• Source: Most-perfect magic square

A most-perfect magic square of order n is a magic square containing the numbers 1 to n2 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.




Examples



Two 12 × 12 most-perfect magic squares can be obtained adding 1 to each element of:

[,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

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




Properties


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 S. 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 squares and most-perfect magic squares.
For n = 36, there are about 2.7 × 1044 essentially different most-perfect magic squares.




References


Kathleen Ollerenshaw, David S. Brée: Most-perfect Pandiagonal Magic Squares: Their Construction and Enumeration, Southend-on-Sea : Institute of Mathematics and its Applications, 1998, 186 pages, ISBN 0-905091-06-X
T.V.Padmakumar, Number Theory and Magic Squares, Sura books Archived 2010-02-25 at the Wayback Machine, India, 2008, 128 pages, ISBN 978-81-8449-321-4




External links


STRONGLY MAGIC SQUARES by T. V. Padmakumar
OEIS: A051235 Number of essentially different most-perfect pandiagonal magic squares of order 4n from The On-Line Encyclopedia of Integer Sequences

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