\begin{definition}[Definition:Perfect Magic Cube]
A '''perfect magic cube''' is an arrangement of the first $n^3$ (strictly) positive integers into an $n \times n \times n$ cubic array such that:
:the sum of the entries in each row in each of the $3$ dimensions
:the sum of the entries along the main diagonal of each plane
:the sum of the entries along the space diagonals
are the same.
\end{definition}