The 12x12 Magic Squares
Technique 1: The Magic Carpet Approach
There are several interesting techniques to make 12x12 magic square. The technique described on this page makes a Pan-Magic square; a set of six magic carpets are combined.
0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | A | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | B | 0 | 2 | 0 | 2 | 1 | 1 | 1 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 1 | 1 | 1 | 1 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 1 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 1 | 1 | 1 | 1 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 1 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 1 | 1 | 1 | 1 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 1 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 1 | 1 | 1 | 1 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 1 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 1 | 1 | 1 | 1 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 1 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 1 | 1 | 1 | 1 | 0 | 2 | 0 | 2 | C |
0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | D | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | E | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | F |
A | B | C | D | E | F | 72 | 36 | 12 | 6 | 3 | 1 | 1 | 2 | 4 | 12 | 24 | 48 | 24 | 12 | 4 | 2 | 1 | 48 | 12 | 6 | 2 | 1 | 72 | 24 | 6 | 3 | 1 | 72 | 36 | 12 | 2 | 1 | 48 | 24 | 12 | 4 | 1 | 72 | 24 | 12 | 6 | 2 | 72 | 1 | 24 | 2 | 12 | 4 | 1 | 72 | 2 | 36 | 6 | 12 |
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Creating the Pan-Magic Squares
The six magic carpets above are composed of three in the top row, each of which has been rotated to provide an orthogonal version in the second row. There are other magic carpets for order 12 but the squares above are the simplest and show how a series of order 12 pan-magic squares can be formed.
Multipliers
The Table on the right shows some examples of a list of multipliers. The numbers in each row multilpy the cells in the appropriate square. The results are summed to produce the squares. The Pan-Magic squares below were obtained using the first two rows.
0 | 71 | 108 | 107 | 12 | 59 | 120 | 95 | 24 | 47 | 132 | 83 | 137 | 78 | 29 | 42 | 125 | 90 | 17 | 54 | 113 | 102 | 5 | 66 | 9 | 62 | 117 | 98 | 21 | 50 | 129 | 86 | 33 | 38 | 141 | 74 | 140 | 75 | 32 | 39 | 128 | 87 | 20 | 51 | 116 | 99 | 8 | 63 | 1 | 70 | 109 | 106 | 13 | 58 | 121 | 94 | 25 | 46 | 133 | 82 | 136 | 79 | 28 | 43 | 124 | 91 | 16 | 55 | 112 | 103 | 4 | 67 | 10 | 61 | 118 | 97 | 22 | 49 | 130 | 85 | 34 | 37 | 142 | 73 | 139 | 76 | 31 | 40 | 127 | 88 | 19 | 52 | 115 | 100 | 7 | 64 | 2 | 69 | 110 | 105 | 14 | 57 | 122 | 93 | 26 | 45 | 134 | 81 | 135 | 80 | 27 | 44 | 123 | 92 | 15 | 56 | 111 | 104 | 3 | 68 | 11 | 60 | 119 | 96 | 23 | 48 | 131 | 84 | 35 | 36 | 143 | 72 | 138 | 77 | 30 | 41 | 126 | 89 | 18 | 53 | 114 | 101 | 6 | 65 |
| 0 | 142 | 3 | 141 | 4 | 138 | 7 | 137 | 8 | 134 | 11 | 133 | 131 | 13 | 128 | 14 | 127 | 17 | 124 | 18 | 123 | 21 | 120 | 22 | 36 | 106 | 39 | 105 | 40 | 102 | 43 | 101 | 44 | 98 | 47 | 97 | 119 | 25 | 116 | 26 | 115 | 29 | 112 | 30 | 111 | 33 | 108 | 34 | 48 | 94 | 51 | 93 | 52 | 90 | 55 | 89 | 56 | 86 | 59 | 85 | 83 | 61 | 80 | 62 | 79 | 65 | 76 | 66 | 75 | 69 | 72 | 70 | 84 | 58 | 87 | 57 | 88 | 54 | 91 | 53 | 92 | 50 | 95 | 49 | 71 | 73 | 68 | 74 | 67 | 77 | 64 | 78 | 63 | 81 | 60 | 82 | 96 | 46 | 99 | 45 | 100 | 42 | 103 | 41 | 104 | 38 | 107 | 37 | 35 | 109 | 32 | 110 | 31 | 113 | 28 | 114 | 27 | 117 | 24 | 118 | 132 | 10 | 135 | 9 | 136 | 6 | 139 | 5 | 140 | 2 | 143 | 1 | 23 | 121 | 20 | 122 | 19 | 125 | 16 | 126 | 15 | 129 | 12 | 130 |
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Copyright © Mar 2010 |
Magic Squares Website |
Updated
Mar 6, 2010
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