More conditions for 3x3 magic squares

Question

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?

"all numbers are distinct, and only natural numbers are used" I think usually one has to use the numbers from 1 to N^2, if N^2 is the number of cells of the magic square — miracle173, Oct 28 '19 at 06:33
@miracle173 that is for a "pure" magic square, I'm talking more generally. — spydragon, Oct 28 '19 at 06:55
"sum of all numbers is equal to three times the magic number (trivial)." I think the sum of all numbers is the number of rows times the magic number. So can you prove your claim? — miracle173, Oct 28 '19 at 07:06
I seems that you are only talking about 3x3 magic squares. Am I right? — miracle173, Oct 28 '19 at 07:11
@miracle173 yes and I have fixed my question to represent that. — spydragon, Oct 28 '19 at 07:12
You are using rules to mean definition, and then later you are using rules to mean theorems (or, facts). You find facts by drawing consequences from the definitions. You can make as many as you like. — Gerry Myerson, Oct 28 '19 at 09:11
@miracle173 yes, I know. you could say that all of these rules come from the eight sums( that's where I got them) — spydragon, Oct 29 '19 at 18:00
@spydragon yes, all of your rules come from the eight sums. But that was not that what I wortee in my comment. I wrote that your rule 7 comes from your rule 3,4 and 6 — miracle173, Oct 30 '19 at 11:52
@miracle173 sorry, I meant to say, yes I know that $7$ comes from $3$, $4$, $6$. but you could also say that all rules are generated by the eight sums, so all rules come from all the other rules. — spydragon, Oct 30 '19 at 17:59

score 1 · Answer 1 · answered Oct 29 '19 at 11:58

1

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)

answered Oct 29 '19 at 11:58

miracle173

11,049

Your link "here" in the first line of the answer, seems to link back to the current question. Was that your intention? – paw88789 Oct 29 '19 at 12:18
@paw88789 Yes, because I think there is not such a collection and such a collection does not make much sense – miracle173 Oct 29 '19 at 12:49
I think that such collections do make sense. (it's very interesting that $n=3e$ and I have never seen that before), – spydragon Oct 29 '19 at 17:58
@spydragon but there are infinitely many that are easy to create. I think if you describe when a set of 9 numbers define a magic square (as I mentioned in my post), then you have said all about magic square numbers what is interesting. One of these necessary "interesting" properties is that the median of these number is the sum of these numbers divided by 9. This median is the number that is assigned to the cell in the center of the square. – miracle173 Oct 30 '19 at 11:49
@spydragon question: when is a property interesting? – miracle173 Oct 30 '19 at 11:50
@miracle173 soon I might post a new question that will include an extremely easy method for finding 3x3 magic squares that I discovered this morning, only by looking at my set of rules. for now I will give an example, give me three positive integers where the largest number is greater than the sum of twice the smallest number and the middle number, then I will give you a magic square. – spydragon Oct 30 '19 at 18:02

More conditions for 3x3 magic squares

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