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.