I am currently studying magic squares and ran into a bit of trouble. The concept I am learning about is a regular square. Below are the conditions of a regular square.
We can say that an $n$-by-$n$ square is regular provided that:
Each of the integers from $0$ to $n^2 − 1$ appears in exactly one cell, and each cell contains only one integer (so that the square is filled), and
If we express the entries in base-$n$ form, each base-$n$ digit occurs exactly once in the units’ position, and exactly once in the $n$’s position.
Example of a Regular Square:
$\begin{pmatrix} 00 & 23 & 32 & 11 \\ 21 & 02 & 13 & 30 \\ 12 & 31 & 20 & 03 \\ 33 & 10 & 01 & 22 \end{pmatrix} = \begin{pmatrix} 0 & 11 & 14 & 5 \\ 9 & 2 & 7 & 12 \\ 6 & 13 & 8 & 3 \\ 15 & 4 & 1 & 10 \end{pmatrix}$
Is every regular square necessarily magic? If so why is it true, or why isn't it true? Can someone please provide me an insightful proof as to why this is or isn't true?