Magic square
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. This sum can be calculated using the formula [N x [N2+1]]/ 2.More formally, a magic square can be defined as an n-by-n matrix containing the numbers 1, 2,..., n2 such that the sum of any row, column or main diagonal yields the same result. All these sums are then necessarily equal to n × (n2 + 1) / 2.
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. There are many ways to arrive at magic squares but the standard and simplest way is to follow certain configurations/formulae which generate regular patterns. More sophisticated magic squares also produce the sum along the two diagonals; some 4x4 squares also give the sum in any small 2x2 block of four numbers.There are also other forms displaying similar characteristics - magic circles, magic polygons, even magic cubes extending the concept to the third dimension.
Chinese literature dating from as early as 2800 BC talks about the legend of Lo Shu or 'scroll of the river Lo'. In the ancient time of 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 an important part of Feng Shui, the ancient Chinese art of geomancy. Feng Shui, literally meaning wind and water, specifies principles of designing environments in tune with elemental forces that may be natural or supernatural. It is beyond the scope of the discussion to bring in more details and comparisons. But an interesting fact is that traditional Chinese cities and temples were laid out in a square broken into nine sections. The odd numbers in the Lo Shu Square are male or yang and even numbers are female or yin. The numbers 1- the beginning of all things and 9- representing completion are considered most auspicious. The number 5 at the centre is the most powerful. The Lo Shu square in the form of a trigram, gives the basis for determining the orientation of buildings. The Lo shu square is also a diagrammatic representation of the seasons showing the ratio of yin- the feminine force and yang- the masculine force in the annual cycle.
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.
In Khajuraho in India was found inscribed the earliest fourth order magic square, dating from the eleventh or twelfth century. This was a peculiar type known as the diabolic or pandiagonal magic square where even the broken diagonals have the same sum.
Magic squares were also frequently found in various cultures, for example, Egypt and India, engraved on stone or metal and worn as talismans, the belief being that magic squares had astrological and divinatory qualities- longevity and disease prevention [against plague etc.,] could be ensured by their usage. 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 4x4 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 2x2 block of numbers, the sum of the four corners, and the sum of the midddle two entries of the two outer columns and rows (eg 5 + 9 + 8 + 12). 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) 4x4 magic squares and 275305224 basic 5x5
magic squares. The number of basic magic squares of any higher degreee 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 6x6 case squares and (3.7982 ± 0.0004) × 1034 in the 7x7 case.
Magic squares are thus universal symbols that are seen in many cultures.
Similar patterns can also be repeated starting from certain other squares, but this is the simplest method.
The starting point could be from any corner and directions can change accordingly. Other regular patterns can also generate a doubly even type but this is the simplest.
'See als:Satanic square, Diabolic square, Prime reciprocal magic square, Magic Star
Brief history of magic squares
Magic Squares have fascinated humanity throughout the ages and have been around for over 4,000 years.Types of magic squares
Magic squares exist for all values of N with only one sole exception- it is impossible to construct a magic square of order 2. Magic squares can be classified into three types – odd, doubly even and singly even. The difference between doubly even and singly even is that whereas in the former N is divisible by four, in the latter N is divisible only by two. The odd number magic square and the doubly even magic square are easy to generate, particularly the former. However, the singly even magic square is more difficult to generate. In this context only the odd number and doubly even types are discussed.Method of constructing a magic square of odd order N
Starting from the central column of the last row with the number 1, the fundamental movement for filling the squares is diagonally down, right- one step at a time. If a filled square is encountered the movement is vertically up one square and then continuing as before. If a move goes outside the square, then it is considered in the first row or column as the case may be. The same pattern can be achieved starting from the central column of the first row. In this case the fundamental movement is diagonally up, left- one step at a time. If a filled square is encountered the movement is vertically down one square and then continuing as before. If a move goes outside the square, then it is considered as last row or column as the case may be.Method of constructing a magic square of order 3
Each square is named as in a matrix-the square constituting the first row, first column is [1,1] and that constituting the last row, last column is [3,3]. Starting from last row, middle column [3,2] enter the number 1. The next square to be filled is diagonally down, right. But this is [4,3] which goes outside the magic square. Instead of this use [1,3], 1 taking the place of 4. So the next number 2 is entered in [1,3]. The next square is [2,4], hence it has to be read as [2,1], enter 3 into this square. The next square is [3,2] which is what was started with in the first place and is already filled. So go up vertically one step from [2,1] instead of diagonal down, right. This leads to [1,1],enter the number 4 here. The next number 5 is entered in [2,2] and 6 in [3,3] both going diagonally down, right. The next square is [4,4] which has to be read as [1,1], but this has already been filled. Hence go up vertically one step from [3,3] to [2,3] and enter the number 7. The next number 8 is entered in [3,4] read as [3,1]. The last number 9 is entered in [4,2] read as [1,2]. Thus now all the squares have been filled up and the total of each row, column and diagonal is 15.
Magic square of order 3
Magic square of order 5
Magic square of order 9
Method of constructing a magic square of doubly even order N
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 N=4, 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 N= 8, the same is done- the 16 central squares and 4 squares at each corner are retained in their places and the rest are switched.Magic square of order 8
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