Shift method

 

You can use this method to construct magic squares of odd order which are no multiple of 3 (= 5x5, 7x7, 11x11, 13x13, 17x17, ... magic squares). Construct the first row of the 7x7 magic square with the digits  0-a-b-c-d-e-f (fill in the digits 1 up to 6 in random order instead of a up to f; that gives 6x5x4x3x2 = 720 different combinations).

 

 

1x digit from shift 2 to the left  + 7x digit from shift 2 to the right+1 = Panmagic 7x7 square

0 1 2 3 4 5 6   0 1 2 3 4 5 6   1 9 17 25 33 41 49
2 3 4 5 6 0 1   5 6 0 1 2 3 4   38 46 5 13 21 22 30
4 5 6 0 1 2 3   3 4 5 6 0 1 2   26 34 42 43 2 10 18
6 0 1 2 3 4 5   1 2 3 4 5 6 0   14 15 23 31 39 47 6
1 2 3 4 5 6 0   6 0 1 2 3 4 5   44 3 11 19 27 35 36
3 4 5 6 0 1 2   4 5 6 0 1 2 3   32 40 48 7 8 16 24
5 6 0 1 2 3 4   2 3 4 5 6 0 1   20 28 29 37 45 4 12

 

 

It is also possible to shift the digits of the first row 3 (instead of 2) places to right/left, and you get more 7x7 panmagic squares (see website: www.grogono.com/magic/7x7.php ). There are 6 possibilities to shift the first row in the first grid / second grid:

  • 2 to the left / 2 to the right
  • 2 to the left / 3 to the right
  • 2 to the left / 3 to the left
  • 3 to the left / 2 to the right
  • 3 to the left / 3 to the right
  • 3 to the left / 2 to the left

 

Construct all 6 (above mentioned shift possibilities) x 720 (combinations first grid) x 720 (combinations second grid) x 49 (shifted version on the 2x2 carpet) x 8 (by rotating and/or mirroring) / 4 (because of duplications) is 304.819.200 panmagic 7x7 squares.

 

 

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7x7, shift method.xls
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7x7, shift method, shift possibilities.x
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