\nSum = 840
\nProduct ={{val2058068231856000}}"}},"i":0}}]}" id="mwDD8">.mwparseroutput .nobold{fontweight:normal}8×8 found by W. W. Horner in 1955
Sum = 840
Product = 2058068231856000
156  18  48  25 
30  144  60  13 
16  20  130  81 
45  65  9  128 
It is unknown if any additivemultiplicative magic squares smaller than 8×8 exist, but it has been proven that no 3×3 or 4×4 additivemultiplicative magic squares and no 3×3 additivemultiplicative semimagic squares exist.^{ [80] }
Magic squares may be constructed which contain geometric shapes instead of numbers. Such squares, known as geometric magic squares, were invented and named by Lee Sallows in 2001.^{ [81] }
In the example shown the shapes appearing are two dimensional. It was Sallows' discovery that all magic squares are geometric, the numbers that appear in numerical magic squares can be interpreted as a shorthand notation which indicates the lengths of straight line segments that are the geometric 'shapes' occurring in the square. That is, numerical magic squares are that special case of a geometric magic square using one dimensional shapes.^{ [82] }
In 2017, following initial ideas of William Walkington and Inder Taneja, the first linear area magic square (LAMS) was constructed by Walter Trump.^{ [83] }
Other two dimensional shapes than squares can be considered. The general case is to consider a design with N parts to be magic if the N parts are labeled with the numbers 1 through N and a number of identical subdesigns give the same sum. Examples include magic circles, magic rectangles, magic triangles ^{ [84] } magic stars, magic hexagons, magic diamonds. Going up in dimension results in magic spheres, magic cylinders, magic cubes, magic parallelepiped, magic solids, and other magic hypercubes.
Possible magic shapes are constrained by the number of equalsized, equalsum subsets of the chosen set of labels. For example, if one proposes to form a magic shape labeling the parts with {1, 2, 3, 4}, the subdesigns will have to be labeled with {1,4} and {2,3}.^{ [84] }
In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into nqueens solutions, and vice versa.^{ [85] }
Magic squares of order 3 through 9, assigned to the seven planets, and described as means to attract the influence of planets and their angels (or demons) during magical practices, can be found in several manuscripts all around Europe starting at least since the 15th century. Among the best known, the Liber de Angelis, a magical handbook written around 1440, is included in Cambridge Univ. Lib. MS Dd.xi.45.^{ [86] } The text of the Liber de Angelis is very close to that of De septem quadraturis planetarum seu quadrati magici, another handbook of planetary image magic contained in the Codex 793 of the Biblioteka Jagiellońska (Ms BJ 793).^{ [87] } The magical operations involve engraving the appropriate square on a plate made with the metal assigned to the corresponding planet,^{ [88] } as well as performing a variety of rituals. For instance, the 3×3 square, that belongs to Saturn, has to be inscribed on a lead plate. It will, in particular, help women during a difficult childbirth.
In about 1510 Heinrich Cornelius Agrippa wrote De Occulta Philosophia, drawing on the Hermetic and magical works of Marsilio Ficino and Pico della Mirandola. In its 1531 edition, he expounded on the magical virtues of the seven magical squares of orders 3 to 9, each associated with one of the astrological planets, much in the same way as the older texts did. This book was very influential throughout Europe until the counterreformation, and Agrippa's magic squares, sometimes called kameas, continue to be used within modern ceremonial magic in much the same way as he first prescribed.^{ [89] }
The most common use for these kameas is to provide a pattern upon which to construct the sigils of spirits, angels or demons; the letters of the entity's name are converted into numbers, and lines are traced through the pattern that these successive numbers make on the kamea. In a magical context, the term magic square is also applied to a variety of word squares or number squares found in magical grimoires, including some that do not follow any obvious pattern, and even those with differing numbers of rows and columns. They are generally intended for use as talismans. For instance the following squares are: The Sator square, one of the most famous magic squares found in a number of grimoires including the Key of Solomon ; a square "to overcome envy", from The Book of Power;^{ [90] } and two squares from The Book of the Sacred Magic of Abramelin the Mage , the first to cause the illusion of a superb palace to appear, and the second to be worn on the head of a child during an angelic invocation:




A magic square in a musical composition is not a block of numbers – it is a generating principle, to be learned and known intimately, perceived inwardly as a multidimensional projection into that vast (chaotic!) area of the internal ear – the space/time crucible – where music is conceived. ... Projected onto the page, a magic square is a dead, black conglomeration of digits; tune in, and one hears a powerful, orbiting dynamo of musical images, glowing with numen and lumen.^{ [92] }
Isomura Kittoku.
Isomura Kittoku.
Isomura Kittoku.
Isomura Kittoku.
The two corresponding prizes are still to be won!
The Parker Square is a mascot for people who give it a go but ultimately fall short.
Moreover, it's a magic square, a pattern in which God supposedly instructed the early Hebrews to gain power from names or their numeric equivalents.
I love when they bring the nerdy FBI guy in to explain the concept of “the magic square,” which he does by telling us that magic squares have been around for a while, and then nothing else. Unless I missed something, all I have at this point is that magic squares are squares that people once thought were magic.
The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, column, or diagonal. The eight queens puzzle is an example of the more general n queens problem of placing n nonattacking queens on an n×n chessboard, for which solutions exist for all natural numbers n with the exception of n = 2 and n = 3.
In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 Latin square is
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.
In mathematics, a magic cube is the 3dimensional 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 socalled 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 combinatorics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality in combinatorics is strongly related to the concept of blocking in statistics, which ensures that independent variables are truly independent with no hidden confounding correlations. "Orthogonal" is thus synonymous with "independent" in that knowing one variable's value gives no further information about another variable's likely value.
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.
Lo Shu Square, or the Nine Halls Diagram, is the unique normal magic square of order three. The Lo Shu is part of the legacy of ancient Chinese mathematical and divinatory traditions, and is an important emblem in Feng Shui (風水), the art of geomancy concerned with the placement of objects in relation to the flow of qi (氣) "natural energy".
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.
A mostperfect magic square of doubly even order n = 4k is a pandiagonal magic square containing the numbers 1 to n^{2} with three additional properties:
An antimagic square of order n is an arrangement of the numbers 1 to n^{2} 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.
The Strachey method for magic squares is an algorithm for generating magic squares of singly even order 4k + 2. An example of magic square of order 6 constructed with the Strachey method:
Location arithmetic is the additive (nonpositional) binary numeral systems, which John Napier explored as a computation technique in his treatise Rabdology (1617), both symbolically and on a chessboardlike grid.
Every magic cube may be assigned to one of six magic cube classes, based on the cube characteristics.
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.
This is a glossary of Sudoku terms and jargon. It is organized thematically, with links to references and example usage provided as ([1]). Sudoku with a 9×9 grid is assumed, unless otherwise noted.
In recreational mathematics and the theory of magic squares, a broken diagonal is a set of n cells forming two parallel diagonal lines in the square. Alternatively, these two lines can be thought of as wrapping around the boundaries of the square to form a single sequence.
The Siamese method, or De la Loubère method, is a simple method to construct any size of nodd magic squares. The method was brought to France in 1688 by the French mathematician and diplomat Simon de la Loubère, as he was returning from his 1687 embassy to the kingdom of Siam. The Siamese method makes the creation of magic squares straightforward.
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.
A geometric magic square, often abbreviated to geomagic square, is a generalization of magic squares invented by Lee Sallows in 2001. A traditional magic square is a square array of numbers whose sum taken in any row, any column, or in either diagonal is the same target number. A geomagic square, on the other hand, is a square array of geometrical shapes in which those appearing in each row, column, or diagonal can be fitted together to create an identical shape called the target shape. As with numerical types, it is required that the entries in a geomagic square be distinct. Similarly, the eight trivial variants of any square resulting from its rotation and/or reflection are all counted as the same square. By the dimension of a geomagic square is meant the dimension of the pieces it uses. Hitherto interest has focused mainly on 2D squares using planar pieces, but pieces of any dimension are permitted.
Wikisource has the text of the 1911 Encyclopædia Britannica article Magic Square . 