12x12 magic square

Composite, Proportional (1) a

 

René Chrétien had noticed the 15x15 composite (4) magic square and showed me it is possible to use the method to construct magic squares of even orders as well.

 

Construct the 12x12 magic square by using 4 proportional 6x6 magic squares. The squares are proportional because all 4 magic 6x6 squares have the same magic sum of (1/2 x 870 = ) 435. We use the method with reflecting grids (6x6) to produce the magic 6x6 squares.  As row coordinates don't use 0 up to 5 but use 0 up to (4x6 -/- 1 = ) 23 instead. Take care that the sum of the row coordinates in each 6x6 square is the same  (0+7+8+15+16+23 = 1+6+9+14+17+22 = 2+5+10+13+18+21 = 3+4+11+12+19+20 = 69) to get proportional squares.

 

 

1x row coordinate                   +24x column coordinate + 1  =    6x6 magic square 

0 7 15 8 16 23   0 5 0 5 5 0   1 128 16 129 137 24
23 7 8 15 16 0   1 1 4 4 1 4   48 32 105 112 41 97
0 16 8 15 7 23   3 2 2 2 3 3   73 65 57 64 80 96
23 16 8 15 7 0   2 3 3 3 2 2   72 89 81 88 56 49
23 7 15 8 16 0   4 4 1 1 4 1   120 104 40 33 113 25
0 16 15 8 7 23   5 0 5 0 0 5   121 17 136 9 8 144
                                       
1 6 14 9 17 22   0 5 0 5 5 0   2 127 15 130 138 23
22 6 9 14 17 1   1 1 4 4 1 4   47 31 106 111 42 98
1 17 9 14 6 22   3 2 2 2 3 3   74 66 58 63 79 95
22 17 9 14 6 1   2 3 3 3 2 2   71 90 82 87 55 50
22 6 14 9 17 1   4 4 1 1 4 1   119 103 39 34 114 26
1 17 14 9 6 22   5 0 5 0 0 5   122 18 135 10 7 143
                                       
2 5 13 10 18 21   0 5 0 5 5 0   3 126 14 131 139 22
21 5 10 13 18 2   1 1 4 4 1 4   46 30 107 110 43 99
2 18 10 13 5 21   3 2 2 2 3 3   75 67 59 62 78 94
21 18 10 13 5 2   2 3 3 3 2 2   70 91 83 86 54 51
21 5 13 10 18 2   4 4 1 1 4 1   118 102 38 35 115 27
2 18 13 10 5 21   5 0 5 0 0 5   123 19 134 11 6 142
                                       
3 4 12 11 19 20   0 5 0 5 5 0   4 125 13 132 140 21
20 4 11 12 19 3   1 1 4 4 1 4   45 29 108 109 44 100
3 19 11 12 4 20   3 2 2 2 3 3   76 68 60 61 77 93
20 19 11 12 4 3   2 3 3 3 2 2   69 92 84 85 53 52
20 4 12 11 19 3   4 4 1 1 4 1   117 101 37 36 116 28
3 19 12 11 4 20   5 0 5 0 0 5   124 20 133 12 5 141

 

 

Put the 4 magic 6x6 squares together.

 

 

12x12 magic square

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

 

 

Each 1/2 row/column/diagonal gives 1/2 of the magic sum and the 12x12 magic square is 6x6 compact.

 

Look at the tight sequence of the digits if you go from sub-square to sub-square forwards and backwards.

 

 

I have used composite method, proportional (1) to construct

8x8, 9x9, 12x12a, 12x12b, 15x15a, 15x15b, 16x16a, 16x16b, 18x18, 20x20a, 20x20b,  21x21a, 21x21b, 24x24a, 24x24b, 24x24c, 27x27a, 27x27b, 28x28a, 28x28b, 30x30a,  30x30b,32x32a, 32x32b and 32x32c

 

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12x12 magic square