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
Example
#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