### 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 13x13 magic square, the first row is 0-a-b-c-d-e-f-g-h-i-j-k-l (fill in 1 up to 12 instead of a up to l; that gives 12x11x10x9x8x7x6x5x4x3x2 = 479.001.600 possibilities).

To construct row 2 up to 13 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 13 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 11 12 2 3 4 5 6 7 8 9 10 11 12 0 1 4 5 6 7 8 9 10 11 12 0 1 2 3 6 7 8 9 10 11 12 0 1 2 3 4 5 8 9 10 11 12 0 1 2 3 4 5 6 7 10 11 12 0 1 2 3 4 5 6 7 8 9 12 0 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 12 0 3 4 5 6 7 8 9 10 11 12 0 1 2 5 6 7 8 9 10 11 12 0 1 2 3 4 7 8 9 10 11 12 0 1 2 3 4 5 6 9 10 11 12 0 1 2 3 4 5 6 7 8 11 12 0 1 2 3 4 5 6 7 8 9 10

+13x number from second grid

 0 1 2 3 4 5 6 7 8 9 10 11 12 11 12 0 1 2 3 4 5 6 7 8 9 10 9 10 11 12 0 1 2 3 4 5 6 7 8 7 8 9 10 11 12 0 1 2 3 4 5 6 5 6 7 8 9 10 11 12 0 1 2 3 4 3 4 5 6 7 8 9 10 11 12 0 1 2 1 2 3 4 5 6 7 8 9 10 11 12 0 12 0 1 2 3 4 5 6 7 8 9 10 11 10 11 12 0 1 2 3 4 5 6 7 8 9 8 9 10 11 12 0 1 2 3 4 5 6 7 6 7 8 9 10 11 12 0 1 2 3 4 5 4 5 6 7 8 9 10 11 12 0 1 2 3 2 3 4 5 6 7 8 9 10 11 12 0 1

= panmagic 13x13 square

 1 15 29 43 57 71 85 99 113 127 141 155 169 146 160 5 19 33 47 61 75 89 103 117 118 132 122 136 150 164 9 23 37 51 65 66 80 94 108 98 112 126 140 154 168 13 14 28 42 56 70 84 74 88 102 116 130 131 145 159 4 18 32 46 60 50 64 78 79 93 107 121 135 149 163 8 22 36 26 27 41 55 69 83 97 111 125 139 153 167 12 158 3 17 31 45 59 73 87 101 115 129 143 144 134 148 162 7 21 35 49 63 77 91 92 106 120 110 124 138 152 166 11 25 39 40 54 68 82 96 86 100 114 128 142 156 157 2 16 30 44 58 72 62 76 90 104 105 119 133 147 161 6 20 34 48 38 52 53 67 81 95 109 123 137 151 165 10 24

It is possible to shift this 13x13 magic square on a 2x2 carpet of the 13x13 magic square and you get 168 more solutions .

Instead of shift 2 to the left and shift 2 to the right, you can also shift 3, 4, 5 or 6 to the right and/or to the left (e.g. in the first grid shift 4 to the right and in the second grid shift 2 to the left ór 2 to the right). In total you can construct all 3,48982 x 1021 panmagic 13x 13 squares.

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

See

13x13, shift method.xls
Microsoft Excel werkblad 59.5 KB