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Recently, I saw a question in see here which is so interesting for me. This question is as follows:

Is it possible to fill the $121$ entries in an $11×11$ square with the values $0,+1,−1$, so that the row sums and column sums are $22$ distinct numbers?

This problem is interesting for me because its definition is reverse of the magic square. I think for solve this problem we must use something like design theory. Also, for $n=6$ this problem have an answer. My next question is:

For which integer number $n$, this problem has a solution and is this problem well known in math?

I will be so thankful for any helpful comments and answers.

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  • $\begingroup$ You should note that if there is an answer, there is one with a row sum of 11 and a row sum of -11. (Why?) Can you use this to figure out what other row and column sums there must be? Gerhard "Can Work With Ternary Matrices" Paseman, 2016.01.15. $\endgroup$
    – Gerhard Paseman
    Jan 16, 2016 at 4:01
  • $\begingroup$ @Meysam: can you say how you obtained the answer for $n=6$? $\endgroup$
    – Shahrooz
    Jan 16, 2016 at 10:18

1 Answer 1

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This problem is so famous. For first trivial reference, you can see:link.

$\it{R. Bodendiek}$ and $\it{G. Burosch}$ studied this problem in a paper with name:

"Solution to the Antimagic 0,1,-1 Matrix Problem."

If there is solution for integer $n$, then we have:

$1)$ $n$ is even,

$2)$ The number in $\{-n,1-n,2-n,...,n\}$ that does not appear as a line sum is either $-n$ or $n$,

$3)$ Of the $n$ largest line sums, half are column sums and half are row sums.

Also, you can search about "Alternating sum matrix", for more information.

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    $\begingroup$ What is the connection between antimagic matrix and alternating sign matrix? Searching for "alternating sum matrix" is not giving any obvious leads towards antimagic matrix problem. $\endgroup$
    – Scipi
    Jan 18, 2016 at 11:56
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    $\begingroup$ @Scipi: I pointed it out since if you see "CRC Handbook of mathematics-2ed", after introducing this subject, it gives some keywords for more research about similar problems. In some texts, when they talk about similar problems, they also talk about antimagic matrix problem. $\endgroup$
    – Shahrooz
    Jan 19, 2016 at 0:38

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