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.

  • $\begingroup$ 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 '16 at 4:01
  • $\begingroup$ @Meysam: can you say how you obtained the answer for $n=6$? $\endgroup$ – Shahrooz Janbaz Jan 16 '16 at 10:18

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.

  • 1
    $\begingroup$ 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 '16 at 11:56
  • 2
    $\begingroup$ @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 Janbaz Jan 19 '16 at 0:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.