so, there are the base rules for 3x3 magic squares, rows, columns, and diagonals add to the same number known as the "magic number", all numbers are distinct, and only natural numbers are used.
But I found a few mathematical rules that I haven't seen anywhere.
- sum of all numbers is equal to three times the magic number (trivial).
- sum of all sums (rows, columns, diagonals) is eight times the magic number (trivial).
- sum of edges is equal to the sum of the magic number and the center ($b+d+f+h=n+e$).
- pairs of corners are even ($0$ corners are even, $2$ corners are even, or $4$ corners are even).
- sum of corners is twice the magic number minus twice the center ($a+c+g+i=2n-2e$) (trivial?).
- the most interesting is: the magic number is 3 times the center ($n=3e$).
- sum of all edges is equal to the sum of all the corners ($a+c+g+i=b+d+f+h$).
I was wondering if there were more such rules and where I might find them?
I think one of the mos comprehensive collections of such formulas in the internet can be found here.
Gerry Myerson already pointed out in a comments that you can derive as many statements as you want from your definition of a magic square. I think such a collection is not really interesting. From your definition
$$\begin{array}[]{} \tag 1 x_{1,1}+&x_{1,2}+&x_{1,3}&&&&&&&=n \\ &&&x_{2,1}+&x_{2,2}+&x_{2,3}&&&&=n \\ &&&&&&x_{3,1}+&x_{3,2}+&x_{3,3}&=n \\ x_{1,1}+&&&x_{2,1}+&&&x_{3,1}&&&=n \\ &x_{1,2}+&&&x_{2,2}+&&&x_{3,2}&&=n \\ &&x_{1,3}+&&&x_{2,3}+&&&x_{3,3}&=n \\ x_{1,1}+&&&&x_{2,2}+&&&&x_{3,3}&=n \\ &&x_{1,3}+&&x_{2,2}+&&x_{3,1}&&&=n \\ \end{array} $$
and additionally $$x_{i,j} \in \mathbb{Z^+} \tag 2$$ $$x_{i,j} = x_{s,t} \implies (i,j)= (s,t) \tag 3$$
The rank of the system of equations $(1)$ is only 7. You can combine multiples of these equation by addition to get a new equation. wich is a new "rule" in your terminology or a new "fact" in Gerry's.
A related, but maybe more interesting question is:
When does a set of 9 numbers can be used to build a magic square?
We know (e.g. from Wikipedia) that the numbers $1,2,3,4,5,6,7,8,9$ can be used to build a magic square. But can the numbers $1,2,3,5,7,11,13,17,19$ be used to build a magic square? No, because there sum is 78 and this is not a multiple of 9. As soon as you have constructed a magic square then you know they will satisfy all the equations that the numbers of a magic square satisfy.
Whenever you have a set of 9 numbers you can try all 9! (= 362880) permutation of these numbers. But is there a feasible way to check this?(Yes, there is)