Magic Square | Even Order
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.
Examples:
Magic Square of order 3: ----------------------- 2 7 6 9 5 1 4 3 8 Magic Square of order 4: ----------------------- 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 Magic Square of order 8: ----------------------- 64 63 3 4 5 6 58 57 56 55 11 12 13 14 50 49 17 18 46 45 44 43 23 24 25 26 38 37 36 35 31 32 33 34 30 29 28 27 39 40 41 42 22 21 20 19 47 48 16 15 51 52 53 54 10 9 8 7 59 60 61 62 2 1
A bit of Theory:
Magic squares are divided into three major categories depending upon order of square.
1) Odd order Magic Square. Example: 3,5,7,… (2*n +1)
2) Doubly-even order Magic Square. Example : 4,8,12,16,.. (4*n)
3) Singly-even order Magic Square. Example : 6,10,14,18,..(4*n +2)
Algorithm for Doubly-even Magic Square :
define an 2-D array of order n*n // fill array with their index-counting // starting from 1 for ( i = 0; i<n; i++) { for ( j = 0; j<n; j++) // filling array with its count value // starting from 1; arr[i][j] = (n*i) + j + 1; } // change value of Array elements // at fix location as per rule // (n*n+1)-arr[i][j] // Top Left corner of Matrix // (order (n/4)*(n/4)) for ( i = 0; i<n/4; i++) { for ( j = 0; j<n/4; j++) arr[i][j] = (n*n + 1) - arr[i][j]; } // Top Right corner of Matrix // (order (n/4)*(n/4)) for ( i = 0; i< n/4; i++) { for ( j = 3* (n/4); j<n; j++) arr[i][j] = (n*n + 1) - arr[i][j]; } // Bottom Left corner of Matrix // (order (n/4)*(n/4)) for ( i = 3* n/4; i<n; i++) { for ( j = 0; j<n/4; j++) arr[i][j] = (n*n + 1) - arr[i][j]; } // Bottom Right corner of Matrix // (order (n/4)*(n/4)) for ( i = 3* n/4; i<n; i++) { for ( j = 3* n/4; j<n; j++) arr[i][j] = (n*n + 1) - arr[i][j]; } // Centre of Matrix (order (n/2)*(n/2)) for ( i = n/4; i<3* n/4; i++) { for ( j = n/4; j<3* n/4; j++) arr[i][j] = (n*n + 1) - arr[i][j]; }
Explanation with an example:(order 4)
1) Define array of order 4*4 and fill it with its count value as:
2) Change value of top-left corner matrix of order (1*1):
3) Change value of top-right corner matrix of order (1*1):
4) Change value of bottom-left corner matrix of order (1*1):
5) Change value of bottom-right corner matrix of order (1*1):
6) Change value of centre matrix of order (2*2):
Implementation for Doubly-even Magic Square:
C++
// C++ Program to print Magic square // of Doubly even order #include<iostream> using namespace std; // Function for calculating Magic square void doublyEven( int n ) { int arr[n][n], i, j; // filling matrix with its count value // starting from 1; for ( i = 0; i < n; i++) for ( j = 0; j < n; j++) arr[i][j] = (n*i) + j + 1; // change value of Array elements // at fix location as per rule // (n*n+1)-arr[i][j] // Top Left corner of Matrix // (order (n/4)*(n/4)) for ( i = 0; i < n/4; i++) for ( j = 0; j < n/4; j++) arr[i][j] = (n*n + 1) - arr[i][j]; // Top Right corner of Matrix // (order (n/4)*(n/4)) for ( i = 0; i < n/4; i++) for ( j = 3 * (n/4); j < n; j++) arr[i][j] = (n*n + 1) - arr[i][j]; // Bottom Left corner of Matrix // (order (n/4)*(n/4)) for ( i = 3 * n/4; i < n; i++) for ( j = 0; j < n/4; j++) arr[i][j] = (n*n+1) - arr[i][j]; // Bottom Right corner of Matrix // (order (n/4)*(n/4)) for ( i = 3 * n/4; i < n; i++) for ( j = 3 * n/4; j < n; j++) arr[i][j] = (n*n + 1) - arr[i][j]; // Centre of Matrix (order (n/2)*(n/2)) for ( i = n/4; i < 3 * n/4; i++) for ( j = n/4; j < 3 * n/4; j++) arr[i][j] = (n*n + 1) - arr[i][j]; // Printing the magic-square for (i = 0; i < n; i++) { for ( j = 0; j < n; j++) cout << arr[i][j] << " " ; cout << "\n" ; } } // driver program int main() { int n=8; doublyEven(n); //Function call return 0; } |
Java
// Java program to print Magic square // of Doubly even order import java.io.*; class GFG { // Function for calculating Magic square static void doublyEven( int n) { int [][] arr = new int [n][n]; int i, j; // filling matrix with its count value // starting from 1; for ( i = 0 ; i < n; i++) for ( j = 0 ; j < n; j++) arr[i][j] = (n*i) + j + 1 ; // change value of Array elements // at fix location as per rule // (n*n+1)-arr[i][j] // Top Left corner of Matrix // (order (n/4)*(n/4)) for ( i = 0 ; i < n/ 4 ; i++) for ( j = 0 ; j < n/ 4 ; j++) arr[i][j] = (n*n + 1 ) - arr[i][j]; // Top Right corner of Matrix // (order (n/4)*(n/4)) for ( i = 0 ; i < n/ 4 ; i++) for ( j = 3 * (n/ 4 ); j < n; j++) arr[i][j] = (n*n + 1 ) - arr[i][j]; // Bottom Left corner of Matrix // (order (n/4)*(n/4)) for ( i = 3 * n/ 4 ; i < n; i++) for ( j = 0 ; j < n/ 4 ; j++) arr[i][j] = (n*n+ 1 ) - arr[i][j]; // Bottom Right corner of Matrix // (order (n/4)*(n/4)) for ( i = 3 * n/ 4 ; i < n; i++) for ( j = 3 * n/ 4 ; j < n; j++) arr[i][j] = (n*n + 1 ) - arr[i][j]; // Centre of Matrix (order (n/2)*(n/2)) for ( i = n/ 4 ; i < 3 * n/ 4 ; i++) for ( j = n/ 4 ; j < 3 * n/ 4 ; j++) arr[i][j] = (n*n + 1 ) - arr[i][j]; // Printing the magic-square for (i = 0 ; i < n; i++) { for ( j = 0 ; j < n; j++) System.out.print(arr[i][j]+ " " ); System.out.println(); } } // driver program public static void main (String[] args) { int n = 8 ; // Function call doublyEven(n); } } // Contributed by Pramod Kumar |
Python3
# Python program to print magic square of double order def DoublyEven(n): # 2-D matrix with all entries as 0 arr = [[(n * y) + x + 1 for x in range (n)] for y in range (n)] # Change value of array elements at fix location # as per the rule (n*n+1)-arr[i][[j] # Corners of order (n/4)*(n/4) # Top left corner for i in range ( 0 ,n / / 4 ): for j in range ( 0 ,n / / 4 ): arr[i][j] = (n * n + 1 ) - arr[i][j]; # Top right corner for i in range ( 0 ,n / / 4 ): for j in range ( 3 * (n / / 4 ),n): arr[i][j] = (n * n + 1 ) - arr[i][j]; # Bottom Left corner for i in range ( 3 * (n / / 4 ),n): for j in range ( 0 ,n / / 4 ): arr[i][j] = (n * n + 1 ) - arr[i][j]; # Bottom Right corner for i in range ( 3 * (n / / 4 ),n): for j in range ( 3 * (n / / 4 ),n): arr[i][j] = (n * n + 1 ) - arr[i][j]; # Centre of matrix,order (n/2)*(n/2) for i in range (n / / 4 , 3 * (n / / 4 )): for j in range (n / / 4 , 3 * (n / / 4 )): arr[i][j] = (n * n + 1 ) - arr[i][j]; # Printing the square for i in range (n): for j in range (n): print ( '%2d ' % (arr[i][j]),end = " " ) print () # Driver Program n = 8 DoublyEven(n) # Contributed by Harshit Agrawal |
C#
// C# program to print Magic square // of Doubly even order using System; class GFG { // Function for calculating Magic square public static void doublyEven( int n) { int [,] arr = new int [n,n]; int i, j; // filling matrix with its count // value starting from 1; for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { arr[i, j] = (n * i) + j + 1; } } // change value of Array elements // at fix location as per rule // (n*n+1)-arr[i][j] // Top Left corner of Matrix // (order (n/4)*(n/4)) for (i = 0; i < n / 4; i++) { for (j = 0; j < n / 4; j++) { arr[i, j] = (n * n + 1) - arr[i, j]; } } // Top Right corner of Matrix // (order (n/4)*(n/4)) for (i = 0; i < n / 4; i++) { for (j = 3 * (n / 4); j < n; j++) { arr[i, j] = (n * n + 1) - arr[i, j]; } } // Bottom Left corner of Matrix // (order (n/4)*(n/4)) for (i = 3 * n / 4; i < n; i++) { for (j = 0; j < n / 4; j++) { arr[i, j] = (n * n + 1) - arr[i, j]; } } // Bottom Right corner of Matrix // (order (n/4)*(n/4)) for (i = 3 * n / 4; i < n; i++) { for (j = 3 * n / 4; j < n; j++) { arr[i, j] = (n * n + 1) - arr[i, j]; } } // Centre of Matrix (order (n/2)*(n/2)) for (i = n / 4; i < 3 * n / 4; i++) { for (j = n / 4; j < 3 * n / 4; j++) { arr[i, j] = (n * n + 1) - arr[i, j]; } } // Printing the magic-square for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { Console.Write(arr[i, j] + " " + " " ); } Console.WriteLine(); } } // Driver Code public static void Main( string [] args) { int n = 8; // Function call doublyEven(n); } } // This code is contributed by Shrikant13 |
PHP
<?php // PHP Program to print Magic square // of Doubly even order // Function for calculating Magic square function doublyEven( $n ) { $arr = array_fill (0, $n , array_fill (0, $n , 0)); // filling matrix with its count // value starting from 1; for ( $i = 0; $i < $n ; $i ++) for ( $j = 0; $j < $n ; $j ++) $arr [ $i ][ $j ] = ( $n * $i ) + $j + 1; // change value of Array elements at fix // location as per rule (n*n+1)-arr[i][j] // Top Left corner of Matrix // (order (n/4)*(n/4)) for ( $i = 0; $i < $n / 4; $i ++) for ( $j = 0; $j < $n / 4; $j ++) $arr [ $i ][ $j ] = ( $n * $n + 1) - $arr [ $i ][ $j ]; // Top Right corner of Matrix // (order (n/4)*(n/4)) for ( $i = 0; $i < $n / 4; $i ++) for ( $j = 3 * ( $n / 4); $j < $n ; $j ++) $arr [ $i ][ $j ] = ( $n * $n + 1) - $arr [ $i ][ $j ]; // Bottom Left corner of Matrix // (order (n/4)*(n/4)) for ( $i = 3 * $n / 4; $i < $n ; $i ++) for ( $j = 0; $j < $n / 4; $j ++) $arr [ $i ][ $j ] = ( $n * $n + 1) - $arr [ $i ][ $j ]; // Bottom Right corner of Matrix // (order (n/4)*(n/4)) for ( $i = 3 * $n / 4; $i < $n ; $i ++) for ( $j = 3 * $n / 4; $j < $n ; $j ++) $arr [ $i ][ $j ] = ( $n * $n + 1) - $arr [ $i ][ $j ]; // Centre of Matrix (order (n/2)*(n/2)) for ( $i = $n / 4; $i < 3 * $n / 4; $i ++) for ( $j = $n / 4; $j < 3 * $n / 4; $j ++) $arr [ $i ][ $j ] = ( $n * $n + 1) - $arr [ $i ][ $j ]; // Printing the magic-square for ( $i = 0; $i < $n ; $i ++) { for ( $j = 0; $j < $n ; $j ++) echo $arr [ $i ][ $j ] . " " ; echo "\n" ; } } // Driver Code $n = 8; doublyEven( $n ); //Function call // This code is contributed by mits ?> |
Javascript
<script> // javascript program to print Magic square // of Doubly even order // Function for calculating Magic square function doublyEven(n) { var arr = Array(n).fill(0).map(x => Array(n).fill(0)); var i, j; // filling matrix with its count value // starting from 1; for ( i = 0; i < n; i++) for ( j = 0; j < n; j++) arr[i][j] = (n*i) + j + 1; // change value of Array elements // at fix location as per rule // (n*n+1)-arr[i][j] // Top Left corner of Matrix // (order (parseInt(n/4))*(parseInt(n/4))) for ( i = 0; i < parseInt(n/4); i++) for ( j = 0; j < parseInt(n/4); j++) arr[i][j] = (n*n + 1) - arr[i][j]; // Top Right corner of Matrix // (order (parseInt(n/4))*(parseInt(n/4))) for ( i = 0; i < parseInt(n/4); i++) for ( j = 3 * (parseInt(n/4)); j < n; j++) arr[i][j] = (n*n + 1) - arr[i][j]; // Bottom Left corner of Matrix // (order (parseInt(n/4))*(parseInt(n/4))) for ( i = 3 * parseInt(n/4); i < n; i++) for ( j = 0; j < parseInt(n/4); j++) arr[i][j] = (n*n+1) - arr[i][j]; // Bottom Right corner of Matrix // (order (parseInt(n/4))*(parseInt(n/4))) for ( i = 3 * parseInt(n/4); i < n; i++) for ( j = 3 * parseInt(n/4); j < n; j++) arr[i][j] = (n*n + 1) - arr[i][j]; // Centre of Matrix (order (n/2)*(n/2)) for ( i = parseInt(n/4); i < 3 * parseInt(n/4); i++) for ( j = parseInt(n/4); j < 3 * parseInt(n/4); j++) arr[i][j] = (n*n + 1) - arr[i][j]; // Printing the magic-square for (i = 0; i < n; i++) { for ( j = 0; j < n; j++) document.write(arr[i][j]+ " " ); document.write( '<br>' ); } } // driver program var n = 8; // Function call doublyEven(n); // This code is contributed by Amit Katiyar </script> |
Output:
64 63 3 4 5 6 58 57 56 55 11 12 13 14 50 49 17 18 46 45 44 43 23 24 25 26 38 37 36 35 31 32 33 34 30 29 28 27 39 40 41 42 22 21 20 19 47 48 16 15 51 52 53 54 10 9 8 7 59 60 61 62 2 1
Time complexity : O(n2)
Auxiliary Space: O(n2). since n2 extra space has been taken.
Reference: http://www.1728.org/magicsq2.html
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