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
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$\begingroup$ 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. $\endgroup$– bobbleDec 9, 2020 at 4:29
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$\begingroup$ Do you really mean "product", which usually means multiplication between numbers? Or do you just want to construct a 4x4 magic square? $\endgroup$– BubblerDec 9, 2020 at 4:29
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$\begingroup$ Apologies, a magic square. $\endgroup$– Yellow XDec 9, 2020 at 4:38
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$\begingroup$ 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. $\endgroup$– BubblerDec 9, 2020 at 4:53
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3 Answers
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1
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.
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$\begingroup$ So basically you're finding two mutually orthogonal latin squares but including the diagonals. $\endgroup$ Dec 9, 2020 at 12:54
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One of the answers:
Let 3=𝑎0, 4=𝑎1, 10=𝑏0... So for the numbers we can get
0231
1320
2013
3102
And adding the 𝑎, 𝑏, 𝑐, 𝑑's (using some sudoku tricks) we have
𝑎0 𝑑2 𝑏3 𝑐1
𝑏1 𝑐3 𝑎2 𝑑0
𝑐2 𝑏0 𝑑1 𝑎3
𝑑3 𝑎1 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.
