 | This article needs additional citations for verification .(July 2021) |
| ||||||||||||||||
| transcription of the indian numerals |
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.
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle.
In mathematics, especially historical and 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".
The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3.
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). 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.
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, and is its chairman.
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.
An antimagic square of order n is an arrangement of the numbers 1 to n2 in a square, such that the sums of the n rows, the n columns and the two diagonals form a sequence of 2n + 2 consecutive integers. The smallest antimagic squares have order 4. Antimagic squares contrast with magic squares, where each row, column, and diagonal sum must have the same value.
In mathematics, a magic cube of order is an grid of natural numbers satisying the property that the numbers in the same row, the same column, the same pillar or the same length- diagonal add up to the same number. It is a -dimensional generalisation of the magic square. A magic cube can be assigned to one of six magic cube classes, based on the cube characteristics. A benefit of this classification is that it is consistent for all orders and all dimensions of magic hypercubes.
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 series is a set of distinct positive integers which add up to the magic constant of a magic square and a magic cube, thus potentially making up lines in magic tesseracts.
168 is the natural number following 167 and preceding 169.
177 is the natural number following 176 and preceding 178.
An associative magic square is a magic square for which each pair of numbers symmetrically opposite to the center sum up to the same value. For an n × n square, filled with the numbers from 1 to n2, this common sum must equal n2 + 1. These squares are also called associated magic squares, regular magic squares, regmagic squares, or symmetric magic squares.
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 entries are members of a fixed, finite, and discrete set, referred to as the "population". The term "Bohemian" was first used to refer to matrices with entries consisting of integers of bounded height, hence the name, derived from the acronym BOunded HEight Matrix of Integers (BOHEMI). The majority of published research on these matrix families studies populations of integers, although this is not strictly true of all possible Bohemian matrices. There is no single family of Bohemian matrices. Instead, a matrix can be said to be Bohemian with respect to a set from which its entries are drawn. Bohemian matrices may possess additional structure. For example, they may be Toeplitz matrices or upper Hessenberg matrices.
| Types |  | |
|---|---|---|
| Related shapes | ||
| Higher dimensional shapes | ||
| Classification | ||
| Related concepts | ||