Magic Square Matrix In C

History

1. Origins Prior to 200 BCE:

• ✅ The Lo Shu Square, the first known magical square, goes back to roughly 2200 BCE in ancient China. The Lo Shu Square is said to have formed on the back of a turtle coming from the Lo River. It has a magic constant of 15 and is a 3x3 magic square. The Lo Shu Square's importance was firmly steeped in Chinese mysticism.

1. Chinese Invasion (200 BCE - 600 CE):

• ✅ During this time, magic squares rose to prominence in Chinese mathematics. Yang Hui (1238-1298), a well-known Chinese mathematician, made substantial contributions to the study of magic squares.

1. Middle Eastern Developments (from the eighth to the thirteenth centuries):

• ✅ Magic squares were known across the Islamic world. Magic squares were examined and written about by scholars such as Al-Kindi and Al-Buni, who emphasized their mathematical and artistic merits.
• ✅ Magic squares were mentioned in Al-Khwarizmi's book "Kitab al-Jabr wal-Muqabala" (The Compendious Book on Calculation by Completion and Balancing).

1. European Renaissance (15th-16th centuries):

• ✅ During the Renaissance, European mathematicians and painters were intrigued by magic squares.
• ✅ In his etching "Melencolia I" (1514), the great German artist and mathematician Albrecht Dürer produced a famous 4x4 magic square. This square is both mathematically and visually interesting.

1. Revolutions in India (17th century):

• ✅ During the 17th century, Indian mathematicians such as Narayana Pandit made significant contributions to the magic square theory.

1. Enlightenment in Europe (17th-18th centuries):

• ✅ European mathematicians such as Leonard Euler and Leonhard Christoph Sturm investigated the characteristics of magic squares, resulting in significant discoveries and contributions.

1. Modern Mathematics (from the nineteenth century to the present):

• ✅ The study of magic squares evolved alongside the development of contemporary mathematics. Significant contributions were made by mathematicians like Édouard Lucas, Gaston Tarry, and John Conway.
• ✅ A key emphasis of research became the categorization of all conceivable orders of magic squares, as well as their attributes.

1. Recreational Mathematics and Art (from the twentieth century to the present):

• ✅ Magic squares have been used in a variety of industries, including recreational mathematics and painting. Magic square puzzles and games are still popular.
• ✅ For some mathematicians, creating enormous, intricate magic squares has become a competitive endeavor.

Properties and characteristics

The sum of numbers in every row, column, and diagonal of a magic square always equals a consistent value, referred to as the "magic sum" or "magic constant." This unique property sets magic squares apart from regular grids of numbers.

The dimension of a magic square grid is known as its order. Commonly encountered magic square orders are 3x3, 4x4, 5x5, 7x7, and other similar variations. While even-order magic squares like 4x4 and 8x8 have been more recently identified and possess distinct characteristics, odd-order magic squares such as 3x3 and 5x5 have been documented for thousands of years.

A typical magic square consists of a sequence of positive numbers starting from 1. These normal magic squares hold significant value in mathematics due to being the fundamental and sophisticated form of magic squares.

Calculating the Enchantment Value: The enchantment value of a specific enchantment grid with a size of n is determined by the formula (n * (n2 + 1)) / 2. This equation ensures that the sum of every row, column, and diagonal remains consistent.

Example: 3 x 3 matrix

2   7   6

  9   5   1

  4   3   8

In this 3x3 magic square:

The arrangement is 3x3, indicating there are 3 rows and 3 columns in total.

The special value (total of every row, column, and diagonal) equals 15.

We can verify that this is a magic square by adding the numbers in each row, column, and diagonal:

• ✅ Row 1: 2 + 7 + 6 = 15
• ✅ Row 2: 9 + 5 + 1 = 15
• ✅ Row 3: 4 + 3 + 8 = 15
• ✅ Column 1: 2 + 9 + 4 = 15
• ✅ Column 2: 7 + 5 + 3 = 15
• ✅ Column 3: 6 + 1 + 8 = 15
• ✅ Diagonal from top-left to bottom-right: 2 + 5 + 8 = 15
• ✅ Diagonal from top-right to bottom-left: 6 + 5 + 4 = 15

Some well-known magic squares

1. Lo Shu Square (3x3)

The origins of the Lo Shu Square can be dated back more than two thousand years in Chinese culture. Legend has it that this mystical 3x3 square was found on the shell of a mystical turtle that surfaced from the Lo River. The credit for uncovering the arrangement of this square goes to the legendary Emperor Yu, believed to have lived around 2200 BCE.

Numerical Configuration: In the Lo Shu Square, the digits 1 to 9 are systematically positioned within the 3x3 matrix. The arrangement ensures that the sum of every row, column, and diagonal consistently adds up to 15.

Example

4   9   2

  3   5   7

  8   1   6

In the Lo Shu Square:

The layout is 3x3, indicating it consists of three rows and three columns.

The mystical sum (total of every row, column, and diagonal) equals 15.

2. Albrecht Dürer's magic square

Albrecht Dürer's enchanting square, created in 1514, is a famous 4x4 magic square crafted by the German artist and mathematician. This magical square by Dürer stands out not only for its mathematical properties but also for its creative artistic expression. Let's delve into the details of Dürer's magical square:

Magic Square by Dürer (4x4):

• ✅ The order is 4x4, which means there are four rows and four columns.
• ✅ Dürer's square comprises the numbers 1 through 16, which are arranged in a 4x4 grid. The unique manner in which the numerals are put is what distinguishes this square.
• ✅ Dürer's square has a magical constant of 34, which means that the total of each row, column, and diagonal equals 34.

Example

16  2   3   13

  5   10  11  8

  9   6   7   12

  4   14  15  1

It is the famous 4x4 magic square created by Dürer. Within this square:

• ✅ Its layout consists of four rows and four columns.
• ✅ The magic sum, which is the sum of each row, column, and diagonal, amounts to 34.

3. Sagrada Família magic square

The magic square found at Sagrada Familia is a unique and modern instance of a magic square, notable for its association with the famous Sagrada Familia basilica in Barcelona. Antoni Gaudí, the architect of the basilica, integrated a mystical square into the design of the basilica's exterior.

The magic square at Sagrada Família is a 4x4 grid linked to the basilica's finishing date. The total of each row, column, and diagonal in the square amounts to 33. This value holds importance in Christian beliefs as it symbolizes the age of Jesus Christ during his crucifixion and resurrection.

10  2   3   25

7   12  14  16

9   11  8   6

4   15  13  5

In this enchanted square

• ✅ Because the total of each row, column, and diagonal equals 33, the magic constant is 33.
• ✅ The incorporation of the plaza in the façade of the Sagrada Famlia serves as a symbolic and significant element, tying the basilica's construction to the Christian religion.
• ✅ Antoni Gaud was a very spiritual man who infused religious and mystical symbolism into his architectural creations.
• ✅ The usage of a magical square in the Sagrada Famlia is a one-of-a-kind example of how mathematics and symbolism may be used in art and architecture.

4. Franklin's Magic Square

The 8x8 magic square by Benjamin Franklin is a spectacular example of a big magic square recognized for its intricacy and unique mathematical features. Benjamin Franklin, one of the United States Founding Fathers, devised this square, which consists of 64 positive and negative integers.

• ✅ The order is 8x8, which implies there are eight rows and eight columns.
• ✅ Franklin's 8x8 magic square comprises the numbers 1 to 64, which are arranged in a complex design. It covers both positive and negative numbers, unlike regular magic squares.
• ✅ Franklin's square has a magic constant of 260, which means that the total of each row, column, and diagonal equals 260.

Example

63   1   62   4   59   3   60   2

11  53  10  54   7  49   8  56

 35  23  37  21  39  25  41  29

 27  45  31  47  33  43  35  19

46  28  44  30  48  32  50  12

 3  59   5  61   1  63   7  55

 19 37  21  39  23  41  25  43

 15  51  13  52  9  58  6  60

In Benjamin Franklin's 8x8 magic square:

The layout consists of 8 rows and 8 columns.

The mystical sum remains at 260, encompassing every row, column, and diagonal.

Approach 1:Magic Squares in Odd Order

Odd-dimensional magic squares are a unique type of magic square characterized by having an odd quantity of both rows and columns. The total of the numbers in every row, column, and diagonal of an odd-dimensional magic square is identical, referred to as the "magic constant." These squares are commonly represented as N × N grids, with N being an odd whole number (like 3x3, 5x5, 7x7, etc.).

Example

Consider a 3x3 magic square with an odd order as a demonstration. The magic sum for a 3x3 magic square is calculated as (322 + 1) / 2, resulting in 5 as the magic constant.

An instance of a 3x3 magic square with odd order is demonstrated as follows:

2  7  6

9  5  1

4  3  8

In this specific square of enchantment, it is evident that the total of every row, every column, and both diagonals is identical to the mystical constant value of 5.

For instance:

• ✅ The sum of the first row (2 + 7 + 6) is 15 (the magic constant).
• ✅ The sum of the second row (9 + 5 + 1) is also 15.
• ✅ The sum of the third row (4 + 3 + 8) is 15.
• ✅ The sum of the first column (2 + 9 + 4) is 15.
• ✅ The sum of the second column (7 + 5 + 3) is 15.
• ✅ The sum of the third column (6 + 1 + 8) is 15.
• ✅ The sum of the main diagonal (2 + 5 + 8) is 15.
• ✅ The sum of the other diagonal (6 + 5 + 4) is 15.

Implementation

Example
Copy
#include <stdio.h>

int main() {

    int n = 3;  

    int magicSquare[n][n];

    for (int i = 0; i< n; i++) {

        for (int j = 0; j < n; j++) {

magicSquare[i][j] = 0;

        }

    }  

    int num = 1;  

    int i = 0, j = n / 2;  

    while (num<= n * n) {

magicSquare[i][j] = num;

num++;

        int nexti = (i - 1 + n) % n;

        int nextj = (j + 1) % n;

        if (magicSquare[nexti][nextj] == 0) {

i = nexti;

            j = nextj;

        } else {

i = (i + 1) % n;

        }

    }

printf("Magic Square of Order %d:\n", n);

    for (i = 0; i< n; i++) {

        for (j = 0; j < n; j++) {

printf("%2d ", magicSquare[i][j]);

        }

printf("\n");

    }

    return 0;

}
Try it Yourself

Output:

[Program Output]

This illustration showcases the fundamental concepts of an odd-order magic square, where the constant sum ensures that the sum of all rows, columns, and diagonals remains identical.

Approach 2: Higher-Dimensional Magic Squares

Higher-dimensional enchantment grids are those that manifest in dimensions beyond the usual 2D square layouts. The concept of magic squares can be extended to higher dimensions, encompassing magic cubes and magic hypercubes. These constructions in three dimensions are also known as "magic hypercubes."

Example

Generating a 3x3x3 magical hypercube poses a complex mathematical challenge, while articulating it verbally can be intricate. Nevertheless, for illustrative purposes, I can offer a fundamental representation of a 3x3x3 magic hypercube. Within this representation, we will employ whole numbers spanning from 1 to 27 arranged in a 3x3x3 matrix.

42 represents the magic constant in a 3x3x3 magic hypercube.

Here is a basic depiction of a 3x3x3 magical hypercube:

Layer 1:

[ 2 7 6]

[ 9 5 1]

[ 4 3 8]

Layer 2:

[11 16 15]

[18 14 10]

[13 12 17]

Layer 3:

[20 25 24]

[27 23 19]

[22 21 26]

Creating a 3x3x3 magic hypercube that adheres to the properties of a magical hypercube, where the total of numbers across various hyperplanes equals the magic constant, involves a significantly complex mathematical process. This basic illustration showcases the appearance of a 3x3x3 magic hypercube.

Implementation

import numpy as np

def is_magic_cube(cube):

    n = cube.shape[0]

magic_constant = np.sum(cube[0, :, :])

    for i in range(n):

        if (np.sum(cube[i, :, :]) != magic_constant or

np.sum(cube[:, i, :]) != magic_constant or

np.sum(cube[:, :, i]) != magic_constant):

            return False

    if (np.sum(np.diagonal(cube, axis1=1, axis2=2)) != magic_constant or

np.sum(np.diagonal(np.flip(cube, axis=2), axis1=1, axis2=2)) != magic_constant):

        return False    

    return True

def generate_3x3x3_magic_cube():

    n = 3

    cube = np.zeros((n, n, n), dtype=int) 

num = 1

i, j, k = 0, n // 2, n // 2 + 1

    while num<= n**3:

cube[i, j, k] = num

num += 1        

i -= 1

        j += 1

        k += 1

        if i< 0:

i = n - 1

        if j == n:

            j = 0

        if k == n:

            k = 0

    return cube

if __name__ == "__main__":

magic_cube = generate_3x3x3_magic_cube() 

print("3x3x3 Magic Cube:")

    print(magic_cube)

    if is_magic_cube(magic_cube):

print("This is a 3x3x3 magic cube.")

    else:

print("This is not a 3x3x3 magic cube.")

Output:

[Program Output]

Approach 3: Symmetry

Symmetry plays a fundamental role in the aesthetics of odd-order magic squares, contributing to their visual appeal. These squares exhibit both reflectional (rotational) and diagonal symmetry, enhancing their overall beauty and mathematical elegance. Let's delve deeper into these specific types of symmetry:

Example

2 7 6

9 5 1

4 3 8

A 3x3 Magic Square with Odd Order exhibiting Symmetry:

Symmetry of Reflection (Vertical and Horizontal):

When the square is folded along its horizontal axis (the middle row), it becomes apparent that the upper and lower sections are mirror images of one another. The numbers in the top half are reflected in the bottom half.

Top Half:   2  7  6

Bottom Half: 9  5  1

4  3  8

When the square is folded vertically along its center column, it becomes apparent that the left and right halves are exact reflections of each other. The numbers on the left side are duplicated in a mirrored fashion on the right side.

Left Half:  2  9  4

Right Half: 7  5  3

6  1  8

1. Symmetrical Rotation:

• ✅ 90-Degree Rotation: If you rotate the square 90 degrees clockwise, its arrangement is preserved. In other words, even if the numbers are rotated 90 degrees, they still form a valid 3x3 odd-order magic square.
Example
Copy
Original:2  7  6

9  5  1

4  3  8

90-Degree Rotation:4  9  2

3  5  7

8  1  6
Try it Yourself

1. Symmetry of Diagonal Lines:

• ✅ The square has diagonal symmetry along its primary diagonal, which goes from the top-left to the bottom-right corner. The numerals on one side of the diagonal are mirror images of the ones on the other.
Example
Copy
Upper Main Diagonal:  2    5

7  1

Lower Main Diagonal:  6    3

4  8
Try it Yourself

Implementation

Example
Copy
def generate_3x3_odd_order_magic_square():

magic_square = [[0, 0, 0], [0, 0, 0], [0, 0, 0]]

    n = 3

num = 1

i, j = 0, n // 2

    while num<= n ** 2:

magic_square[i][j] = num

num += 1

next_i, next_j = (i - 1) % n, (j + 1) % n

        if magic_square[next_i][next_j] == 0:

i, j = next_i, next_j

        else:

i = (i + 1) % n

    return magic_square

def print_magic_square(square):

    for row in square:

print(" ".join(map(str, row)))

if __name__ == "__main__":

magic_square = generate_3x3_odd_order_magic_square()

print("3x3 Odd-Order Magic Square:")

print_magic_square(magic_square)
Try it Yourself

Output:

[Program Output]

Applications

• ✅ Coding and Cryptography: Magic squares are used in coding theory and cryptography. They may be used to encrypt and decode messages, adding an extra degree of protection.
• ✅ Sudoku: Sudoku puzzles, which have grown in popularity across the world, are simply 9x9 magic squares. Solving sudoku problems entails arranging numbers so that they fulfill precise limitations, similar to building a magic square.
• ✅ Game Design: Magic squares have appeared in a variety of board games and puzzles. They provide a distinct and difficult aspect to the gameplay.
• ✅ Recreational Mathematics: Magic squares captivate recreational mathematicians and puzzle aficionados. The task of solving and constructing magic squares is both enjoyable and intellectually engaging.
• ✅ Art and Design: Magic squares have been utilized to produce visually beautiful patterns in art and design. Magic squares were used by artists such as Albrecht Dürer to enhance the visual attractiveness of their work.