Within a city, consider 3 districts $A$, $B$, and $C$, each with $n$ houses. Each house is contained in exactly 1 district (i.e. there is no overlap between districts). Construct a network of roads as...

Within a city, consider 3 districts $A$, $B$, and $C$, each with $n$ houses. Each house is contained in exactly 1 district (i.e. there is no overlap between districts). Construct a network of roads as follows:

 \begin{itemize}

     \item There is a road between every house in $A$ and every house in $B$.

     \item There is a road between every house in $B$ and every house in $C$.

     \item There is a road between every house in $C$ and every house in $A$.

     \item There are no other roads except those specified above.

 \end{itemize}

 \begin{enumerate}

     \item Express the number of roads in the city, in terms of $n$. Justify your answer.

     \item Prove that you can walk to any house from any other house through some sequence of roads.

     \item Express the length of the longest cycle of roads in the city in terms of $n$. Justify your answer.

 \end{enumerate}