Question 32
In the UC Berkeley bureaucratic hierarchy, certain administrators report to certain other administrators. It so happens that if $A$ reports to $B$ and $B$ reports to $C$, then $C$ reports to $A$. Also, administrators do not report to themselves. Prove that all the administrators can be divided into three disjoint groups $X, Y, Z$ so that, if $A$ reports to $B$, then either $A$ is in $X$ and $B$ is in $Y$, or $A$ is in $Y$ and $B$ is in $Z$, or $A$ is in $Z$ and $B$ is in $X$.