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Arrange the following numbers in a way such that all rows, columns and the diagonals add up to the same number.

6   5   4   3
13  12  11  10
20  19  18  17
27  26  25  24
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4
  • $\begingroup$ Welcome to Puzzling! Is this a puzzle you found elsewhere? If so, please edit the source into your question. We have an attribution policy here. $\endgroup$
    – bobble
    Dec 9, 2020 at 4:29
  • $\begingroup$ Do you really mean "product", which usually means multiplication between numbers? Or do you just want to construct a 4x4 magic square? $\endgroup$
    – Bubbler
    Dec 9, 2020 at 4:29
  • $\begingroup$ Apologies, a magic square. $\endgroup$
    – Yellow X
    Dec 9, 2020 at 4:38
  • $\begingroup$ While it's not too hard to construct a magic square with these numbers, I don't think it's a textbook problem, and I can't locate any duplicates. $\endgroup$
    – Bubbler
    Dec 9, 2020 at 4:53

3 Answers 3

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One can easily make a size 4 normal magic square by adding one with 4 times 1,2,3,4 and one with 4 times 0,4,8,12:
1: Put the numbers on the diagonal in any order. 2: Put the top row element on the other diagonal; but use the opposite available position for the two squares. 3: Fill in both squares, avoiding same numbers in all rows/columns 4: Add them up.

The requested numbers form 4 blocks of 4 , and replacing 0,4,8,12 with 2,9,16,23 (thus) leads to $24*24*2$ valid solutions this way, e.g. the blue one in the picture. enter image description here

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One of the answers:
Let 3=π‘Ž0, 4=π‘Ž1, 10=𝑏0... So for the numbers we can get
0231
1320
2013
3102
And adding the π‘Ž, 𝑏, 𝑐, 𝑑's (using some sudoku tricks) we have
π‘Ž0 𝑑2 𝑏3 𝑐1
𝑏1 𝑐3 π‘Ž2 𝑑0
𝑐2 𝑏0 𝑑1 π‘Ž3
𝑑3 π‘Ž1 𝑐0 𝑏2
Or
3 26 13 18
11 20 5 24
19 10 25 6
27 4 17 12

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There are already two good answers. Here's the laziest approach, which only works because we are feeling lucky.

Step 1: Take any 4x4 magic square with the property that the numbers 1-4 don't appear in the same row, column, or diagonal, and the same is true for blocks 5-8, 9-12, and 13-16. The first completed grid in google image search will do nicely:

 8 11 14  1
13  2  7 12
 3 16  9  6
10  5  4 15

Step 2: add the block offsets 2, 5, 8, and 11 to blocks 1-4, 5-8, 9-12, and 13-16 respectively

 13 19 25  3
 24  4 12 20
  5 27 17 11
 18 10  6 26

This adds 2 + 5 + 8 + 11 = 26 to every row, column, and diagonal, so the result is still magic.

Step 3 (optional): Permute to taste.

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