Magic SquareMeeran Ali AhmadClass IXth –AR.no- 08
This presentation aims to explore the possibleapplications of magic squares in everyday life.In doing this, different types of magic squareswill be investigated and the methods used toconstruct them. How magic squares haveevolved and where they originally came fromshould also be considered. Also, variations onthe basic magic square will be looked at to seeif these have any practical applications.
In recreational mathematics, a magic square of order n is an arrangementof n2 numbers, usually distinct integers, in a square, such that the nnumbers in all rows, all columns, and both diagonals sum to the sameconstant. A normal magic square contains the integers from 1 to n2. Theterm "magic square" is also sometimes used to refer to any of varioustypes of word square. Normal magic squares exist for all orders n ≥ 1 except n = 2, although thecase n = 1 is trivial, consisting of a single cell containing the number 1.The smallest nontrivial case, shown below, is of order 3. The constant sum in every row, column and diagonal is called the magicconstant or magic sum, M. The magic constant of a normal magic squaredepends only on n and has the value. In the following magic square , you mayobserve that the 5 numbers in all rows,all columns, and both diagonals sum to 205.9 77 43 75 137 57 63 21 2723 3 41 79 5955 61 19 25 4581 7 39 5 73
Magic squares were known to Chinesemathematicians, as early as 650 BCE and Arabmathematicians, possibly as early as the 7th century,when the Arabs conquered northwestern parts of theIndian subcontinent and learned Indian mathematicsand astronomy, including other aspects ofcombinatorial mathematics. The first magic squares oforder 5 and 6 appear in an encyclopedia from Baghdadcirca 983 CE, the Encyclopedia of the Brethren of Purity(Rasail Ihkwan al-Safa); simpler magic squares wereknown to several earlier Arab mathematicians. Some ofthese squares were later used in conjunction withmagic letters as in (Shams Al-maarif) to assist Arabillusionists and magicians.
There are many ways to construct magic squares, but the standard(and most simple) way is to follow certainconfigurations/formulas which generate regular patterns. Magicsquares exist for all values of n, with only one exception: it isimpossible to construct a magic square of order 2. Magic squarescan be classified into three types: odd, doubly even (n divisible byfour) and singly even (n even, but not divisible by four). Odd anddoubly even magic squares are easy to generate; the constructionof singly even magic squares is more difficult but several methodsexist, including the LUX method for magic squares (due to JohnHorton Conway) and the Strachey method for magic squares. Group theory was also used for constructing new magic squaresof a given order from one of them, please see. The number of different n×n magic squares for n from 1 to 5, notcounting rotations and reflections: 1, 0, 1, 880, 275305224 (sequence A006052).The number for n = 6has been estimated to 1.7745×1019.
Magic squares were used in a number of ways in the ancientperiod such as- MusicThe main area of the application of magic squares to music is in rhythm,rather than notes. Indian musicians seem to have applied them to theirmusic and they seem to be useful in time cycles and additive rhythm. .This is because for rhythm, consecutive numbers 1 to are not used to fillthe cells of the magic square. SudokuSudoku was first introduced in 1979 and became popular in Japan during the1980’s (Pegg & Weisstein, 2006). It has recently become a very popularpuzzle in Europe, but it is actually a form of Latin square. A Sudokusquare is a 9x9 grid, split into 9 3x3 sub-squares. Each sub-square is filledin with the numbers 1 to where , so that the 9x9 grid becomes a Latinsquare. This means each row and column contain the numbers 1 to 9only once. Therefore each row, column and sub-square will sum to thesame amount.
Mathematicians today do not need tospeculate and attach meaning to magic squaresto make them important, as has been done inthe past with Chinese and other myths. Thesquares were thought to be mysterious andmagic, although now it is clear that they arejust ways of arranging numbers and symbolsusing certain rules. They can be applied tomusic and Sudoku as has been discussed butare mainly of interest in mathematics for their“magic” properties rather than their practicalapplications.