Any two vertices $A,B$ of a regular $n$-gon are connected by an oriented segment (i.e. either $A\to B$ or $B\to A$). Find the maximum possible number of quadruples $(A,B,C,D)$ of vertices such that $A\to B\to C\to D\to A$.
Source: Mongolia 1999 Teachers secondary level P2
Tags: combinatorics
Any two vertices $A,B$ of a regular $n$-gon are connected by an oriented segment (i.e. either $A\to B$ or $B\to A$). Find the maximum possible number of quadruples $(A,B,C,D)$ of vertices such that $A\to B\to C\to D\to A$.