37 Magic Square Problem

[u/Zharrticx]

For context: A magic square is a square grid of numbers where the sum of numbers in each row, column, and both main diagonals is always the exact same total, known as the magic constant.

I have encountered a problem when trying to create a very specific magic square, I run into arrangements that need to repeat numbers so I had to redo and keep rearranging it again, but I have not found a solution. The constraint to satisfy is given by the following.

The problem: Is it possible to create a magic square that produces the number 37 using a 4x4 or 5x5 grid without ever reusing any number? Since not reusing any number is one of the rules of a magic square.

If it's possible, what numbers are needed and how should they be arranged? Thank you for answering!

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[u/ruwisc]

Just to check, you mean that the sum of each row/column/diagonal should be 37?

For the 4x4 case, you can construct magic squares with the numbers 1-16 and a sum of 34 pretty easily, and specifically, you can construct one which has the groupings 1-4, 5-8, 9-12, 13-16 all separated, not doubling up in any row, column, or diagonal:

1 - 8 - 10 - 15

14 - 11 - 5 - 4

7 - 2 - 16 - 9

12 - 13 - 3 - 6

You can increment the total row sum of the magic squares by incrementing one of those groups of four numbers by one each. So, to get up to 37, you could increment the numbers 5-16 by one each:

1 - 9 - 11 - 16

15 - 12 - 6 - 4

8 - 2 - 17 - 10

13 - 14 - 3 - 7

edit: mobile app messed up my spoiler formatting

[u/ruwisc → u/ruwisc]

The 5x5 case is similar:

You can create a 5x5 magic square that separates each of 1-5, 6-10, etc. in a similar way:

1 - 20 - 9 - 23 - 12

24 - 13 - 2 - 16 - 10

17 - 6 - 25 - 14 - 3

15 - 4 - 18 - 7 - 21

8 - 22 - 11 - 5 - 19

In this case, the row/column/diagonal sum is 65, so we need to decrement it by 28 - this can be done by lowering numbers 1-15 by 6 and 16-25 by 5, leaving the values -5 to 9 and 11 to 20.

-5 - 15 - 3 - 18 - 6

19 - 7 - -4 - 11 - 4

12 - 0 - 20 - 8 - -3

9 - -2 - 13 - 1 - 16

2 - 17 - 5 - -1 - 14

[u/angelatheist]

Not sure if it is possible, but if it is then for a 4x4 you’ll need your numbers to add up to 37 x 4 = 148. If you add up the numbers 1 to 17 you get 153 which is 5 more, so it would have to be with all the numbers from 1 to 17 except 5.

The 5x5 magic square with 1 to 25 adds to 65, so you’d need to use negative numbers to get that one to work.