n - 23
   \]
   since \( A \) has 22 friends and \( n - 23 \) strangers (excluding \( A \) itself).

2. Consider a person \( S \in S(A) \). Since \( S \) does not know \( A \), \( S \) and \( A \) must have exactly 6 friends in common. Thus, each person \( S \in S(A) \) knows exactly 6 people in \( F(A) \).

3. The total number of relationships between \( F(A) \) and \( S(A) \) is given by:
   \[
   6(n - 23)
   \]
   because each of the \( n - 23 \) strangers knows exactly 6 friends of \( A \).

4. On the other hand, every two people in \( F(A) \) do not know each other (since they all know \( A \)), therefore, each person in \( F(A) \) knows exactly 21 people in \( S(A) \). This means that the number of relationships between \( F(A) \) and \( S(A) \) is:
   \[
   21 \cdot 22
   \]
   because there are 22 friends, and each friend knows 21 strangers.

5. Equating the two expressions for the number of relationships between \( F(A) \) and \( S(A) \), we get:
   \[
   6(n - 23) = 21 \cdot 22
   \]

6. Solving for \( n \):
   \[
   6(n - 23) = 462
   \]
   \[
   n - 23 = 77
   \]
   \[
   n = 100
   \]

Thus, the number of people who attended the party is \( \boxed{100} \).