### Shift method

Use this method to construct magic squares of odd order which are no multiple of 3 (= 5x5, 7x7, 11x11, 13x13, 17x17, ... magic squares). To construct an 11x11 magic square, the first row is 0-a-b-c-d-e-f-g-h-i-j (fill in 1 up to 10 instead of a up to j; that gives 10x9x8x7x6x5x4x3x2 = 3.628.800 possibilities).

To construct row 2 up to 11 of the first grid shift the first row of the first grid each time two places to the left. To construct row 2 up to 11 of the second grid shift the first row of the second grid each time two places to the right.

Take 1x number from first grid +1

 0 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 0 1 4 5 6 7 8 9 10 0 1 2 3 6 7 8 9 10 0 1 2 3 4 5 8 9 10 0 1 2 3 4 5 6 7 10 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 0 3 4 5 6 7 8 9 10 0 1 2 5 6 7 8 9 10 0 1 2 3 4 7 8 9 10 0 1 2 3 4 5 6 9 10 0 1 2 3 4 5 6 7 8

+11x number from second grid

 0 1 2 3 4 5 6 7 8 9 10 9 10 0 1 2 3 4 5 6 7 8 7 8 9 10 0 1 2 3 4 5 6 5 6 7 8 9 10 0 1 2 3 4 3 4 5 6 7 8 9 10 0 1 2 1 2 3 4 5 6 7 8 9 10 0 10 0 1 2 3 4 5 6 7 8 9 8 9 10 0 1 2 3 4 5 6 7 6 7 8 9 10 0 1 2 3 4 5 4 5 6 7 8 9 10 0 1 2 3 2 3 4 5 6 7 8 9 10 0 1

= panmagic 11x11 square

 1 13 25 37 49 61 73 85 97 109 121 102 114 5 17 29 41 53 65 77 78 90 82 94 106 118 9 21 33 34 46 58 70 62 74 86 98 110 111 2 14 26 38 50 42 54 66 67 79 91 103 115 6 18 30 22 23 35 47 59 71 83 95 107 119 10 112 3 15 27 39 51 63 75 87 99 100 92 104 116 7 19 31 43 55 56 68 80 72 84 96 108 120 11 12 24 36 48 60 52 64 76 88 89 101 113 4 16 28 40 32 44 45 57 69 81 93 105 117 8 20

It is possible to shift this 11x11 magic square on a 2x2 carpet of the 11x11 magic square and you get 120 more solutions .

Instead of shift 2 to the left and shift 2 to the right, you can also shift 3, 4 or 5 to the right and/or to the left (e.g. in the first grid shift 3 to the right and in the second grid shift 4 to the left ór 4 to the right). In total you can construct all 89.227.651.645.440.000 panmagic 11x 11 squares (= more than 89 milion x billion possibilities).

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

See

11x11, Shift method.xls