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# The 12x12 Magic Squares

 12x12 Index

### Technique 3: The Bree/Ollerenshaw Approach

The magic Square below has been created using the technique descibed by Bree and Ollerenshaw.

 0 133 2 135 4 137 11 142 9 140 7 138 131 22 129 20 127 18 120 13 122 15 124 17 24 109 26 111 28 113 35 118 33 116 31 114 107 46 105 44 103 42 96 37 98 39 100 41 48 85 50 87 52 89 59 94 57 92 55 90 83 70 81 68 79 66 72 61 74 63 76 65 132 1 134 3 136 5 143 10 141 8 139 6 23 130 21 128 19 126 12 121 14 123 16 125 108 25 110 27 112 29 119 34 117 32 115 30 47 106 45 104 43 102 36 97 38 99 40 101 84 49 86 51 88 53 95 58 93 56 91 54 71 82 69 80 67 78 60 73 62 75 64 77
Bree/Ollerensshaw 12x12

### Construction

The following two squares illustrate their technique. In the first square, an simple number array has been modified so that the right columns and lower rows have been arranged in reverse order. The cells have also been colored to facilitate following the changes in the second square.

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

### Features

The square demonstrates the features they describe. For order n where S = n2 - 1:

1. All diagonals sum to nS/2 (the magic sum) - including the broken ones
2. Every 2x2 block of cells sums to 2S
3. Any pair of integers separate diagonally by n/2 will sum to S

A spreadsheet is available for Download. This spreadsheet will create Bree/Ollerenshaw Magic squares for order 4 to order 40 in steps of 4.

 12x12 Index

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