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A magic square is an n-dimensional matrix in which each row, column, and main diagonal sums to the same 'magic number' (denoted by s). A normal magic square uses each of the numbers from 1 to n exactly once.

The normal magic square below is Albrecht Durer's famous magic square in which not only the rows, columns, and diagonals add up to s=34, but also the four corners and the inner opposite sides. Of particular note is that Mr. Durer created this magic square in 1514 and the middle of the bottom row also reads 1514.

Albrecht Durer's Famous Magic Square

So, in that spirit, I have decided to create a normal 5x5 magic square (s=65) with the number 2019 in the middle of the bottom row. Here it is:

2019 Magic Square

Note: This was generated using Loubere rule with normal vector (-2,-1) and jump vector(2,2) then permuting columns 1 and 4

I have two questions:

(1) There are 275,305,224 valid normal 5x5 magic squares. How many of those contain (2019) in the middle of the bottom row?

(2) Without moving 20, 1, or 9: Can you give me an example of a normal 5x5 magic square where the inner cross (15, 6, 17, 25, 24 in this example) also adds up to 65?

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  • $\begingroup$ The property in your second question is satisfied by any 5x5 pandiagonal magic square. $\endgroup$
    – noedne
    Commented Mar 9, 2019 at 23:16
  • $\begingroup$ Oh, I must be tired I didn't realize I messed up on one of my diagonals! Thanks, I'm going to load this up in excel and see if I can be a little more careful this time. $\endgroup$
    – Diatche
    Commented Mar 9, 2019 at 23:35
  • $\begingroup$ Did you forget to add a reference for the number in point (1)? I'm adding one for you, but feel free to add you own if there's some place you'd prefer. $\endgroup$
    – boboquack
    Commented Mar 11, 2019 at 9:21

1 Answer 1

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There are

$30619$

5x5 squares with 20,1,9 in the indicated position. I used a modified version of the code found here to brute force search for 5x5s with 20,1,9 in the middle of the bottom row.

This appears to meet (2):

$\begin{array}{|c|c|c|c|c|}\hline4&7&13&25&16\\\hline15&21&19&2&8\\\hline17&3&10&11&24\\\hline6&14&22&18&5\\\hline23&20&1&9&12\\\hline\end{array}$

(There are 3 other pandiagonal magic squares with 20,1,9 in the middle of the bottom row.)

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