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Magic Square: An nxn square where every horizontal,vertical, and diagonal line all add up to the same number.

Magic Constant: The number which every line with a magic square adds up to

So most magic squares, have a magic constant. But some do not. A fairly trivial answer is 0x0. Where no numbers are present, therefore it cannot sum up into any number.

A less than trivial answer but not complicated answer is 2x2. No possible configuration of 1,2,3,4 within a 2x2 block can have all lines add up to the same number. If it were possible it would theoretically be 5

So besides 0, and 2 are there any other positive dimensions for a magic square that do not have a magic constant? Technically making all possible configurations invalid magic squares?

Mathematical Proof is Appreciated!

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    $\begingroup$ I downvoted because the answer is easy to find. The Wikipedia article on magic squares says in the intro "Normal magic squares of all sizes except 2 × 2 (that is, where n = 2) can be constructed." And it gives constructions. I see no point in having an SE answer to this. $\endgroup$
    – xnor
    Commented Dec 5, 2014 at 1:38
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    $\begingroup$ Magic squares usually require each of the numbers from 1 to n^2 to appear exactly once. It happens not to matter for your question though, since this is an additional restriction. Also, a 0x0 empty square is magic by a formal definition if n>0 is not assumed. The set of lines of empty, and so the multiset of sums is empty, and so all sums are equal. This is true whether you formalize "all equal" as "there exists a value x so that all elements in the multiset equal x" or "any pair of value in the multiset are equal". $\endgroup$
    – xnor
    Commented Dec 5, 2014 at 1:48
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    $\begingroup$ @warspyking The constructions are right in the article. You can (and should) check them yourself. Please do some basic research before you post a question. $\endgroup$
    – xnor
    Commented Dec 5, 2014 at 2:06
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    $\begingroup$ @warspyking You don't get it. This is math. You don't know it's right because it's from an authority, you know it's right because you read it and check it yourself. $\endgroup$
    – xnor
    Commented Dec 5, 2014 at 2:10
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    $\begingroup$ @xnor Point taken. Still couldn't hurt to have the info on the site (look at all the other easily searchable questions on this site.) $\endgroup$
    – warspyking
    Commented Dec 5, 2014 at 2:16

1 Answer 1

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From Wikipedia (my emphasis):

Normal magic squares of all sizes except 2 × 2 (that is, where n = 2) can be constructed. The 1 × 1 magic square, with only one cell containing the number 1, is trivial.

For a mathematical proof of this, see the Nrich article here. The idea is to turn an n x n square into a diamond (an n x n square standing on one corner), fill in that diamond with the numbers 1 to n^2 in the natural way, and then fold everything into the original square. Some pictures to demonstrate (also taken from the Nrich article):

3x3 case

5x5 case

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  • $\begingroup$ And that's why it doesn't work for 2x2 - the only squares are the corner squares! $\endgroup$
    – No Name
    Commented Nov 2 at 6:08

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