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?
For example, the sum is 450 on the picture above
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4$\begingroup$ 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 '20 at 17:16
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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.
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
I haven't been on here lately and have kinda forgotten how to format so will edit while I figure things out
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1$\begingroup$ But isn't 2 x 4 x 9 = 72? So the minimum should really be 436. $\endgroup$ – Peter Shor May 27 '20 at 19:45
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$\begingroup$ 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]] $\endgroup$ – gszavae May 28 '20 at 2:36
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I think this may be maximum
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
