A. 魔方 n阶是n的排列 2. 一个正方形中的数，通常是不同的整数，所有行、列和两条对角线中的n个数之和为同一常数。幻方包含从1到n的整数 2. . 每行、每列和每对角线中的常数和称为 魔常数或魔和 ，M.正规幻方的幻方常数仅取决于n，且具有以下值： M=n（n 2. +1)/2
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For normal magic squares of order n = 3, 4, 5, ...,the magic constants are: 15, 34, 65, 111, 175, 260, ...
在这篇文章中，我们将讨论如何通过编程生成大小为n的幻方。这种方法只考虑n的奇数，而不适用于偶数。在进一步研究之前，请考虑下面的例子：
Magic Square of size 3----------------------- 2 7 6 9 5 1 4 3 8Sum in each row & each column = 3*(32+1)/2 = 15Magic 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 17Sum in each row & each column = 5*(52+1)/2 = 65Magic 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 30Sum in each row & each column = 7*(72+1)/2 = 175
你有没有发现数字的存储模式？
在任何幻方中，第一个数字即1存储在位置（n/2，n-1）。让这个位置为（i，j）。下一个数字存储在位置（i-1，j＋1），在这里我们可以把每个行和列看作圆形数组，即它们环绕。
有三个条件： 1.上一个数字的行号减1，上一个数字的列号增1，计算出下一个数字的位置。在任何时候，如果计算的行位置变为-1，它将环绕到n-1。类似地，如果计算出的列位置变为n，它将环绕为0。 2.如果幻方在计算位置已经包含一个数字，则计算列位置将减少2，计算行位置将增加1。 3.如果计算的行位置为-1，计算的列位置为n，则新位置为：（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 around9. position of number 9=(2-1,2+1)=(1,3)=(1,0) //wrap around
基于上述方法，以下是工作代码：
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 << ":Sum of " "each row or column " << n * (n * n + 1) / 2 << ":" ; 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:Sum of " "each row or column %d:" , n, n * (n * n + 1) / 2); for (i = 0; i < n; i++) { for (j = 0; j < n; j++) printf ( "%3d " , magicSquare[i][j]); printf ( "" ); } } // 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 , "" ) 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 . ":Sum of each row or column " . $n * ( $n * $n + 1) / 2; echo "" ; for ( $i = 0; $i < $n ; $i ++) { for ( $j = 0; $j < $n ; $j ++) echo $magicSquare [ $i ][ $j ] . " " ; echo "" ; } } // 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> |
输出
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
时间复杂性： O（n） 2. )
辅助空间： O（n） 2. )
注： 这种方法只适用于n的奇数值。
参考资料： http://en.wikipedia.org/wiki/Magic_square
本文由 阿希什·巴恩瓦尔 并由Geeksforgeks团队审核。如果您发现任何不正确的地方，或者您想分享有关上述主题的更多信息，请写评论
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