Magic Square Function in C

In this article, we will discuss the magic square function in C with its examples.

A magic square of order n is an arrangement of n2 numbers, often different integers, in a square such that the total of the n numbers in the square's diagonals, rows, and columns is a fixed value. Integers from 1 to n2 are contained within a magic square.

The magic constant, or magic sum, M, is the total that is constant in each row, column, and diagonal. A typical magic square's magic constant, which only depends on n, has the following value:

M = n(n2+1)/2

Let's consider the normal magic squares of order n = 6, 7, 8, ...

For these orders, the magic constants are approximately: 372, 509, 660, 825, ...

Magic Square of size 4
6 11 16
7 2 17
13 18 3
9 14 19
Sum in each row & each column = 4*(34+1)/2 = 70

Magic Square of size 6
1 6 26 19 24
32 7 21 23 25
9 2 22 27 20
28 33 8 17 14
11 36 15 13 30
18 34 5 16 4
Sum in each row & each column = 6*(111+1)/2 = 666

Magic Square of size 8
37 14 53 30 7 46 23
16 57 34 11 50 27 4
61 38 15 54 31 8 47
40 17 56 33 10 49 26
19 63 35 12 55 32 9
58 36 13 52 29 6 45
21 42 64 25 2 44 28
43 65 24 1 48 15 51
Sum in each row & each column = 8*(260+1)/2 = 2080

The first integer, 1, is kept at position (n/2, n-1) in any magic square. Let the coordinates be (i,j). The following number is kept at position (i-1, j+1), where each row and column can be thought of as a circular array because they wrap around.

Three requirements are accomplished:

The position of the subsequent number is determined by decrementing the previous number's row number by 1, and increasing the previous number's column number by 1. The estimated row position will always wrap around to n-1 if it ever gets negative. The calculated column position will also wrap around to zero if it rises to n.

Example:

Magic Square of size 4
15 14 4
6 7 9
10 11 5
3 2 16

Steps:

position of number 1 = (4/2, 4-1) = (2, 3)
position of number 2 = (2-1, 3+1) = (1, 4)
position of number 3 = (1-1, 4+1) = (0, 5)
position of number 4 = (0-1, 5+1) = (-1, 6)
new position = (0, 6-4) = (0, 2)
position of number 5 = (0-1, 2+1) = (-1, 3)
new position = (0, 3-2) = (0, 1)
position of number 6 = (0-1, 1+1) = (-1, 2)
new position = (0, 2-2) = (0, 0)
position of number 7 = (0-1, 0+1) = (-1, 1)
new position = (0, 1-2) = (0, -1)
new position = (1, -1+4) = (1, 3)
position of number 8 = (1-1, 3+1) = (0, 4)
position of number 9 = (0-1, 4+1) = (-1, 5)
new position = (0, 5-4) = (0, 1)
position of number 10 = (0-1, 1+1) = (-1, 2)
new position = (0, 2-2) = (0, 0)
position of number 11 = (0-1, 0+1) = (-1, 1)
new position = (0, 1-2) = (0, -1)
new position = (1, -1+4) = (1, 3)
position of number 12 = (1-1, 3+1) = (0, 4)
position of number 13 = (0-1, 4+1) = (-1, 5)
new position = (0, 5-4) = (0, 1)
position of number 14 = (0-1, 1+1) = (-1, 2)
new position = (0, 2-2) = (0, 0)
position of number 15 = (0-1, 0+1) = (-1, 1)
new position = (0, 1-2) = (0, -1)
new position = (1, -1+4) = (1, 3)
position of number 16 = (1-1, 3+1) = (0, 4)

Example:

Let's take a program to understand the use of the magic square function in C.

#include <stdio.h>
#include <string.h>

void generateSquare(int n)
{
    int magicSquare[n][n];

memset(magicSquare, 0, sizeof(magicSquare));

    int i = n / 2;
    int j = n - 1;

    for (int num = 1; num<= n * n;) {
        if (i == -1 && j == n) // 3rd condition
        {
            j = n - 2;
i = 0;
        }
        else {

            if (j == n)
                j = 0;

            // 1st condition helper if next number
            // is goes to out of square's upper side
            if (i< 0)
i = n - 1;
        }
        if (magicSquare[i][j]) // 2nd condition
        {
            j -= 2;
i++;
            continue;
        }
        else
magicSquare[i][j] = num++; // set number

j++;
i--; // 1st condition
    }

    // Print magic square
printf("The Magic Square for n=%d:\nSum of "
           "each row or column %d:\n\n",
           n, n * (n * n + 1) / 2);
    for (i = 0; i< n; i++) {
        for (j = 0; j < n; j++)
printf("%3d ", magicSquare[i][j]);
printf("\n");
    }
}

// Driver program to test above function
int main()
{
    int n = 5; // Works only when n is odd
generateSquare(n);
    return 0;
}

Output:

The Magic Square for n=5:
Sum of each row or column 65:

17  24   1   8  15 
 23   5   7  14  16 
  4   6  13  20  22 
10  12  19  21   3 
11  18  25   2   9

Complexities:

Time Complexity: O(n2)

Auxiliary Space: O(n2)

NOTE: This method only functions for odd values of nWork just with n's ODD values.

Explanation:

Start at the square's middle cell in the final column.
Put the number 1
Go to the cell in the top-right corner.
If it is not within square bounds, use modulo with n.
Go to the cell that follows taking the modulo.
Move to the left cell once we have reached a multiple of n.
If it is not within square bounds, use modulo with n.
Put the following number, and then repeat the process.

Example:

Let's take another program to understand the use of the magic square function in C.

#include <stdio.h>

int main() {
    int n = 5;
    int magicSquare[n][n];
    for (int i = 0; i< n; i++) {
        for (int j = 0; j < n; j++) {
magicSquare[i][j] = -1;
        }
    }
    // Indices for element to be inserted
    int x = (n / 2), y = n - 1;
    // Loop till n ^ 2 times
    for (int i = 1; i<= n * n; i++) {
        // Put the current element at (x, y)
magicSquare[x][y] = i;

        if (i % n == 0) {
            // If we get multiple of n, move left
            y--;
        } else {
            // Else, move top-right
            x--; y++;
        }
        // Take modulo to avoid out of bounds
        x += n; x %= n;
        y += n; y %= n;
    }

    // Print magic square
    for (int i = 0; i< n; i++) {
        for (int j = 0; j < n; j++) {
printf("%3d ", magicSquare[i][j]);
        }
printf("\n");
    }

    return 0;
}

Output:

17  24   1   8  15 
 23   5   7  14  16 
  4   6  13  20  22 
10  12  19  21   3 
11  18  25   2   9

Conclusion:

In conclusion, magic squares are unique configurations of numbers that have the special characteristic of having equal sums along all of the rows, columns, and diagonals. Throughout history, they have enthralled mathematicians, artists, and societies. Magic squares offer a rich junction of mathematics and other fields, from their mathematical complexity and aesthetic appeal in art to their cultural meaning and practical uses in encryption. These configurations continue to demonstrate mathematics' great beauty and adaptability as a topic for investigation and creation.

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