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Most-perfect Magic SquareA most-perfect magic square of order n is a magic square containing the numbers 0 to n² − 1 with two additional properties: - Each 2×2 subsquare sums to 2s, where s = n² − 1.
- All pairs of integers distant n/2 along a (major) diagonal sum to s.
All most-perfect magic squares are panmagic squares. Apart from the trivial case of the 1st order square, most-perfect magic squares are all of order 4n. In their book, Kathleen Ollerenshaw and David Bre give a method of construction and enumeration of all most-perfect magic squares. They also show that there is a one-to-one correspondence between reversible magic squares and most-perfect magic squares. For n = 36, there are about 2.7 × 1044 essentially different most-perfect magic squares.
Example 12-by-12 most-perfect square follows. ,1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 ,9 ,10 ,11 ,12 1, 64 92 81 94 48 77 67 63 50 61 83 78 2, 31 99 14 97 47 114 28 128 45 130 12 113 3, 24 132 41 134 8 117 27 103 10 101 43 118 4, 23 107 6 105 39 122 20 136 37 138 4 121 5, 16 140 33 142 0 125 19 111 2 109 35 126 6, 75 55 58 53 91 70 72 84 89 86 56 69 7, 76 80 93 82 60 65 79 51 62 49 95 66 8, 115 15 98 13 131 30 112 44 129 46 96 29 9, 116 40 133 42 100 25 119 11 102 9 135 26 10, 123 7 106 5 139 22 120 36 137 38 104 21 11, 124 32 141 34 108 17 127 3 110 1 143 18 12, 71 59 54 57 87 74 68 88 85 90 52 73
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