How many combination has a magic square Let's assume that I have magic square 3x3 and I know middle number. How many combinations can I create if I try to solve that? Is there a way how to find them all?
 A: If you have a magic square of sum $N$:
$$\begin{array}{|c|c|c|}
\hline
a & b & c \\ \hline
d & e & f \\ \hline
g & h & i \\ \hline
\end{array}$$
then if you add or substract the same number from all the cells, you also get a magic square, particularly:
$$\begin{array}{|c|c|c|}
\hline
a-e & b-e & c-e \\ \hline
d-e & 0 & f-e \\ \hline
g-e & h-e & i-e \\ \hline
\end{array}$$
is a magic square of sum $N-3e$.
So WLOG we can assume that the central number is $0$. Then
$$\begin{array}{|c|c|c|}
\hline
-a & b & -a+c \\ \hline
c & 0 & -c \\ \hline
-a+b & -b & a \\ \hline
\end{array}$$
is a square number of sum $0$ as long as $2a = b + c$ and $b$ and $c$ are different nonzero integers of the same parity. If you choose $b > c > 0$  with you automatically remove most of the rotations and reflexions.
Particularly, you have this infinite set of solutions:
$$\begin{array}{|c|c|c|}
\hline
1-b & b & -1 \\ \hline
b-2 & 0 & 2-b \\ \hline
1 & -b & b-1 \\ \hline
\end{array}$$
For any $b > 3$. If you want to exclude $0$, just add $n$ to every cell, where $n\not\in \{b,b-1,b-2\}$. Add $n > b-2$ to every cell to only have positive values in the cells.
