View source for Most-perfect magic square

[[File:2152085cab.png|thumb|right|Most-perfect magic square from the Parshvanath Jain temple in Khajuraho, India]]

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| align="center" | transcription of<br />the [[Indian numerals|indian numerals]]
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A '''most-perfect magic square''' of order ''n'' is a [[Magic square|magic square]] containing the numbers 1 to ''n''<sup>2</sup> with two additional properties:

# Each 2 × 2 subsquare sums to 2''s'', where ''s''&nbsp;=&nbsp;''n''<sup>2</sup>&nbsp;+&nbsp;1.
# All pairs of integers distant ''n''/2 along a (major) diagonal sum to ''s''.

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==Examples==
[[File:SeiRamaChakramMagicSquare.jpg|200px|thumb|Image of [[Sriramachakra|Sriramachakra]] as a most-perfect magic square given in the Panchangam published by Sringeri Sharada Peetham.]]
[[File:most_perfect_magic_square_construction.svg|200px|thumb|Construction of a fourth-order most-perfect magic square from a [[Latin square|Latin square]] with distinct diagonals, M, and its [[Transpose|transpose]], M<sup>T</sup>.]]
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 4''n''. In their book, [[Biography:Kathleen Ollerenshaw|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 [[Bijection|one-to-one correspondence]] between reversible squares and most-perfect magic squares.

For ''n''&nbsp;=&nbsp;36, there are about 2.7&nbsp;×&nbsp;10<sup>44</sup> [[Frénicle standard form|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'', [http://www.surabooks.com/ProductDisplay.asp?RefID=2800 Sura books] {{Webarchive|url=https://web.archive.org/web/20100225233941/http://www.surabooks.com/ProductDisplay.asp?RefID=2800 |date=2010-02-25 }}, India, 2008, 128 pages, ISBN 978-81-8449-321-4

==External links==
* [http://www.fq.math.ca/Scanned/35-3/padmakumar.pdf STRONGLY MAGIC SQUARES by T. V. Padmakumar]
* {{OEIS2C|A051235}} Number of essentially different most-perfect pandiagonal magic squares of order 4n from The [[On-Line Encyclopedia of Integer Sequences]]
{{Magic polygons}}
[[Category:Magic squares]]

{{Sourceattribution|Most-perfect magic square}}