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A most-perfect magic square of order n is a magic square containing the numbers 1 to n2 with two additional properties:
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
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.
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 'order' of the magic square is the number of integers along one side (n), 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.
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures.
In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns.
9 (nine) is the natural number following 8 and preceding 10.
111 is the natural number following 110 and preceding 112.
In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an 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, the so-called magic constant of the cube, denoted M3(n). It can be shown that if a magic cube consists of the numbers 1, 2, ..., n3, then it has magic constant (sequence A027441 in the OEIS)
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 kth powers for 1 ≤ k ≤ P. 2-multimagic squares are called bimagic, 3-multimagic squares are called trimagic, 4-multimagic squares tetramagic, and 5-multimagic squares pentamagic.
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
A pandiagonal magic square or panmagic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is the chairman of the OEIS Foundation.
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 pantriagonal magic cube is a magic cube where all 4m2 pantriagonals sum correctly. There are 4 one-segment pantriagonals, 12(m − 1) two-segment pantriagonals, and 4(m − 2)(m − 1) three-segment pantriagonals. This class of magic cubes may contain some simple magic squares and/or pandiagonal magic squares, but not enough to satisfy any other classifications.
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.
A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant M. A normal magic hexagon contains the consecutive integers from 1 to 3n2 − 3n + 1. It turns out that normal magic hexagons exist only for n = 1 and n = 3. Moreover, the solution of order 3 is essentially unique. Meng also gave a less intricate constructive proof.
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers.
In mathematics, a power of three is a number of the form 3n where n is an integer, that is, the result of exponentiation with number three as the base and integer n as the exponent.
Sriramachakra is a mystic diagram or a yantra given in Tamil almanacs as an instrument of astrology for predicting one's future. The geometrical diagram consists of a square divided into smaller squares by equal numbers of lines parallel to the sides of the square. Certain integers in well defined patterns are written in the various smaller squares. In some almanacs, for example, in the Panchangam published by the Sringeri Sharada Peetham or the Pnachangam published by Srirangam Temple, the diagram takes the form of a magic square of order 4 with certain special properties. This magic square belongs to a certain class of magic squares called strongly magic squares which has been so named and studied by T V Padmakumar, an amateur mathematician from Thiruvananthapuram, Kerala. In some almanacs, for example, in the Pambu Panchangam, the diagram consists of an arrangement of 36 small squares in 6 rows and 6 columns in which the digits 1, 2, ..., 9 are written in that order from left to right starting from the top-left corner, repeating the digits in the same direction once the digit 9 is reached.
A Bohemian matrix family is a set of matrices whose free entries come from a single discrete, usually finite population, denoted here by P. That is, each entry of any matrix from this particular Bohemian matrix family must be an element of P. Such a matrix is called a Bohemian matrix. Bohemian matrices can have other structures as well, such as being a Toeplitz matrix or an upper Hessenberg matrix. Usually, only one Bohemian matrix family with a fixed population P is studied at a time, and so one can classify any given matrix as being Bohemian or not, without significant ambiguity.
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