Most-perfect magic square
A most-perfect magic square of doubly even order n = 4k is a pan-diagonal magic square containing the numbers 1 to n2 with three additional properties:
- Each 2×2 subsquare, including wrap-round, sums to s/k, where s = n(n2 + 1)/2 is the magic sum.
- All pairs of integers distant n/2 along any diagonal (major or broken) are complementary (i.e. they sum to n2 + 1).
![]() |
| |||||||||||||||||||||||||||||||||||
Most-perfect magic square from the Parshvanath Jain temple in Khajuraho |
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] Only 16 of the 64 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]
|
|
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.
The number of essentially different most-perfect magic squares of order n for 4n = 1, 2, ... form the sequence:
- 48, 368640, 22295347200, 932242784256000, 144982397807493120000, ... (sequence A051235 in the OEIS).
For example, there are about 2.7 × 1044 essentially different most-perfect magic squares of order 36.
All order four panmagic squares are most-perfect magic squares. The second property implies that each pair of the integers with the same background color 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 |
Physical properties
The image below shows areas completely surrounded by larger numbers with a blue background. A water retention topographical model is one example of the physical properties of magic squares. The water retention model progressed from the specific case of the magic square to a more generalized system of random levels. A quite interesting counter-intuitive finding that a random two-level system will retain more water than a random three-level system when the size of the square is greater than 51 X 51 was discovered. This was reported in the Physical Review Letters in 2012 and referenced in the Nature article in 2018.[3][4]
![](../I/m/Most-perfect_magic_square.jpg)
Generalizations
Most-perfect magic cubes
There are 108 of these 2x2 subsquares that have the same sum for the 4x4x4 most-perfect cube.[5]
![](../I/m/Most-perfect_magic_cube.png)
See also
- Sriramachakra
- Pandiagonal magic square (diabolic square)
Notes
- F1 Compiler http://www.f1compiler.com/samples/Most%20Perfect%20Magic%20Square%208x8.f1.html
- http://budshaw.ca/Reversible.html Reversible Squares, S. Harry White, 2014
- Knecht, Craig; Walter Trump; Daniel ben-Avraham; Robert M. Ziff (2012). "Retention capacity of random surfaces". Physical Review Letters. 108 (4): 045703. arXiv:1110.6166. Bibcode:2012PhRvL.108d5703K. doi:10.1103/PhysRevLett.108.045703. PMID 22400865.
- https://oeis.org/A201126 OEIS A201126
- 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
- OEIS sequence A051235 (Number of essentially different most-perfect pandiagonal magic squares of order 4n)
- OEIS sequence A270205 (Number of 2 X 2 planar subsets in an n X n X n cube)
- OEIS sequence A275359 (Maximum incarceration of numbers in an n X n X n number cubes with full incarceration volumes) -- Incarcetration
- B. Burger, J. S. Andrade Jr., H. J. Herrmann (2018). "A comparison of hydrological and topological watersheds". Scientific Reports. 8 (1): 10586. Bibcode:2018NatSR...810586B. doi:10.1038/s41598-018-28470-2. PMC 6043487. PMID 30002379.CS1 maint: multiple names: authors list (link)
- Pattern overlay of every other cell being surrounded by four larger cell values introduced