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I was reading about magic squares and suppose that we speak here only of the magic squares that have in itself numbers from $1$ to $n^2$.

It is easy to see that we cannot have $2$x$2$ magic square because we cannot arrange numbers $1,2,3,4$ in such a square so that the sum of numbers in every row, column and diagonals is the same number.

But the natural questions that comes is:

Is it true (if it is proven, can someone point me to some references?) that for every $n \in \mathbb N \setminus \{2\}$ there exists at least one $n$x$n$ magic square?

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  • $\begingroup$ Yes. There are algorithms given here, for example. $\endgroup$ – rogerl Feb 9 '16 at 1:49
  • $\begingroup$ @rogerl Thank you. Do you know does there exists any unifying proof which deals with all cases at once? $\endgroup$ – Farewell Feb 9 '16 at 1:54
  • $\begingroup$ It's not hard to see, just by setting up $2n+2$ homogeneous equations in $n^2$ unknowns, that there is a magic square with common sum $0$ if $n>2$. Adding a constant to each square then produces a magic square with a nonzero sum. It is not obvious (to me), however, that the elements can be chosen so that the entries are $1$ through $n^2$. $\endgroup$ – rogerl Feb 9 '16 at 2:06
  • $\begingroup$ @rogerl: That looks like a very weak argument to me! You seem to consider the zero matrix as a magic square.Or perhaps I have misunderstood? $\endgroup$ – TonyK Apr 14 '16 at 0:02
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Sierpinski in his classical book, Elementary Number theory, explains :

It is proved that there exist magic squares for any n >=3 (cf. L.Bieberbach).

referring to an article (in german) by Bieberbach (1954).

Moreover, Sierpinski claims to present a proof due to Makowski but the proof seems to depend on a conjecture (the so-called Schinzel conjecture or conjecture H in his book, cf. page 133).

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Wikipedia gives a technique for constructing examples. For odd order, the elements from $kn+1$ to $(k+1)n$ are written on a (broken) diagonal so that each row and column contain one of each residue $\bmod n$ and one of each block $[1,n], [n+1,2n], \dots [n^2-n+1,n^2]$. The ones that are a multiple of $4$ are constructed by reflecting some of the numbers through the center. Those of the form $4n+2$ are formed from a square of size $2n+1$

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