Make it fifteen: Answer

   8 1 6
   3 5 7
   4 9 2
This can of course be rotated and mirrored at will.

Heluk Tekbulut:

The answer for 5x5 (numbers 1-5²):

  3 16  9 22 15
 20  8 21 14  2
  7 25 13  1 19
 24 12  5 18  6
 11  4 17 10 23
For any m x m matrix with numbers 1 to m² where m is odd there is a solution and the sum is m²*(m²+1)/(2*m).

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