Most-perfect magic square
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Most-perfect magic square from the Parshvanath Jain temple in Khajuraho |
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
Specific examples of most-perfect magic squares that begin with the 2015 date demonstrate how theory and computer science are able to define this group of magic squares. [1] [2] Only 16 of the 49 2x2 cell blocks that sum to 130 are accented by the different colored fonts in the 8x8 example.
![](../I/m/Magic_Square_2015.jpeg)
The 12x12 square below was found by making all the 42 principal reversible squares with ReversibleSquares, running Transform1 2All on all 42, making 23040 of each, (of the 23040 x 23040 total each), then making the most-perfect squares from these with ReversibleMost-Perfect. These squares were then scanned for squares with 20,15 in the proper cells for any of the 8 rotations. The 2015 squares all originated with principal reversible square number #31. This square has values that sum to 35 on opposite sides of the vertical midline in the first two rows.[2]
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Physical Properties
The image below shows numbers completely surrounded by larger numbers with a blue background.
![](../I/m/Most-perfect_magic_square.jpg)
Most-perfect magic square upgraded to a cube
There are 108 of these 2x2 subsquares that have the same sum for the 4x4x4 most-perfect cube.[3]
![](../I/m/Most-perfect_magic_cube.png)
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.
The second property above implies that each pair of the integers with the same background colour in the 4×4 square below have the same sum, and hence any 2 such pairs sum to the magic constant.
7 | 12 | 1 | 14 |
2 | 13 | 8 | 11 |
16 | 3 | 10 | 5 |
9 | 6 | 15 | 4 |
See also
- Sriramachakra
- Pandiagonal magic square (diabolic square)
Notes
- ↑ F1 Compiler http://www.f1compiler.com/samples/Most%20Perfect%20Magic%20Square%208x8.f1.html
- 1 2 Harry White, http://budshaw.ca/Most-perfect.html
- ↑ https://oeis.org/A270205 OEIS A270205
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
- T. V. Padmakumar, Strongly magic squares
- Harvey Heinz: Most-perfect Magic Squares
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A051235, number of essentially different most-perfect pandiagonal magic squares of order 4n, at the On-Line Encyclopedia of Integer Sequences
- Most-perfect space,
- Incarceration,