Most-perfect magic square
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A 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.
For example, a 12×12 most-perfect magic square could be:
[,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
All most-perfect magic squares are panmagic squares.
Apart from the trivial case of the first order square, most-perfect magic squares are all of order 4n. In their book, Kathleen Ollerenshaw and David Brée 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.