order="1" cellpadding="3" cellspacing="0" align="center" style="background:#efefef color:black">
7	12	1	14
2	13	8	11
16	3	10	5
9	6	15	4

transcription

Transcription (linguistics)

Transcription in the linguistic sense is the systematic representation of language in written form. The source can either be utterances or preexisting text in another writing system, although some linguists only consider the former as transcription.Transcription should not be confused with...

 of
the indian numerals

Indian numerals

Most of the positional base 10 numeral systems in the world have originated from India, where the concept of positional numeration was first developed...

A most-perfect magic square of order n is a magic square

Magic square

In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...

 containing the numbers 1 to n² with two additional properties:

1. Each 2×2 subsquare sums to 2s, where s = n² + 1.
2. 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 square

Panmagic square

A pandiagonal magic square or panmagic square is a magic square with the additional property that the broken diagonals, i.e...

s.

Apart from the trivial case of the first order square, most-perfect magic squares are all of order 4n. In their book, Kathleen Ollerenshaw

Kathleen Ollerenshaw

Dame Kathleen Mary Ollerenshaw, née Timpson, DBE is a British mathematician and politician. Deaf since the age of eight, she loved doing arithmetic problems as a child. As a young woman, she attended St Leonards School and Sixth Form College in St Andrews, Scotland where today the house of young...

 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

Bijection

A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

 between reversible magic squares and most-perfect magic squares.

For n = 36, there are about 2.7 × 10⁴⁴ essentially different

Frénicle standard form

A magic square is in Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions apply:# the element at position [1,1] is the smallest of the four corner elements; and...

most-perfect magic squares.

External links

• A051235: Number of essentially different most-perfect pandiagonal magic squares of order 4n from The On-Line Encyclopedia of Integer Sequences