# N Queen Problem

Data StructureAlgorithmsBacktracking Algorithms

This problem is to find an arrangement of N queens on a chess board, such that no queen can attack any other queens on the board.

The chess queens can attack in any direction as horizontal, vertical, horizontal and diagonal way.

A binary matrix is used to display the positions of N Queens, where no queens can attack other queens.

#### Input and Output

Input:
The size of a chess board. Generally, it is 8. as (8 x 8 is the size of a normal chess board.)
Output:
The matrix that represents in which row and column the N Queens can be placed.
If the solution does not exist, it will return false.

1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0

In this output, the value 1 indicates the correct place for the queens.
The 0 denotes the blank spaces on the chess board.

#### Algorithm

isValid(board, row, col)

#### Input: The chess board, row and the column of the board.

Output: True when placing a queen in row and place position is a valid or not.

Begin
if there is a queen at the left of current col, then
return false
if there is a queen at the left upper diagonal, then
return false
if there is a queen at the left lower diagonal, then
return false;
return true //otherwise it is valid place
End

solveNQueen(board, col)

Input: The chess board, the col where the queen is trying to be placed.

Output: The position matrix where queens are placed.

Begin
if all columns are filled, then
return true
for each row of the board, do
if isValid(board, i, col), then
set queen at place (i, col) in the board
if solveNQueen(board, col+1) = true, then
return true
otherwise remove queen from place (i, col) from board.
done
return false
End

#### Source Code (C++)

#include<iostream>
using namespace std;
#define N 8

void printBoard(int board[N][N]) {
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++)
cout << board[i][j] << " ";
cout << endl;
}
}

bool isValid(int board[N][N], int row, int col) {
for (int i = 0; i < col; i++)    //check whether there is queen in the left or not
if (board[row][i])
return false;
for (int i=row, j=col; i>=0 && j>=0; i--, j--)
if (board[i][j])       //check whether there is queen in the left upper diagonal or not
return false;
for (int i=row, j=col; j>=0 && i<N; i++, j--)
if (board[i][j])      //check whether there is queen in the left lower diagonal or not
return false;
return true;
}

bool solveNQueen(int board[N][N], int col) {
if (col >= N)           //when N queens are placed successfully
return true;
for (int i = 0; i < N; i++) {     //for each row, check placing of queen is possible or not
if (isValid(board, i, col) ) {
board[i][col] = 1;      //if validate, place the queen at place (i, col)
if ( solveNQueen(board, col + 1))    //Go for the other columns recursively
return true;

board[i][col] = 0;        //When no place is vacant remove that queen
}
}
return false;       //when no possible order is found
}

bool checkSolution() {
int board[N][N];
for(int i = 0; i<N; i++)
for(int j = 0; j<N; j++)
board[i][j] = 0;      //set all elements to 0

if ( solveNQueen(board, 0) == false ) {     //starting from 0th column
cout << "Solution does not exist";
return false;
}
printBoard(board);
return true;
}

int main() {
checkSolution();
}

#### Output

1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0
Published on 09-Jul-2018 13:09:03
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