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A most-perfect magic square of order ''n'' is a
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''
,'' 64 92 81 94 48 77 67 63 50 61 83 78
,'' 31 99 14 97 47 114 28 128 45 130 12 113
,'' 24 132 41 134 8 117 27 103 10 101 43 118
,'' 23 107 6 105 39 122 20 136 37 138 4 121
,'' 16 140 33 142 0 125 19 111 2 109 35 126
,'' 75 55 58 53 91 70 72 84 89 86 56 69
,'' 76 80 93 82 60 65 79 51 62 49 95 66
,'' 115 15 98 13 131 30 112 44 129 46 96 29
,'' 116 40 133 42 100 25 119 11 102 9 135 26
0,'' 123 7 106 5 139 22 120 36 137 38 104 21
1,'' 124 32 141 34 108 17 127 3 110 1 143 18
2,'' 71 59 54 57 87 74 68 88 85 90 52 73
1 2 3 4 5 6 7 8 9 10 11 12''
,'' 4 113 14 131 3 121 31 138 21 120 32 130
,'' 136 33 126 15 137 25 109 8 119 26 108 16
,'' 73 44 83 62 72 52 100 69 90 51 101 61
,'' 64 105 54 87 65 97 37 80 47 98 36 88
,'' 1 116 11 134 0 124 28 141 18 123 29 133
,'' 103 66 93 48 104 58 76 41 86 59 75 49
,'' 112 5 122 23 111 13 139 30 129 12 140 22
,'' 34 135 24 117 35 127 7 110 17 128 6 118
,'' 43 74 53 92 42 82 70 99 60 81 71 91
0,'' 106 63 96 45 107 55 79 38 89 56 78 46
1,'' 115 2 125 20 114 10 142 27 132 9 143 19
2,'' 67 102 57 84 68 94 40 77 50 95 39 85
Sura books
, India, 2008, 128 pages, ISBN 978-81-8449-321-4
STRONGLY MAGIC SQUARES by T. V. Padmakumar
* Number of essentially different most-perfect pandiagonal magic squares of order 4n from The
A most-perfect magic square of order ''n'' is a magic square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
containing the numbers 1 to ''n''2 with two additional properties:
# Each 2 × 2 subsquare sums to 2''s'', where ''s'' = ''n''2 + 1.
# All pairs of integers distant ''n''/2 along a (major) diagonal sum to ''s''.
__TOC__
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''
,'' 64 92 81 94 48 77 67 63 50 61 83 78
,'' 31 99 14 97 47 114 28 128 45 130 12 113
,'' 24 132 41 134 8 117 27 103 10 101 43 118
,'' 23 107 6 105 39 122 20 136 37 138 4 121
,'' 16 140 33 142 0 125 19 111 2 109 35 126
,'' 75 55 58 53 91 70 72 84 89 86 56 69
,'' 76 80 93 82 60 65 79 51 62 49 95 66
,'' 115 15 98 13 131 30 112 44 129 46 96 29
,'' 116 40 133 42 100 25 119 11 102 9 135 26
0,'' 123 7 106 5 139 22 120 36 137 38 104 21
1,'' 124 32 141 34 108 17 127 3 110 1 143 18
2,'' 71 59 54 57 87 74 68 88 85 90 52 73
1 2 3 4 5 6 7 8 9 10 11 12''
,'' 4 113 14 131 3 121 31 138 21 120 32 130
,'' 136 33 126 15 137 25 109 8 119 26 108 16
,'' 73 44 83 62 72 52 100 69 90 51 101 61
,'' 64 105 54 87 65 97 37 80 47 98 36 88
,'' 1 116 11 134 0 124 28 141 18 123 29 133
,'' 103 66 93 48 104 58 76 41 86 59 75 49
,'' 112 5 122 23 111 13 139 30 129 12 140 22
,'' 34 135 24 117 35 127 7 110 17 128 6 118
,'' 43 74 53 92 42 82 70 99 60 81 71 91
0,'' 106 63 96 45 107 55 79 38 89 56 78 46
1,'' 115 2 125 20 114 10 142 27 132 9 143 19
2,'' 67 102 57 84 68 94 40 77 50 95 39 85
Properties
All most-perfect magic squares arepanmagic square A pandiagonal magic square or panmagic square (also diabolic square, diabolical square or diabolical magic square) is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the squar ...
s.
Apart from the trivial case of the first order square, most-perfect magic squares are all of order 4''n''. In their book, Kathleen Ollerenshaw
Dame Kathleen Mary Ollerenshaw, (''née'' Timpson; 1 October 1912 – 10 August 2014) was a British mathematician and politician who was Lord Mayor of Manchester from 1975 to 1976 and an advisor on educational matters to Margaret Thatcher's go ...
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between reversible square
Reversibility can refer to:
* Time reversibility, a property of some mathematical or physical processes and systems for which time-reversed dynamics are well defined
:* Reversible diffusion, an example of a reversible stochastic process
* Reversibl ...
s 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
, India, 2008, 128 pages, ISBN 978-81-8449-321-4
External links
STRONGLY MAGIC SQUARES by T. V. Padmakumar
* Number of essentially different most-perfect pandiagonal magic squares of order 4n from The
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
{{Magic polygons
Magic squares