Magic Square | ODD Order

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 magic square contains the integers from 1 to n². 
The constant sum in every row, column and diagonal are called the magic constant or magic sum, M. The magic constant of a normal magic square depends only on n and has the following value: 
M = n(n²+1)/2

For normal magic squares of order n = 3, 4, 5, ...,
the magic constants are: 15, 34, 65, 111, 175, 260, ... 

In this post, we will discuss how programmatically we can generate a magic square of size n. This approach only takes into account odd values of n and doesn’t work for even numbers. Before we go further, consider the below examples:

Magic Square of size 3
-----------------------
  2   7   6
  9   5   1
  4   3   8
Sum in each row & each column = 3*(32+1)/2 = 15


Magic Square of size 5
----------------------
  9   3  22  16  15
  2  21  20  14   8
 25  19  13   7   1
 18  12   6   5  24
 11  10   4  23  17
Sum in each row & each column = 5*(52+1)/2 = 65


Magic Square of size 7
----------------------
 20  12   4  45  37  29  28
 11   3  44  36  35  27  19
  2  43  42  34  26  18  10
 49  41  33  25  17   9   1
 40  32  24  16   8   7  48
 31  23  15  14   6  47  39
 22  21  13   5  46  38  30
Sum in each row & each column = 7*(72+1)/2 = 175

Did you find any pattern in which the numbers are stored? 

In any magic square, the first number i.e. 1 is stored at position (n/2, n-1). Let this position be (i,j). The next number is stored at position (i-1, j+1) where we can consider each row & column as circular array i.e. they wrap around.

Three conditions hold:

1. The position of next number is calculated by decrementing row number of the previous number by 1, and incrementing the column number of the previous number by 1. At any time, if the calculated row position becomes -1, it will wrap around to n-1. Similarly, if the calculated column position becomes n, it will wrap around to 0.
2. If the magic square already contains a number at the calculated position, calculated column position will be decremented by 2, and calculated row position will be incremented by 1.
3. If the calculated row position is -1 & calculated column position is n, the new position would be: (0, n-2). 

Example:
Magic Square of size 3
----------------------
 2  7  6
 9  5  1
 4  3  8 

Steps:
1. position of number 1 = (3/2, 3-1) = (1, 2)
2. position of number 2 = (1-1, 2+1) = (0, 0)
3. position of number 3 = (0-1, 0+1) = (3-1, 1) = (2, 1)
4. position of number 4 = (2-1, 1+1) = (1, 2)
   Since, at this position, 1 is there. So, apply condition 2.
   new position=(1+1,2-2)=(2,0)
5. position of number 5=(2-1,0+1)=(1,1)
6. position of number 6=(1-1,1+1)=(0,2)
7. position of number 7 = (0-1, 2+1) = (-1,3) // this is tricky, see condition 3 
   new position = (0, 3-2) = (0,1)
8. position of number 8=(0-1,1+1)=(-1,2)=(2,2) //wrap around
9. position of number 9=(2-1,2+1)=(1,3)=(1,0) //wrap around

Based on the above approach, the following is the working code: 

C++

// C++ program to generate odd sized magic squares
#include <bits/stdc++.h>
using namespace std;

// A function to generate odd sized magic squares
void generateSquare(int n)
{
    int magicSquare[n][n];

    // set all slots as 0
    memset(magicSquare, 0, sizeof(magicSquare));

    // Initialize position for 1
    int i = n / 2;
    int j = n - 1;

    // One by one put all values in magic square
    for (int num = 1; num <= n * n;) {
        if (i == -1 && j == n) // 3rd condition
        {
            j = n - 2;
            i = 0;
        }
        else {
            // 1st condition helper if next number
            // goes to out of square's right side
            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
    cout << "The Magic Square for n=" << n
         << ":\nSum of "
            "each row or column "
         << n * (n * n + 1) / 2 << ":\n\n";
    for (i = 0; i < n; i++) {
        for (j = 0; j < n; j++)

            // setw(7) is used so that the matrix gets
            // printed in a proper square fashion.
            cout << setw(4) << magicSquare[i][j] << " ";
        cout << endl;
    }
}

// Driver code
int main()
{

    // Works only when n is odd
    int n = 7;
    generateSquare(n);
    return 0;
}

// This code is contributed by rathbhupendra

C

// C program to generate odd sized magic squares
#include <stdio.h>
#include <string.h>

// A function to generate odd sized magic squares
void generateSquare(int n)
{
    int magicSquare[n][n];

    // set all slots as 0
    memset(magicSquare, 0, sizeof(magicSquare));

    // Initialize position for 1
    int i = n / 2;
    int j = n - 1;

    // One by one put all values in magic square
    for (int num = 1; num <= n * n;) {
        if (i == -1 && j == n) // 3rd condition
        {
            j = n - 2;
            i = 0;
        }
        else {
            // 1st condition helper if next number
            // goes to out of square's right side
            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 = 7; // Works only when n is odd
    generateSquare(n);
    return 0;
}

Java

// Java program to generate odd sized magic squares
import java.io.*;

class GFG {
    // Function to generate odd sized magic squares
    static void generateSquare(int n)
    {
        int[][] magicSquare = new int[n][n];

        // Initialize position for 1
        int i = n / 2;
        int j = n - 1;

        // One by one put all values in magic square
        for (int num = 1; num <= n * n;) {
            if (i == -1 && j == n) // 3rd condition
            {
                j = n - 2;
                i = 0;
            }
            else {
                // 1st condition helper if next number
                // goes to out of square's right side
                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;
            }

            // 2nd condition
            if (magicSquare[i][j] != 0) {
                j -= 2;
                i++;
                continue;
            }
            else
                // set number
                magicSquare[i][j] = num++;

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

        // print magic square
        System.out.println("The Magic Square for " + n
                           + ":");
        System.out.println("Sum of each row or column "
                           + n * (n * n + 1) / 2 + ":");
        for (i = 0; i < n; i++) {
            for (j = 0; j < n; j++)
                System.out.print(magicSquare[i][j] + " ");
            System.out.println();
        }
    }

    // driver program
    public static void main(String[] args)
    {
        // Works only when n is odd
        int n = 7;
        generateSquare(n);
    }
}

// Contributed by Pramod Kumar

Python3

# Python program to generate
# odd sized magic squares
# A function to generate odd
# sized magic squares


def generateSquare(n):

    # 2-D array with all
    # slots set to 0
    magicSquare = [[0 for x in range(n)]
                   for y in range(n)]

    # initialize position of 1
    i = n // 2
    j = n - 1

    # Fill the magic square
    # by placing values
    num = 1
    while num <= (n * n):
        if i == -1 and j == n:  # 3rd condition
            j = n - 2
            i = 0
        else:

            # next number goes out of
            # right side of square
            if j == n:
                j = 0

            # next number goes
            # out of upper side
            if i < 0:
                i = n - 1

        if magicSquare[int(i)][int(j)]:  # 2nd condition
            j = j - 2
            i = i + 1
            continue
        else:
            magicSquare[int(i)][int(j)] = num
            num = num + 1

        j = j + 1
        i = i - 1  # 1st condition

    # Printing magic square
    print("Magic Square for n =", n)
    print("Sum of each row or column",
          n * (n * n + 1) // 2, "\n")

    for i in range(0, n):
        for j in range(0, n):
            print('%2d ' % (magicSquare[i][j]),
                  end='')

            # To display output
            # in matrix form
            if j == n - 1:
                print()

# Driver Code


# Works only when n is odd
n = 7
generateSquare(n)

# This code is contributed
# by Harshit Agrawal

C#

// C# program to generate odd sized magic squares
using System;

class GFG {

    // Function to generate odd sized magic squares
    static void generateSquare(int n)
    {
        int[, ] magicSquare = new int[n, n];

        // Initialize position for 1
        int i = n / 2;
        int j = n - 1;

        // One by one put all values in magic square
        for (int num = 1; num <= n * n;) {
            if (i == -1 && j == n) // 3rd condition
            {
                j = n - 2;
                i = 0;
            }
            else {
                // 1st condition helper if next number
                // goes to out of square's right side
                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;
            }

            // 2nd condition
            if (magicSquare[i, j] != 0) {
                j -= 2;
                i++;
                continue;
            }
            else
                // set number
                magicSquare[i, j] = num++;

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

        // print magic square
        Console.WriteLine("The Magic Square for " + n
                          + ":");
        Console.WriteLine("Sum of each row or column "
                          + n * (n * n + 1) / 2 + ":");

        for (i = 0; i < n; i++) {
            for (j = 0; j < n; j++)
                Console.Write(magicSquare[i, j] + " ");
            Console.WriteLine();
        }
    }

    // driver program
    public static void Main()
    {

        // Works only when n is odd
        int n = 7;

        generateSquare(n);
    }
}

// This code is contributed by Sam007.

PHP

<?php
// php program to generate odd sized
// magic squares

// A function to generate odd sized
// magic squares
function generateSquare($n)
{

    // set all slots as 0
    $magicSquare;
    for ($i = 0; $i < $n; $i++)
        for ($j = 0; $j < $n; $j++)
            $magicSquare[$i][$j] = 0;

    // Initialize position for 1
    $i = (int)$n / 2;
    $j = $n - 1;

    // One by one put all values in
    // magic square
    for ($num = 1; $num <= $n * $n; )
    {

        // 3rd condition
        if ($i == -1 && $j == $n)
        {
            $j = $n-2;
            $i = 0;
        }
        else
        {
            // 1st condition helper if
            // next number goes to out
            // of square's right side
            if ($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;
        }

        // 2nd condition
        if ($magicSquare[$i][$j])
        {
            $j -= 2;
            $i++;
            continue;
        }
        else

            // set number
            $magicSquare[$i][$j] = $num++;

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

    // Print magic square
    echo "The Magic Square for n = " . $n
        . ":\nSum of each row or column "
        . $n * ($n * $n + 1) / 2;

    echo "\n\n";
    for ($i = 0; $i < $n; $i++)
    {
        for ($j = 0; $j < $n; $j++)
            echo $magicSquare[$i][$j] . " ";
        echo "\n";
    }
}

// Driver program to test above function

// Works only when n is odd
$n = 7;
generateSquare ($n);

// This code is contributed by mits.
?>

Javascript

<script>

// javascript program to generate odd sized magic squares
   // Function to generate odd sized magic squares

function generateSquare(n)
{
    magicSquare = Array(n).fill(0).map(x => Array(n).fill(0));

    // Initialize position for 1
    var i = parseInt(n / 2);
    var j = n - 1;

    // One by one put all values in magic square
    for (num = 1; num <= n * n;) {
        if (i == -1 && j == n) // 3rd condition
        {
            j = n - 2;
            i = 0;
        }
        else {
            // 1st condition helper if next number
            // goes to out of square's right side
            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;
        }

        // 2nd condition
        if (magicSquare[i][j] != 0) {
            j -= 2;
            i++;
            continue;
        }
        else
            // set number
            magicSquare[i][j] = num++;

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

    // print magic square
    document.write("The Magic Square for " + n
                       + ":<br>");
    document.write("Sum of each row or column "
                       + parseInt(n * (n * n + 1) / 2) + ":<br>");
    for (i = 0; i < n; i++) {
        for (j = 0; j < n; j++)
            document.write(magicSquare[i][j] + " ");
        document.write("<br>");
    }
}

// driver program
 // Works only when n is odd
var n = 7;
generateSquare(n);

// This code is contributed by 29AjayKumar

</script>

Output

The Magic Square for n=7:
Sum of each row or column 175:

  20   12    4   45   37   29   28 
  11    3   44   36   35   27   19 
   2   43   42   34   26   18   10 
  49   41   33   25   17    9    1 
  40   32   24   16    8    7   48 
  31   23   15   14    6   47   39 
  22   21   13    5   46   38   30 

Time Complexity: O(n²)
Auxiliary Space: O(n²)

NOTE: This approach works only for odd values of n.