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

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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.

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