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In mathematics, magic squares consist of a number of
integers arranged in the form of a square in such a way that the sum of the numbers
in every row, column and diagonal are the same. A magic square may have odd or even number of rows and columns. Usually the magic
square is filled up by consecutive numbers from one to N2 where N is the number of rows or columns.
A magic square is designated with reference to this. Thus a magic square of order N will have N number of rows
and columns and will be filled by numbers ranging from one to N2.
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
Brief history of magic squares
The Lo Shu Square
Chinese literature dating from as early as 2800 BC tells the legend of Lo Shu or "scroll of the river Lo": in ancient China, there was a
huge flood. The people tried to offer some sacrifice to the river god of one of the flooding rivers, the Lo river, to calm his
anger. Then, there emerged from the water a turtle with a curious figure/pattern on its shell; there were circular dots of
numbers that were arranged in a three by three nine-grid pattern such that the sum of the numbers in each row, column and
diagonal was the same: 15. This number is also equal to the 15 days in each of the 24 cycles of the Chinese solar year. This
pattern, in a certain way, helped in controlling the river.
The Lo Shu Square, as the magic square on the turtle shell is called,
is a unique normal magic square of order three.
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.
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.
Albrecht Dürer's magic square
The 4×4 magic square in Albrecht Dürer's engraving Melancholia I is believed to be the first seen in European 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: 1514.
It has been known since 1693 that there exist 880 basic (excluding those obtained by rotation and reflection) 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 simulation and methods from statistical mechanics to be (1.7745 ± 0.0016) × 1019
in the 6×6 case squares and (3.7982 ± 0.0004) × 1034 in the 7×7 case.
The Passion façade of the Sagrada Família church in Barcelona, designed by sculptor Josep Subirachs, features a 4×4
magic square:
4x4 magic square in architecture
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.
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, originally due to
John Conway. Only odd and doubly even magic squares are discussed below.
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 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 column, respectively.
Similar patterns can also be obtained by starting from other squares.
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.
Related Problems
Latin Squares
Euler showed how to derive magic squares from latin squares.
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 polygons, and magic cubes.
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.
See also
External links
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)
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