Printing all solutions in N-Queen Problem

The N Queen is the problem of placing N chess queens on an N×N chessboard so that no two queens attack each other. For example, following is a solution for 4 Queen problem.

The N Queen is the problem of placing N chess queens on an N×N chessboard so that no two queens attack each other. For example, following are two solutions for 4 Queen problem.


nQueen-solution2



In previous post, we have discussed an approach that prints only one possible solution, so now in this post the task is to print all solutions in N-Queen Problem. The solution discussed here is an extension of same approach.

Backtracking Algorithm
The idea is to place queens one by one in different columns, starting from the leftmost column. When we place a queen in a column, we check for clashes with already placed queens. In the current column, if we find a row for which there is no clash, we mark this row and column as part of the solution. If we do not find such a row due to clashes then we backtrack and return false.

1) Start in the leftmost column
2) If all queens are placed
    return true
3) Try all rows in the current column.  Do following
   for every tried row.
    a) If the queen can be placed safely in this row
       then mark this [row, column] as part of the 
       solution and recursively check if placing  
       queen here leads to a solution.
    b) If placing queen in [row, column] leads to a
       solution then return true.
    c) If placing queen doesn't lead to a solution 
       then unmark this [row, column] (Backtrack) 
       and go to step (a) to try other rows.
3) If all rows have been tried and nothing worked, 
   return false to trigger backtracking.

There is only a slight modification in above algorithm that is highlighted in the code.

C++

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/* C/C++ program to solve N Queen Problem using
backtracking */
#include<bits/stdc++.h>
#define N 4
  
/* A utility function to print solution */
void printSolution(int board[N][N])
{
    static int k = 1;
    printf("%d-\n",k++);
    for (int i = 0; i < N; i++)
    {
        for (int j = 0; j < N; j++)
            printf(" %d ", board[i][j]);
        printf("\n");
    }
    printf("\n");
}
  
/* A utility function to check if a queen can
be placed on board[row][col]. Note that this
function is called when "col" queens are
already placed in columns from 0 to col -1.
So we need to check only left side for
attacking queens */
bool isSafe(int board[N][N], int row, int col)
{
    int i, j;
  
    /* Check this row on left side */
    for (i = 0; i < col; i++)
        if (board[row][i])
            return false;
  
    /* Check upper diagonal on left side */
    for (i=row, j=col; i>=0 && j>=0; i--, j--)
        if (board[i][j])
            return false;
  
    /* Check lower diagonal on left side */
    for (i=row, j=col; j>=0 && i<N; i++, j--)
        if (board[i][j])
            return false;
  
    return true;
}
  
/* A recursive utility function to solve N
Queen problem */
bool solveNQUtil(int board[N][N], int col)
{
    /* base case: If all queens are placed
    then return true */
    if (col == N)
    {
        printSolution(board);
        return true;
    }
  
    /* Consider this column and try placing
    this queen in all rows one by one */
    bool res = false;
    for (int i = 0; i < N; i++)
    {
        /* Check if queen can be placed on
        board[i][col] */
        if ( isSafe(board, i, col) )
        {
            /* Place this queen in board[i][col] */
            board[i][col] = 1;
  
            // Make result true if any placement
            // is possible
            res = solveNQUtil(board, col + 1) || res;
  
            /* If placing queen in board[i][col]
            doesn't lead to a solution, then
            remove queen from board[i][col] */
            board[i][col] = 0; // BACKTRACK
        }
    }
  
    /* If queen can not be place in any row in
        this column col then return false */
    return res;
}
  
/* This function solves the N Queen problem using
Backtracking. It mainly uses solveNQUtil() to
solve the problem. It returns false if queens
cannot be placed, otherwise return true and
prints placement of queens in the form of 1s.
Please note that there may be more than one
solutions, this function prints one of the
feasible solutions.*/
void solveNQ()
{
    int board[N][N];
    memset(board, 0, sizeof(board));
  
    if (solveNQUtil(board, 0) == false)
    {
        printf("Solution does not exist");
        return ;
    }
  
    return ;
}
  
// driver program to test above function
int main()
{
    solveNQ();
    return 0;
}

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/* Java program to solve N Queen   


Problem using backtracking */


  


class GfG  


{ 


  


static int N = 4;  


static int k = 1; 


  


/* A utility function to print solution */


static void printSolution(int board[][])  


{  


    System.out.printf("%d-\n", k++);  


    for (int i = 0; i < N; i++)  


    {  


        for (int j = 0; j < N; j++)  


            System.out.printf(" %d ", board[i][j]);  


        System.out.printf("\n");  


    }  


    System.out.printf("\n");  


}  


  


/* A utility function to check if a queen can  


be placed on board[row][col]. Note that this  


function is called when "col" queens are  


already placed in columns from 0 to col -1.  


So we need to check only left side for  


attacking queens */


static boolean isSafe(int board[][], int row, int col)  


{  


    int i, j;  


  


    /* Check this row on left side */


    for (i = 0; i < col; i++)  


        if (board[row][i] == 1)  


            return false;  


  


    /* Check upper diagonal on left side */


    for (i = row, j = col; i >= 0 && j >= 0; i--, j--)  


        if (board[i][j] == 1)  


            return false;  


  


    /* Check lower diagonal on left side */


    for (i = row, j = col; j >= 0 && i < N; i++, j--)  


        if (board[i][j] == 1)  


            return false;  


  


    return true;  


}  


  


/* A recursive utility function   


to solve N Queen problem */


static boolean solveNQUtil(int board[][], int col)  


{  


    /* base case: If all queens are placed  


    then return true */


    if (col == N)  


    {  


        printSolution(board);  


        return true;  


    }  


  


    /* Consider this column and try placing  


    this queen in all rows one by one */


    boolean res = false;  


    for (int i = 0; i < N; i++)  


    {  


        /* Check if queen can be placed on  


        board[i][col] */


        if ( isSafe(board, i, col) )  


        {  


            /* Place this queen in board[i][col] */


            board[i][col] = 1;  


  


            // Make result true if any placement  


            // is possible  


            res = solveNQUtil(board, col + 1) || res;  


  


            /* If placing queen in board[i][col]  


            doesn't lead to a solution, then  


            remove queen from board[i][col] */


            board[i][col] = 0; // BACKTRACK  


        }  


    }  


  


    /* If queen can not be place in any row in  


        this column col then return false */


    return res;  


}  


  


/* This function solves the N Queen problem using  


Backtracking. It mainly uses solveNQUtil() to  


solve the problem. It returns false if queens  


cannot be placed, otherwise return true and  


prints placement of queens in the form of 1s.  


Please note that there may be more than one  


solutions, this function prints one of the  


feasible solutions.*/


static void solveNQ()  


{  


    int board[][] = new int[N][N];  


  


    if (solveNQUtil(board, 0) == false)  


    {  


        System.out.printf("Solution does not exist");  


        return ;  


    }  


  


    return ;  


}  


  


// Driver code  


public static void main(String[] args) 


{ 


    solveNQ(); 


} 


} 


  


// This code has been contributed 


// by 29AjayKumar 



















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Python3














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''' Python3 program to solve N Queen Problem using  


backtracking '''


k = 1


  


# A utility function to print solution  


def printSolution(board):  


  


    global k 


    print(k, "-\n") 


    k = k + 1


    for i in range(4):  


        for j in range(4): 


            print(board[i][j], end = " ") 


        print("\n") 


    print("\n")  


  


''' A utility function to check if a queen can  


be placed on board[row][col]. Note that this  


function is called when "col" queens are  


already placed in columns from 0 to col -1.  


So we need to check only left side for  


attacking queens '''


def isSafe(board, row, col) : 


      


    # Check this row on left side  


    for i in range(col):  


        if (board[row][i]):  


            return False


  


    # Check upper diagonal on left side  


    i = row 


    j = col 


    while i >= 0 and j >= 0: 


        if(board[i][j]): 


            return False; 


        i -= 1


        j -= 1


  


    # Check lower diagonal on left side  


    i = row 


    j = col 


    while j >= 0 and i < 4: 


        if(board[i][j]): 


            return False


        i = i + 1


        j = j - 1


  


    return True


  


''' A recursive utility function to solve N  


Queen problem '''


def solveNQUtil(board, col) : 


      


    ''' base case: If all queens are placed  


    then return true '''


    if (col == 4):  


        printSolution(board)  


        return True


  


    ''' Consider this column and try placing  


    this queen in all rows one by one '''


    res = False


    for i in range(4): 


      


        ''' Check if queen can be placed on  


        board[i][col] '''


        if (isSafe(board, i, col)):  


          


            # Place this queen in board[i][col]  


            board[i][col] = 1;  


  


            # Make result true if any placement  


            # is possible  


            res = solveNQUtil(board, col + 1) or res;  


  


            ''' If placing queen in board[i][col]  


            doesn't lead to a solution, then  


            remove queen from board[i][col] '''


            board[i][col] = 0 # BACKTRACK  


          


    ''' If queen can not be place in any row in  


        this column col then return false '''


    return res 


  


''' This function solves the N Queen problem using  


Backtracking. It mainly uses solveNQUtil() to  


solve the problem. It returns false if queens  


cannot be placed, otherwise return true and  


prints placement of queens in the form of 1s.  


Please note that there may be more than one  


solutions, this function prints one of the  


feasible solutions.'''


def solveNQ() : 


  


    board = [[0 for j in range(10)]  


                for i in range(10)] 


  


    if (solveNQUtil(board, 0) == False):  


      


        print("Solution does not exist")  


        return


    return


  


# Driver Code  


solveNQ()  


  


# This code is contributed by YatinGupta 



















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C#














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/* C# program to solve N Queen  


Problem using backtracking */


using System; 


  


class GfG  


{  


  


static int N = 4;  


static int k = 1;  


  


/* A utility function to print solution */


static void printSolution(int [,]board)  


{  


    Console.Write("{0}-\n", k++);  


    for (int i = 0; i < N; i++)  


    {  


        for (int j = 0; j < N; j++)  


            Console.Write(" {0} ", board[i, j]);  


        Console.Write("\n");  


    }  


    Console.Write("\n");  


}  


  


/* A utility function to check if a queen can  


be placed on board[row,col]. Note that this  


function is called when "col" queens are  


already placed in columns from 0 to col -1.  


So we need to check only left side for  


attacking queens */


static bool isSafe(int [,]board, int row, int col)  


{  


    int i, j;  


  


    /* Check this row on left side */


    for (i = 0; i < col; i++)  


        if (board[row, i] == 1)  


            return false;  


  


    /* Check upper diagonal on left side */


    for (i = row, j = col; i >= 0 && j >= 0; i--, j--)  


        if (board[i, j] == 1)  


            return false;  


  


    /* Check lower diagonal on left side */


    for (i = row, j = col; j >= 0 && i < N; i++, j--)  


        if (board[i, j] == 1)  


            return false;  


  


    return true;  


}  


  


/* A recursive utility function  


to solve N Queen problem */


static bool solveNQUtil(int [,]board, int col)  


{  


    /* base case: If all queens are placed  


    then return true */


    if (col == N)  


    {  


        printSolution(board);  


        return true;  


    }  


  


    /* Consider this column and try placing  


    this queen in all rows one by one */


    bool res = false;  


    for (int i = 0; i < N; i++)  


    {  


        /* Check if queen can be placed on  


        board[i,col] */


        if ( isSafe(board, i, col) )  


        {  


            /* Place this queen in board[i,col] */


            board[i,col] = 1;  


  


            // Make result true if any placement  


            // is possible  


            res = solveNQUtil(board, col + 1) || res;  


  


            /* If placing queen in board[i,col]  


            doesn't lead to a solution, then  


            remove queen from board[i,col] */


            board[i,col] = 0; // BACKTRACK  


        }  


    }  


  


    /* If queen can not be place in any row in  


        this column col then return false */


    return res;  


}  


  


/* This function solves the N Queen problem using  


Backtracking. It mainly uses solveNQUtil() to  


solve the problem. It returns false if queens  


cannot be placed, otherwise return true and  


prints placement of queens in the form of 1s.  


Please note that there may be more than one  


solutions, this function prints one of the  


feasible solutions.*/


static void solveNQ()  


{  


    int [,]board = new int[N, N];  


  


    if (solveNQUtil(board, 0) == false)  


    {  


        Console.Write("Solution does not exist");  


        return ;  


    }  


  


    return ;  


}  


  


// Driver code  


public static void Main()  


{  


    solveNQ();  


}  


}  


  


/* This code contributed by PrinciRaj1992 */



















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Output:


1-
 0  0  1  0 
 1  0  0  0 
 0  0  0  1 
 0  1  0  0 

2-
 0  1  0  0 
 0  0  0  1 
 1  0  0  0 
 0  0  1  0 



This article is contributed by Sahil Chhabra. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.


Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



          
          
          

          

    

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