# Magic Square

A magic square of order n is an arrangement of n^2 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 n^2.

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 following value:
M = n(n^2+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. 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*(3^2+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*(5^2+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*(7^2+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 previous number by 1, and incrementing the column number of 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, following is the working code:

## C/C++

 `// C++ program to generate odd sized magic squares ` `#include ` `#include ` ` `  `// 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: Sum of "``` `       ````"each row or column %d: "````,  n, n*(n*n+1)/2); ` `    ``for` `(i=0; i

## 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

## Python

 `# 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 Squre for n ="``, n) ` `    ``print` `(``"Sum of each row or column"``,  ` `            ``n ``*` `(n ``*` `n ``+` `1``) ``/` `2``, ````" "````) ` `     `  `    ``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

 ` `

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
```

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

References:
http://en.wikipedia.org/wiki/Magic_square

## tags:

Mathematical Matrix Mathematical Matrix