Magic Square Brief History Of Magic Squares The Lo Shu Square

More formally, a magic square can be defined as an n-by-n matrix such that the sum of any row, column or main diagonal yields the same result (the square's magic constant, denoted M₂(n)); if these numbers are 1, 2,..., n², then

1 Brief history of magic squares

1.1 The Lo Shu Square

The Lo Shu Square, as the magic square on the turtle shell is called, is a unique normal magic square of order three.

1.2 The earliest square of order four

The earliest magic square of order four was found inscribed Khajuraho, India, dating from the eleventh or twelfth century; it is also a so-called diabolic or panmagic (pandiagonal magic square) where, in addition to the rows, columns and main diagonals, the broken diagonals also have the same sum.

1.3 Cultural significance of magic squares

Magic Squares have fascinated humanity throughout the ages, and have been around for over 4,000 years. They were frequently found in a number of cultures, including Egypt and India, engraved on stone or metal and worn as talismans, the belief being that magic squares had astrological and divinatory qualities, their usage ensuring longevity, and prevention against diseases.

The Kubera-Kolam is a floor painting used in India which is in the form of a magic square of order three. It begins with the number twenty and ends with the number twenty-eight.

The magic square figures in Greek writings dating from about 1300 BC and was used by Arabian astrologers in the ninth century when drawing up horoscopes.

1.4 Albrecht Dürer's magic square

The 4×4 magic square in Albrecht DürerAlbrecht Durer ( May 21, 1471 April 6, 1528) was a German painter, wood carver and engraver. He is best known for his woodcuts in series, including the Apocalypse ( 1498), two series on the crucifixion of Christ, the Great Passion ( 1498- 1510) and the Li's engraving Melancholia IAlbrecht Durer's engraving Melancholia I (originally known by Durer as Melencolia I is notable for being an allegorical depiction of the main symptoms of Melancholy, now better known as depression. Allusions and imagery used by Durer in Melancholia I The is believed to be the first seen in EuropeFor the band of the same name, see Europe (band . Europe is a continent forming the westermost part of the Eurasian supercontinent. Europe is bounded to the north by the Arctic Ocean, to the west by the Atlantic Ocean, to the south by the Mediterranean Sean art. The sum 34 can be found in the rows, columns, diagonals, any 2×2 block of numbers, the sum of the four corners, the sums of the four outer numbers clockwise from the corners (3 + 8 + 14 + 9) and likewise the four counter-clockwise, and the sum of the middle two entries of the two outer columns and rows (e.g. 5 + 9 + 8 + 12), as well as several kite-shaped quartets, e.g. 3 + 5 + 11 + 15; the two numbers in the middle of the bottom row give the date of the engraving: 1514Events March Louis XII of France makes peace with Emperor Maximilian. July Peace between England and France. Albrecht Durer makes his famous engraving Melancholia I''. Births March 8 Amago Haruhisa, Japanese samurai and warlord Shimazu Takahisa, Japanese.

It has been known since 1693 that there exist 880 basic (excluding those obtained by rotationThis article is about rotation as a movement of a physical body. For other meanings, see rotation (disambiguation). Rotation is the movement of a body in such a way that any given point of that body remains at a constant distance from some other fixed poi and reflectionThe term reflection (also spelt reflexion can refer to several different concepts: In mathematics, reflection is the transformation of a space. In physics, reflection is a wave phenomenon. In computer science, reflection is a programming language feature) 4×4 magic squares and 275,305,224 basic 5×5 magic squares. The number of basic magic squares of any higher degree is not yet known but it was estimated by Klaus Pinn and C. Wieczerkowski (1998) using Monte Carlo simulationMonte Carlo methods are algorithms for solving various kinds of computational problems by using random numbers (or more often pseudo-random numbers), as opposed to deterministic algorithms. Monte Carlo methods are extremely important in computational phys and methods from statistical mechanicsStatistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. It provides a fr to be (1.7745 ± 0.0016) × 10¹⁹ in the 6×6 case squares and (3.7982 ± 0.0004) × 10³⁴ in the 7×7 case.

1.5 The Sagrada Família magic square

The Passion façade of the Sagrada Família church in Barcelona, designed by sculptor Josep Subirachs , features a 4×4 magic square:

The magic sum of the square is 33, the age of Jesus at the time of the Passion. Structurally, it is very similar to the Melancholia magic square, but it has had the numbers in four of the cells reduced by 1.

2 Types of magic squares and their construction

There are many ways to construct magic squares, but the standard (and most simple) way is to follow certain configurations / formulas which generate regular patterns. Magic squares exist for all values of n, with only one exception - it is impossible to construct a magic square of order 2. Magic squares can be classified into three types: odd, doubly even (n divisible by four) and singly even (n even, but not divisible by four). Odd and doubly even magic squares are easy to generate; the construction of singly even magic squares is more difficult but several methods exist, including the LUX method for magic squares (due to John Conway) and the Strachey method for magic squares. Only odd and doubly even magic squares are discussed below.

2.1 A method for constructing a magic square of odd order

Starting from the central column of the last row with the number 1, the fundamental movement for filling the squares is diagonally down and right, one step at a time. If a filled square is encountered, one moves vertically up one square, then continuing as before. When a move would leave the square, it is wrapped around to the first row or last column, respectively.

The same pattern can be achieved starting from the central column of the first row; In this case the fundamental movement is diagonally up and left, one step at a time, and if a filled square is encountered, one moves vertically down one square, then continuing as before. When a move would leave the square, it is wrapped around the last row or first column, respectively.

Similar patterns can also be obtained by starting from other squares.


Order 3	Order 5	Order 9

2.2 A method of constructing a magic square of doubly even order

All the numbers are written in order from right to left across each row in turn, starting from the top right hand corner. Numbers are then either retained in the same place or interchanged with their diametrically opposite numbers in a certain regular pattern. In the magic square of order four, the numbers in the four central squares and one square at each corner are retained in the same place and the others are interchanged with their diametrically opposite numbers. In the magic square of order eight, the same is done; the 16 central squares and 4 squares at each corner are retained in their places and the rest are switched.

A general rule: If n represents the order of the doubly even square, retain numbers in the following pattern. The central square with sides of legnth n/2 should be retained. Also retain the squares with sides of legnth n/4 in each of the four corners.

Order 8

3 Related Problems

3.1 Latin Squares

Euler showed how to derive magic squares from latin squares.

3.2 Magic Problems

Certain other restrictions can be imposed on magical squares, resulting, for example, in bimagic, trimagic and multimagic squares, and there are also other forms displaying similar characteristics, including magic circles, magic polygon s, and magic cubes.

3.3 n-Queens Problem

Paul Muljadi discovered and proved the n-Queens Problem is related to Magic squares because the Magic constant of n Queens Problem is also the Magic constant of Magic Squares of order n > 3.

In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into N-queens solutions, and vice versa.

4 See also

5 External links

6 References

W. S. Andrews, Magic Squares and Cubes. (New York: Dover, 1960), originally printed in 1917
John Lee Fults, Magic Squares. (La Salle, Illinois: Open Court, 1974).
Cliff Pickover, The Zen of Magic Squares, Circles, and Stars (Princeton, New Jersey: Princeton University Press)

Recreational mathematics *

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