# Composite magic squares

A magic square of order m · n is called composite, when it can be decomposed into m2 magic subsquares, each of order n.

It is to be noted that the minimal composite magic square must be of order n=9. This is the minimal number with two divisors, for which magic squares exist.

This means that the fundamental magic square can be decomposed into m2=32=9 magic subsquares, each of them with order n=. Of course it is impossible that all subsquares are normalized, i.e. are composed of consecutive integers 1, 2, … , n2.

 71 66 67 20 25 24 29 34 33 64 68 72 27 23 19 36 32 28 69 70 65 22 21 26 31 30 35 8 3 4 40 39 44 74 79 78 1 5 9 45 41 37 81 77 73 6 7 2 38 43 42 76 75 80 47 54 49 56 63 58 11 16 15 52 50 48 61 59 57 18 14 10 51 46 53 60 55 62 13 12 17

The next composite magic square is found for order m · n=12. This order can be divided in the following manner:

This means that the composite square can be decomposed in m2=32=9 magic subsquares of order n=4.

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

But, also another decomposition is possible.

This composite magic square is decomposed into m2=42=16 magic subsquares, each of order n=3.

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