If I place all the numbers from 1 to 9 in a 3x3 grid and I add the products of each row and column, then what is the minimal and the maximal sum?enter image description here

For example, the sum is 450 on the picture above

  • 5
    $\begingroup$ Given the no-computers tag, do you have a special trick in mind that allows us to prove that our results are optimal, or are you just looking for answers without any proof? $\endgroup$
    – Magma
    May 27, 2020 at 17:16

2 Answers 2


For the maximum sum, I saw SteveV's answer and noticed that he wasn't

combining the highest numbers with nine. I messed around in a table and found a way that also combines 5 and 6 with nine, not just eight and seven because they will make a higher number than 3 and 6 when multiplied.

With that in mind, I came up with a new solution.

Graph (Maximum)

For the minimum sum,

I realized that you should just try and get the lowest by putting all the big numbers with small numbers when possible and the rest of the time putting them with one to minimize products. The way I did this is by putting one in the middle and having the rest of the small numbers in the corners so the only times that a big number would be in the same row or column as another big number when there was also a one in the row or column.

Here is my solution for the minimum sum

Minimum graph

I haven't been on here lately and have kinda forgotten how to format so will edit while I figure things out

  • $\begingroup$ nice! i said it but i didn't do it :) $\endgroup$
    – SteveV
    May 27, 2020 at 17:53
  • 1
    $\begingroup$ But isn't 2 x 4 x 9 = 72? So the minimum should really be 436. $\endgroup$
    – Peter Shor
    May 27, 2020 at 19:45
  • $\begingroup$ @PeterShor oops! ill fix that $\endgroup$
    – Yout Ried
    May 27, 2020 at 19:54
  • $\begingroup$ I came to the same conclusion by maximizing the value of the high numbers, I think it's probably the maximum [[9,8,7],[6,4,2],[5,3,1]] $\endgroup$ May 28, 2020 at 2:36

I think this may be maximum

enter image description here
For a score of 900

But i provide no proof, just

a feeling that multiplying the largest numbers you can will be the right strategy


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