Magic Square A magic square is a square grid (where n is the number of cells on each side) filled with distinct positive integers in the range such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. The sum is called the magic constant or magic sum of the magic square. A square grid with n-cells on each side is said to have order n. India The oldest datable magic square in India occurs in Varāhamihira's encyclopedic work on divination, Bṛhatsaṃhitā (AD 550). He utilized a modified magic square of order four in order to prescribe combinations and quantities of ingredients of perfume. It consists of two sets of the natural numbers 1–8, and its constant sum (p) is 18. It is, so to speak, pan‐diagonal, that is, not only the two main diagonals but also all “broken” diagonals have the same constant sum. Utpala, the commentator (AD 967), also points out many other quadruplets that have the same sum. Magic Square in Parshavnath temple The 3x3 magic square was used as part of rituals in India from Vedic times, and continues to be used to this day. The Ganesh yantra is a 3x3 magic square. There is a well-known 12th-century 4×4 magic square inscribed on the wall of the Parshvanath temple in Khajuraho, India.
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This is known as the Chautisa Yantra since its magic sum is 34. It is one of the three 4×4 pandiagonal magic squares and is also an instance of the most perfect magic square. The study of this square led to the appreciation of pandiagonal squares by European mathematicians in the late 19th century. Pandiagonal squares were referred to as Nasik squares or Jain squares in older English literature. Bhaskaracharya Magic Square No zeros Numbers 1 to 9 are repeating frequently Traditionally this is being written on rice for Aksharabhyas
Ramanujan Magic Square Ramanujan created a super magic square. The top row is his birthdate (December 22, 1887). This is a super magic square because not only do the rows, columns, and diagonals add up to the same number, but the four corners, the four middle squares (17, 9, 24, 89), the first and last rows two middle numbers (12, 18, 86, 23), and the first and last columns two middle numbers (88, 10, 25, 16) all add up to the sum of 139. Check out his magic square below. Look at these possibilities sum of identical colored boxes is also 139.
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Create a magic square using your date of birth,
Lo Shu Magic Square Magic squares were known to Chinese mathematicians, and Arab mathematicians, possibly as early as the 7th century, when the Arabs conquered northwestern parts of the Indian subcontinent and learned Indian mathematicians and astronomers, including other aspects of combinatorial mathematics. ‘Cornelius Agrippa’ (1486 B.C. to 1535 B.C.) of China is believed to be the first for construction of magic squares. 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 number of days in each of the 24 cycles of the Chinese solar year. This pattern, in a certain way, was used by the people in controlling the river.
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Melancholia Magic Square Durer's magic square is a magic square with magic constant 34 used in an engraving entitled Melancholia-I by Albrecht Durer (The British Museum, Burton 1989, Gellert et al. 1989).
The engraving shows a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Durer's magic square is located in the upper right-hand corner of the engraving. The numbers 15 and 14 appear in the middle of the bottom row, indicating the date of the engraving, 1514.
Types of magic squares While the classification of magic squares can be done in many ways, some useful categories are given below. An n×n square array of integers 1, 2, ..., n² is called: Simple magic square when its rows, columns, and two diagonals sum to give magic constant and no more. They are also known as ordinary magic squares. Lo Shu is example for simple magic square
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Semi-magic square when its rows and columns sum to give the magic constant. Example for semi magic square of order 5 2 8 14 20 26
18 24 5 11 12
9 15 21 22 3
25 6 7 13 19
16 17 23 4 10
Even/Odd Order Magic Squares Magic squares of even order contain an even number of rows and an even number of columns while magic squares of odd order contain an odd number of rows and an odd number of columns. Odd order – When n is odd, i.e. n = 3, 5, 7…; Doubly Even Magic Squares Doubly even magic squares are all those, whose order (for an n by n, the order is n) is a multiple of 4. Doubly even order – When n is multiple of 4, i.e. n = 4, 8, 12…; Singly Even Magic Squares Singly even magic squares are all other even order magic squares. Singly even order – When n is even but is not divisible by 4, i.e. n = 6, 10, 14…. Pan-diagonal magic square when it is a magic square with a further property that the broken diagonals sum to the magic constant. They are also called panmagic squares, perfect squares, diabolic squares, Jain squares, or Nasik squares. Panmagic squares do not exist for singly even orders. However, singly even non-normal squares can be panmagic. Durer's Melancholia is one example of a pandiagonal magic square 16 5 9 4
3 10 6 15
2 11 7 14
13 8 12 1
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Construction of General Magic Square There is no specific method or limited algorithm to build or construct all the types of magic square. So the algorithm which works for even squares order will not work for odd order without some additional work or modification, except for the trial and developing computer software by using programming languages. Various methods for constructing magic squares have been evolved through the ages. i) Square of Order 3 Z-X
Z+X+Y
Z-Y
Z+X-Y
Z
Z-X+Y
Z+Y
Z-X-Y
Z+X
This is the general method, by which we can get infinite magic square of order 3 with the condition, 0 < X < Y < Z-X & Y â&#x2030; 2X so that we get distinct number & the Magic Sum, S = 3Z For example X = 2, Y = 3, & Z = 7 Where 0 < X < Y < Z-X & Y â&#x2030; 2X & Magic Sum = S = 2 5 12 4 6 7 8 10 2 9 ii) Normal Magic Square of Order 4 The first normal magic square of order 4 was discovered in the first centaury in India by mathematician named Nagarajuna whose Magic Sum S = 34. The sum 34 can also be found in the four center squares, the four corner squares, the two squares in the middle of the top and bottom row, and the two entries in the middle of the left and right columns. 4 9 5 16
15 6 10 3
14 7 11 2
1 12 8 13 Page 7 of 12
There are 880 distinct 4*4 magic squares of the first 16 integers (normal), and the 275,305,224 distinct 5*5 magic squares of the first 25 integers, the latter were first counted by computer in 1973. Properties of Magic Square There are several fascinating features and marvelous properties for the magic square which rule and determine the general observations that can be listed below The sum, difference & the product of 2 Semi-magic squares is also Semimagic Square. A magic square can be expressed as the sum as well as the difference of two semi magic squares in infinitely many ways. The addition of certain number to each number in the square keeps the square magic. The multiplication of certain number by each number in the square keeps the square magic. The exchange of two columns or two rows from the center of square equidistantly keeps the square magic. The magic square of an even order with interchanged quadrants keeps the square magic. Applications of Magic Square Music: In India, magic squares are used in music composition. The numbers in the squares are replaced with musical notes. Sudoku: It’s become a very popular puzzle. Sudoku square is a 9X9 grid, split into nine 3X3 sub-squares. Each sub square is filled in with number 1 to n where n=9, so that 9X9 grid becomes a Latin square. This means each row and column contains the number 1 to 9 only once. Therefore each row, column and sub-square will sum to the same number. Magic squares are used in Modern Physics (Moment of Inertia…..).
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Magic square was used as part of rituals in India from Vedic times. Ganesh yantras is a 3x3 magic square. Magic square of order four was used to prescribe combinations and quantities of ingredients of perfume. Magic squares are rich with mathematical properties related to many branches of mathematics.
Magic Squares- Challenge In a magic square, the numbers in each row, column, and diagonal have the same sum—the magic sum. Complete each magic square. Use any whole numbers 1–15. Each number can be used no more than two times in each magic square.
Magic sum= 21
Magic sum= 18
4 7
9
6
8 7
Magic sum= 12
Magic sum= 15
5 4
5
3
6
4
Magic sum= 24 7 8
Magic sum= 30 9 10
15
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Magic Squares- Challenge Answers
Magic sum= 21
Magic sum= 18
6
11
4
5
10
3
5
7
9
4
6
8
10
3
8
9
2
7
Magic sum= 12
Magic sum= 15
2
5
5
7
2
6
7
4
1
4
5
6
3
3
6
4
8
3
Magic sum= 24
Magic sum= 30
6
11
7
9
7
14
9
8
7
15
10
15
9
5
10
6
13
11
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Mathematics is the language with which GOD wrote the Universe. - Galileo Number rules the universe. - Pythagoras An equation means nothing to me, unless it expresses a
thought of GOD.
- Srinivasa Ramanujan Numbers have life; they are not just symbols on paper.
- Shakuntala Devi Pure Mathematics is the poetry of logical ideas.
- Albert Einstein Mathematics makes the invisible visible. -Devlin
The essence of mathematics is not make simple things
complicated, but to make the complicated things simple.
- S.Gudder
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