### 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 numbers 0-a-b-c-d-e-f (fill in the numbers 1 up to 6 in random order instead of a up to f; that gives 6x5x4x3x2 = 720 different combinations).

1x number from shift 2 to the left + 7x number 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 numbers 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.

Use the shift method to construct magic squares of odd order from 5x5 to infinity.

See

7x7, shift method.xls