Given the dimension of a chess board (N x M), determine the minimum number of queens required to cover all the squares of the board. A queen can attack any square along its row, column or diagonals.
Examples:
Input : N = 8, M = 8
Output : 5
Layout : Q X X X X X X X
X X Q X X X X X
X X X X Q X X X
X Q X X X X X X
X X X Q X X X X
X X X X X X X X
X X X X X X X X
X X X X X X X X Input : N = 3, M = 5
Output : 2
Layout : Q X X X X
X X X X X
X X X Q X
This article attempts to solve the problem in a very simple way without much optimization.
- Step 1: Starting from any corner square of the board, find an 'uncovered' square (Uncovered square is a square which isn't attacked by any of the queens already placed). If none found, goto Step 4.
- Step 2: Place a Queen on this square and increment variable 'count' by 1.
- Step 3: Repeat step 1.
- Step 4: Now, you've got a layout where every square is covered. Therefore, the value of 'count' can be the answer. However, you might be able to do better, as there might exist a better layout with lesser number of queens. So, store this 'count' as the best value till now and proceed to find a better solution.
- Step 5: Remove the last queen placed and place it in the next 'uncovered' cell. Step 6: Proceed recursively and try out all the possible layouts. Finally, the one with the least number of queens is the answer.
Implementation: Dry run the following code for better understanding.
#include <bits/stdc++.h>
using namespace std;
// The chessboard is represented by a 2D array.
bool board[8][8];
// N x M is the dimension of the chess board.
const int N = 8, M = 8;
// The minimum number of queens required.
// Initially, set to MAX_VAL.
int minCount = INT_MAX;
string layout; // Stores the best layout.
// Stores the current layout in 'layout'
// variable as String.
void storeLayout()
{
stringstream ss;
for (int i = 0; i < N; i++) {
for (int j = 0; j < M; j++)
ss << (board[i][j] ? "Q " : "X ");
ss << "\n";
}
layout = ss.str();
}
// Returns 'true' if the square (row, col) is
// being attacked by at least one queen.
bool isAttacked(int row, int col)
{
int i, j;
// Check the 'col'th column for any queen.
for (i = 0; i < N; ++i)
if (board[i][col])
return true;
// Check the 'row'th row for any queen.
for (j = 0; j < M; ++j)
if (board[row][j])
return true;
// Check the diagonals for any queen.
for (i = 0; i < min(N, M); ++i)
if (row - i >= 0 && col - i >= 0
&& board[row - i][col - i])
return true;
else if (row - i >= 0 && col + i < M
&& board[row - i][col + i])
return true;
else if (row + i < N && col - i >= 0
&& board[row + i][col - i])
return true;
else if (row + i < N && col + i < M
&& board[row + i][col + i])
return true;
// This square is unattacked. Hence return 'false'.
return false;
}
// Finds minimum count of queens needed and places them.
void placeQueen(int countSoFar)
{
int i, j;
if (countSoFar >= minCount)
// We've already obtained a solution with lesser or
// same number of queens. Hence, no need to proceed.
return;
bool flag = false;
while (!flag) {
for (i = 0; i < N; ++i)
for (j = 0; j < M; ++j) {
if (!isAttacked(i, j)) {
i = N;
flag = true;
break;
}
}
if (flag)
break;
// All squares all covered. Hence, this
// is the best solution till now.
minCount = countSoFar;
storeLayout();
break;
}
// Checks if there exists any unattacked cells.
for (i = 0; i < N; ++i)
for (j = 0; j < M; ++j) {
if (!isAttacked(i, j)) {
// Square (i, j) is unattacked.
// Therefore, place a queen here.
board[i][j] = true;
// Increment 'count' and proceed
// recursively.
placeQueen(countSoFar + 1);
// Remove this queen and attempt to
// find a better solution.
board[i][j] = false;
}
}
}
// Driver code
int main()
{
memset(board, false, sizeof(board));
placeQueen(0);
cout << minCount << endl;
cout << "\nLayout: \n" << layout;
}
// Java program to find minimum number of queens needed
// to cover a given chess board.
public class Backtracking {
// The chessboard is represented by a 2D array.
static boolean[][] board;
// N x M is the dimension of the chess board.
static int N, M;
// The minimum number of queens required.
// Initially, set to MAX_VAL.
static int minCount = Integer.MAX_VALUE;
static String layout; // Stores the best layout.
// Driver code
public static void main(String[] args)
{
N = 8;
M = 8;
board = new boolean[N][M];
placeQueen(0);
System.out.println(minCount);
System.out.println("\nLayout: \n" + layout);
}
// Finds minimum count of queens needed and places them.
static void placeQueen(int countSoFar)
{
int i, j;
if (countSoFar >= minCount)
// We've already obtained a solution with lesser
// or same number of queens. Hence, no need to
// proceed.
return;
// Checks if there exists any unattacked cells.
findUnattackedCell : {
for (i = 0; i < N; ++i)
for (j = 0; j < M; ++j)
if (!isAttacked(i, j))
// Square (i, j) is unattacked.
break findUnattackedCell;
// All squares all covered. Hence, this
// is the best solution till now.
minCount = countSoFar;
storeLayout();
return;
}
for (i = 0; i < N; ++i)
for (j = 0; j < M; ++j) {
if (!isAttacked(i, j)) {
// Square (i, j) is unattacked.
// Therefore, place a queen here.
board[i][j] = true;
// Increment 'count' and proceed
// recursively.
placeQueen(countSoFar + 1);
// Remove this queen and attempt to
// find a better solution.
board[i][j] = false;
}
}
}
// Returns 'true' if the square (row, col) is
// being attacked by at least one queen.
static boolean isAttacked(int row, int col)
{
int i, j;
// Check the 'col'th column for any queen.
for (i = 0; i < N; ++i)
if (board[i][col])
return true;
// Check the 'row'th row for any queen.
for (j = 0; j < M; ++j)
if (board[row][j])
return true;
// Check the diagonals for any queen.
for (i = 0; i < Math.min(N, M); ++i)
if (row - i >= 0 && col - i >= 0
&& board[row - i][col - i])
return true;
else if (row - i >= 0 && col + i < M
&& board[row - i][col + i])
return true;
else if (row + i < N && col - i >= 0
&& board[row + i][col - i])
return true;
else if (row + i < N && col + i < M
&& board[row + i][col + i])
return true;
// This square is unattacked. Hence return 'false'.
return false;
}
// Stores the current layout in 'layout'
// variable as String.
static void storeLayout()
{
StringBuilder sb = new StringBuilder();
for (boolean[] row : board) {
for (boolean cell : row)
sb.append(cell ? "Q " : "X ");
sb.append("\n");
}
layout = sb.toString();
}
}
# The chessboard is represented by a 2D array.
board = [[False] * 8 for _ in range(8)]
# N x M is the dimension of the chess board.
N, M = 8, 8
# The minimum number of queens required.
# Initially, set to MAX_VALUE.
minCount = float('inf')
layout = "" # Stores the best layout.
# Stores the current layout in 'layout'
# variable as String.
def storeLayout():
global layout
str_layout = ""
for i in range(N):
for j in range(M):
if board[i][j]:
str_layout += "Q "
else:
str_layout += "X "
str_layout += "\n"
layout = str_layout
# Returns 'True' if the square (row, col) is
# being attacked by at least one queen.
def isAttacked(row, col):
# Check the 'col'th column for any queen.
for i in range(N):
if board[i][col]:
return True
# Check the 'row'th row for any queen.
for j in range(M):
if board[row][j]:
return True
# Check the diagonals for any queen.
for i in range(min(N, M)):
if row - i >= 0 and col - i >= 0 and board[row - i][col - i]:
return True
elif row - i >= 0 and col + i < M and board[row - i][col + i]:
return True
elif row + i < N and col - i >= 0 and board[row + i][col - i]:
return True
elif row + i < N and col + i < M and board[row + i][col + i]:
return True
return False
# Finds minimum count of queens needed and places them.
def placeQueen(countSoFar):
global minCount
if countSoFar >= minCount:
# We've already obtained a solution with lesser or
# same number of queens. Hence, no need to proceed.
return
flag = False
while not flag:
for i in range(N):
for j in range(M):
if not isAttacked(i, j):
i = N
flag = True
break
if flag:
break
if flag:
break
# All squares are covered. Hence, this
# is the best solution till now.
minCount = countSoFar
storeLayout()
break
# Checks if there exists any unattacked cells.
for i in range(N):
for j in range(M):
if not isAttacked(i, j):
# Square (i, j) is unattacked.
# Therefore, place a queen here.
board[i][j] = True
# Increment 'count' and proceed
# recursively.
placeQueen(countSoFar + 1)
# Remove this queen and attempt to
# find a better solution.
board[i][j] = False
# Driver code
placeQueen(0)
print(minCount)
print("\nLayout: \n" + layout)
using System;
using System.Text;
class Program {
// The chessboard is represented by a 2D array
static bool[, ] board = new bool[8, 8];
// N * M is the dimension of the chess board
const int N = 8, M = 8;
// The minimum number of queens required.
// Initially, set to MAX_VALUE.
static int minCount = int.MaxValue;
// Stores the best layout.
static string layout;
// Stores the current layout in 'layout'
// variable as String.
static void StoreLayout()
{
StringBuilder sb = new StringBuilder();
for (int i = 0; i < N; i++) {
for (int j = 0; j < M; j++) {
sb.Append(board[i, j] ? "Q " : "X ");
}
sb.Append("\n");
}
layout = sb.ToString();
}
// Returns 'true' if the square (row, col) is
// being attacked by at least one queen.
static bool IsAttacked(int row, int col)
{
int i, j;
// Check the 'col'th column for any queen.
for (i = 0; i < N; ++i) {
if (board[i, col])
return true;
}
// Check the 'row'th row for any queen.
for (j = 0; j < M; ++j) {
if (board[row, j])
return true;
}
// Check the diagonals for any queen.
for (i = 0; i < Math.Min(N, M); ++i) {
if (row - i >= 0 && col - i >= 0
&& board[row - i, col - i])
return true;
else if (row - i >= 0 && col + i < M
&& board[row - i, col + i])
return true;
else if (row + i < N && col - i >= 0
&& board[row + i, col - i])
return true;
else if (row + i < N && col + i < M
&& board[row + i, col + i])
return true;
}
// This square is unattacked. Hence return 'false'.
return false;
}
// Finds minimum count of queens needed and places them.
static void PlaceQueen(int countSoFar)
{
int i, j;
// We've already obtained a solution with lesser or
// same number of queens. Hence, no need to proceed.
if (countSoFar >= minCount)
return;
bool flag = false;
while (!flag) {
for (i = 0; i < N; ++i) {
for (j = 0; j < M; ++j) {
if (!IsAttacked(i, j)) {
i = N;
flag = true;
break;
}
}
if (flag)
break;
}
if (flag)
break;
// All squares all covered. Hence, this
// is the best solution till now.
minCount = countSoFar;
StoreLayout();
break;
}
// Checks if there exists any unattacked cells.
for (i = 0; i < N; ++i) {
for (j = 0; j < M; ++j) {
if (!IsAttacked(i, j)) {
// Square (i, j) is unattacked.
// Therefore, place a queen here.
board[i, j] = true;
// Increment 'count' and proceed
// recursively.
PlaceQueen(countSoFar + 1);
// Remove this queen and attempt to
// find a better solution.
board[i, j] = false;
}
}
}
}
static void Main()
{
Array.Clear(board, 0, board.Length);
PlaceQueen(0);
Console.WriteLine(minCount);
Console.WriteLine("\nLayout: \n" + layout);
}
}
// The chessboard is represented by a 2D array.
const board = Array.from(Array(8), () => new Array(8).fill(false));
// N x M is the dimension of the chess board.
const N = 8, M = 8;
// The minimum number of queens required.
// Initially, set to MAX_VALUE.
let minCount = Number.MAX_VALUE;
let layout = ""; // Stores the best layout.
// Stores the current layout in 'layout'
// variable as String.
function storeLayout() {
let str = "";
for (let i = 0; i < N; i++) {
for (let j = 0; j < M; j++) {
if (board[i][j]) {
str += "Q ";
} else {
str += "X ";
}
}
str += "\n";
}
layout = str;
}
// Returns 'true' if the square (row, col) is
// being attacked by at least one queen.
function isAttacked(row, col) {
let i, j;
// Check the 'col'th column for any queen.
for (i = 0; i < N; ++i) {
if (board[i][col]) {
return true;
}
}
// Check the 'row'th row for any queen.
for (j = 0; j < M; ++j) {
if (board[row][j]) {
return true;
}
}
// Check the diagonals for any queen.
for (i = 0; i < Math.min(N, M); ++i) {
if ( row - i >= 0 && col - i >= 0 && board[row - i][col - i]) {
return true;
} else if ( row - i >= 0 && col + i < M && board[row - i][col + i] ) {
return true;
} else if ( row + i < N && col - i >= 0 && board[row + i][col - i] ) {
return true;
} else if ( row + i < N && col + i < M && board[row + i][col + i] ) {
return true;
}
}
// This square is unattacked. Hence return 'false'.
return false;
}
// Finds minimum count of queens needed and places them.
function placeQueen(countSoFar) {
let i, j;
if (countSoFar >= minCount) {
// We've already obtained a solution with lesser or
// same number of queens. Hence, no need to proceed.
return;
}
let flag = false;
while (!flag) {
for (i = 0; i < N; ++i) {
for (j = 0; j < M; ++j) {
if (!isAttacked(i, j)) {
i = N;
flag = true;
break;
}
}
if (flag) {
break;
}
}
if (flag) {
break;
}
// All squares are covered. Hence, this
// is the best solution till now.
minCount = countSoFar;
storeLayout();
break;
}
// Checks if there exists any unattacked cells.
for (i = 0; i < N; ++i) {
for (j = 0; j < M; ++j) {
if (!isAttacked(i, j)) {
// Square (i, j) is unattacked.
// Therefore, place a queen here.
board[i][j] = true;
// Increment 'count' and proceed
// recursively.
placeQueen(countSoFar + 1);
// Remove this queen and attempt to
// find a better solution.
board[i][j] = false;
}
}
}
}
// Driver code
placeQueen(0);
console.log(minCount);
console.log("\nLayout: \n" + layout);
Output
5 Layout: Q X X X X X X X X X Q X X X X X X X X X Q X X X X Q X X X X X X X X X Q X X X X X X X X X X X X X X X X X X X X X X X X X X X X