Definition:Magic Cube
Definition
A 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 space diagonals
are the same.
It is not guaranteed that the entries along each of the main diagonals of each plane also sum to the same constant.
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.
Order
An $n \times n \times n$ magic cube is called an order $n$ magic cube.
Examples
Order $1$
The Order $1$ magic cube is trivial:
- $\begin{array}{|c|}
\hline 1 \\ \hline \end{array}$
Order $3$
- $\begin{array}{|c|c|c|}
\hline 2 & 13 & 27 \\ \hline 22 & 9 & 11 \\ \hline 18 & 20 & 4 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|} \hline 16 & 21 & 5 \\ \hline 3 & 14 & 25 \\ \hline 23 & 7 & 12 \\ \hline \end{array} \qquad \begin{array}{|c|c|c|} \hline 24 & 8 & 16 \\ \hline 17 & 19 & 6 \\ \hline 1 & 15 & 26 \\ \hline \end{array}$