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A most-perfect magic square of order n is a magic square containing the numbers 1 to n with two additional properties:

1. Each 2 × 2 subsquare sums to 2s, where s = n + 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:

                  
    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
  123    7  106    5  139   22  120   36  137    38   104    21
  124   32  141   34  108   17  127    3  110     1   143    18
   71   59   54   57   87   74   68   88   85    90    52    73

                  
     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
  106   63   96   45  107   55   79   38   89    56    78    46
  115    2  125   20  114   10  142   27  132     9   143    19
   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 × 10 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

v t e Magic polygons
Types	Magic circle Magic hexagon Magic hexagram Magic square Magic star Magic triangle
Related shapes	Alphamagic square Antimagic square Geomagic square Heterosquare Pandiagonal magic square Prime reciprocal magic square Most-perfect magic square
Higher dimensional shapes	Magic cube classes Magic hypercube Magic hyperbeam
Classification	Associative magic square Pandiagonal magic square Multimagic square
Related concepts	Latin square Word square Number Scrabble Eight queens puzzle Magic constant Magic graph Magic series

• Magic squares