2. This presentation aims to explore the possible
applications of magic squares in everyday life.
In doing this, different types of magic squares
will be investigated and the methods used to
construct them. How magic squares have
evolved and where they originally came from
should also be considered. Also, variations on
the basic magic square will be looked at to see
if these have any practical applications.
3. In recreational mathematics, a magic square of order n is an arrangement
of n2 numbers, usually distinct integers, in a square, such that the n
numbers in all rows, all columns, and both diagonals sum to the same
constant. A normal magic square contains the integers from 1 to n2. The
term "magic square" is also sometimes used to refer to any of various
types of word square.
Normal magic squares exist for all orders n ≥ 1 except n = 2, although the
case 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 magic
constant or magic sum, M. The magic constant of a normal magic square
depends only on n and has the value.
In the following magic square , you may
observe that the 5 numbers in all rows
,
all columns, and both diagonals sum to 205.
9 77 43 75 1
37 57 63 21 27
23 3 41 79 59
55 61 19 25 45
81 7 39 5 73
4. Magic squares were known to Chinese
mathematicians, as early as 650 BCE and Arab
mathematicians, possibly as early as the 7th century,
when the Arabs conquered northwestern parts of the
Indian subcontinent and learned Indian mathematics
and astronomy, including other aspects of
combinatorial mathematics. The first magic squares of
order 5 and 6 appear in an encyclopedia from Baghdad
circa 983 CE, the Encyclopedia of the Brethren of Purity
(Rasa'il Ihkwan al-Safa); simpler magic squares were
known to several earlier Arab mathematicians. Some of
these squares were later used in conjunction with
magic letters as in (Shams Al-ma'arif) to assist Arab
illusionists and magicians.
5. 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
Horton Conway) and the Strachey method for magic squares.
Group theory was also used for constructing new magic squares
of a given order from one of them, please see.
The number of different n×n magic squares for n from 1 to 5, not
counting rotations and reflections:
1, 0, 1, 880, 275305224 (sequence A006052).The number for n = 6
has been estimated to 1.7745×1019.
6. Magic squares were used in a number of ways in the ancient
period such as-
Music
The main area of the application of magic squares to music is in rhythm,
rather than notes. Indian musicians seem to have applied them to their
music 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 fill
the cells of the magic square.
Sudoku
Sudoku was first introduced in 1979 and became popular in Japan during the
1980’s (Pegg & Weisstein, 2006). It has recently become a very popular
puzzle in Europe, but it is actually a form of Latin square. A Sudoku
square is a 9x9 grid, split into 9 3x3 sub-squares. Each sub-square is filled
in with the numbers 1 to where , so that the 9x9 grid becomes a Latin
square. This means each row and column contain the numbers 1 to 9
only once. Therefore each row, column and sub-square will sum to the
same amount.
7. Mathematicians today do not need to
speculate and attach meaning to magic squares
to make them important, as has been done in
the past with Chinese and other myths. The
squares were thought to be mysterious and
magic, although now it is clear that they are
just ways of arranging numbers and symbols
using certain rules. They can be applied to
music and Sudoku as has been discussed but
are mainly of interest in mathematics for their
“magic” properties rather than their practical
applications.
