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Most-perfect magic square from the Parshvanath Jain temple in Khajuraho |

A **most-perfect magic square** of doubly even order *n* = 4*k* is a pan-diagonal magic square containing the numbers 1 to *n*^{2} with three additional properties:

- Examples
- Properties
- Physical properties
- Generalizations
- Most-perfect magic cubes
- See also
- Notes
- References
- External links

- Each 2×2 subsquare, including wrap-round, sums to
*s*/*k*, where*s*=*n*(*n*^{2}+ 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*n*^{2}+ 1).

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.

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] }

The 2021 update below shows how the 2x2 cell block sums are preserved in a row / col translation.

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, 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 × 10^{44} 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 |

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] }

There are 108 of these 2x2 subsquares that have the same sum for the 4x4x4 most-perfect cube.^{ [5] }^{ }

- Sriramachakra
- Pandiagonal magic square (diabolic square)

- ↑ 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

In recreational mathematics, a square array of numbers, usually positive integers, is called a **magic square** if the sums of the numbers in each row, each column, and both main diagonals are the same. The integer is the **order** of the magic square and the constant sum is called the **magic constant**. If the array includes just the positive integers , the magic square is said to be **normal**. Some authors take magic square to mean normal magic square.

In mathematics, a **magic cube** is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a *n* × *n* × *n* pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal to the same number, the so-called magic constant of the cube, denoted *M*_{3}(*n*). It can be shown that if a magic cube consists of the numbers 1, 2, ..., *n*^{3}, then it has magic constant

In mathematics, a **perfect magic cube** is a magic cube in which not only the columns, rows, pillars, and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant.

In mathematics, a ** P-multimagic square** is a magic square that remains magic even if all its numbers are replaced by their

**35** (**thirty-five**) is the natural number following 34 and preceding 36.

**1000** or **one thousand** is the natural number following 999 and preceding 1001. In most English-speaking countries, it is often written with a comma separating the thousands unit: **1,000**.

**300** is the natural number following 299 and preceding 301.

**2000** is a natural number following 1999 and preceding 2001.

**126** is the natural number following 125 and preceding 127.

The **magic constant** or **magic sum** of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. In general where is the side length of the square.

**3000** is the natural number following 2999 and preceding 3001. It is the smallest number requiring thirteen letters in English.

**4000** is the natural number following 3999 and preceding 4001. It is a decagonal number.

**8000** is the natural number following 7999 and preceding 8001.

**Dame Kathleen Mary Ollerenshaw**, was a British mathematician and politician who was Lord Mayor of Manchester from 1975 to 1976 and an advisor on educational matters to Margaret Thatcher's government in the 1980s.

Every magic cube may be assigned to one of six **magic cube classes**, based on the cube characteristics.

A **Diagonal Magic Cube** is an improvement over the *simple* magic cube. It is the second of six magic cube classes when ranked by the number of lines summing correctly.

In recreational mathematics, a **pandiagonal magic cube** is a magic cube with the additional property that all broken diagonals have the same sum as each other. Pandiagonal magic cubes are extensions of diagonal magic cubes and generalize pandiagonal magic squares to three dimensions.

**Walter Trump** is a German mathematician. He is known for his work in recreational mathematics.

A * Nasik magic hypercube* is a magic hypercube with the added restriction that all possible lines through each cell sum correctly to where

**Water retention on mathematical surfaces** is the catching of water in ponds on a surface of cells of various heights on a regular array such as a square lattice, where water is rained down on every cell in the system. The boundaries of the system are open and allow water to flow out. Water will be trapped in ponds, and eventually all ponds will fill to their maximum height, with any additional water flowing over spillways and out the boundaries of the system. The problem is to find the amount of water trapped or retained for a given surface. This has been studied extensively for two mathematical surfaces: magic squares and random surfaces. The model can also be applied to the triangular grid.

- 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

- T. V. Padmakumar, Strongly magic squares
- Harvey Heinz: Most-perfect Magic Squares
- OEIS sequenceA051235(Number of essentially different most-perfect pandiagonal magic squares of order 4n)
- OEIS sequenceA270205(Number of 2 X 2 planar subsets in an n X n X n cube)
- OEIS sequenceA275359(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

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