N Queens Problem in DSA Java - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of N Queens solution count for N=4

What is the output of the following Java code that counts the number of solutions to the 4-Queens problem?

DSA Java

public class NQueens {
    private int count = 0;
    private boolean isSafe(int row, int col, int[] board) {
        for (int i = 0; i < row; i++) {
            if (board[i] == col || Math.abs(board[i] - col) == row - i) {
                return false;
            }
        }
        return true;
    }
    private void solve(int row, int n, int[] board) {
        if (row == n) {
            count++;
            return;
        }
        for (int col = 0; col < n; col++) {
            if (isSafe(row, col, board)) {
                board[row] = col;
                solve(row + 1, n, board);
            }
        }
    }
    public int totalNQueens(int n) {
        int[] board = new int[n];
        solve(0, n, board);
        return count;
    }
    public static void main(String[] args) {
        NQueens nq = new NQueens();
        System.out.println(nq.totalNQueens(4));
    }
}

A 4

B 2

C 1

D 0

💡 Hint

Recall the known number of solutions for 4-Queens problem.

The 4-Queens problem has exactly 2 distinct solutions where 4 queens can be placed on a 4x4 board without threatening each other.

🧠 Conceptual

intermediate

Understanding the safety check in N Queens

In the N Queens problem, what does the safety check condition Math.abs(board[i] - col) == row - i ensure?

A It checks if two queens are on the same diagonal.

B It checks if two queens are in adjacent columns.

C It checks if two queens are in the same row.

D It checks if two queens are in the same column.

💡 Hint

Think about how diagonals relate to row and column differences.

Two queens are on the same diagonal if the difference in their columns equals the difference in their rows.

🔧 Debug

advanced

Identify the error in this N Queens solver snippet

What error will occur when running this code snippet for N Queens?

DSA Java

private void solve(int row, int n, int[] board) {
    if (row == n) {
        count++;
        return;
    }
    for (int col = 0; col <= n; col++) {
        if (isSafe(row, col, board)) {
            board[row] = col;
            solve(row + 1, n, board);
        }
    }
}

A ArrayIndexOutOfBoundsException

B StackOverflowError

C No error, runs correctly

D NullPointerException

💡 Hint

Check the loop boundary for columns.

The loop runs with col <= n, but valid columns are 0 to n-1. Accessing board[row] = col is safe, but isSafe uses col which can be n, causing index errors.

🚀 Application

advanced

Number of solutions for 5 Queens problem

Using a correct N Queens solver, what is the number of distinct solutions for N=5?

A 10

B 12

C 5

D 8

💡 Hint

The 5-Queens problem has a known solution count.

The 5-Queens problem has exactly 10 solutions, but only 8 are distinct when considering symmetry.

❓ Predict Output

expert

Output of printing one solution board for N=4

What is the printed output of the following Java code that prints one valid 4-Queens board solution?

DSA Java

public class NQueensPrint {
    private boolean isSafe(int row, int col, int[] board) {
        for (int i = 0; i < row; i++) {
            if (board[i] == col || Math.abs(board[i] - col) == row - i) {
                return false;
            }
        }
        return true;
    }
    private boolean solve(int row, int n, int[] board) {
        if (row == n) {
            return true;
        }
        for (int col = 0; col < n; col++) {
            if (isSafe(row, col, board)) {
                board[row] = col;
                if (solve(row + 1, n, board)) {
                    return true;
                }
            }
        }
        return false;
    }
    public void printBoard(int[] board) {
        int n = board.length;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (board[i] == j) {
                    System.out.print("Q ");
                } else {
                    System.out.print(". ");
                }
            }
            System.out.println();
        }
    }
    public static void main(String[] args) {
        NQueensPrint nq = new NQueensPrint();
        int n = 4;
        int[] board = new int[n];
        nq.solve(0, n, board);
        nq.printBoard(board);
    }
}

A

Q . . . 
. Q . . 
. . . Q 
. . Q . 

B

. Q . . 
. . . Q 
Q . . . 
. . Q . 

C

Q . . . 
. . Q . 
. Q . . 
. . . Q 

D

. . Q . 
Q . . . 
. . . Q 
. Q . . 

💡 Hint

The solver returns the first valid solution it finds.

The backtracking solver places queens row by row starting from column 0, so the first solution printed matches option C.

Practice

1. What is the main goal of the N Queens Problem?

easy

A. Place queens only on the diagonal cells of the board

B. Place N queens on an NxN chessboard so that no two queens attack each other

C. Count the total number of queens on the board

D. Find the shortest path for a queen to move across the board

Solution

1. Step 1: Understand the problem statement

   The N Queens Problem requires placing N queens on an NxN chessboard so that no two queens threaten each other.

2. Step 2: Identify the correct goal

   This means no two queens share the same row, column, or diagonal.

3. Final Answer:

   Place N queens on an NxN chessboard so that no two queens attack each other -> Option B

4. Quick Check:

   Goal = Place queens safely [OK]

Hint: Remember: no two queens can share row, column, or diagonal [OK]

Common Mistakes:

• Thinking queens can share the same row or column
• Confusing the problem with counting queens
• Assuming queens only attack diagonally

2. Which of the following Java method signatures is correct for a recursive function solving the N Queens Problem?

easy

A. static int solveNQueens(int[][] board, int n, int row)

B. public void solveNQueens(int n, int row, int[][] board)

C. public static boolean solveNQueens(int row, int n, int[][] board)

D. boolean solveNQueens(int n, int[][] board, int row)

Solution

1. Step 1: Check method signature correctness

   The recursive method should return a boolean indicating success/failure, take current row, board size, and board state.

2. Step 2: Validate parameter order and modifiers

   public static boolean solveNQueens(int row, int n, int[][] board) matches this: public static boolean solveNQueens(int row, int n, int[][] board)

3. Final Answer:

   public static boolean solveNQueens(int row, int n, int[][] board) -> Option C

4. Quick Check:

   Correct signature = public static boolean solveNQueens(int row, int n, int[][] board) [OK]

Hint: Recursive method returns boolean and takes current row first [OK]

Common Mistakes:

• Wrong return type (e.g., void instead of boolean)
• Incorrect parameter order
• Missing static keyword if required

3. What will be the output of the following Java code snippet for N=4?

public class NQueensTest {
    static int count = 0;
    public static void main(String[] args) {
        int n = 4;
        int[][] board = new int[n][n];
        solveNQueens(0, n, board);
        System.out.println(count);
    }
    static boolean solveNQueens(int row, int n, int[][] board) {
        if (row == n) {
            count++;
            return true;
        }
        boolean res = false;
        for (int col = 0; col < n; col++) {
            if (isSafe(row, col, board, n)) {
                board[row][col] = 1;
                res = solveNQueens(row + 1, n, board) || res;
                board[row][col] = 0;
            }
        }
        return res;
    }
    static boolean isSafe(int row, int col, int[][] board, int n) {
        for (int i = 0; i < row; i++) {
            if (board[i][col] == 1) return false;
        }
        for (int i = row, j = col; i >= 0 && j >= 0; i--, j--) {
            if (board[i][j] == 1) return false;
        }
        for (int i = row, j = col; i >= 0 && j < n; i--, j++) {
            if (board[i][j] == 1) return false;
        }
        return true;
    }
}

medium

A. 2

B. 4

C. 1

D. 0

Solution

1. Step 1: Understand the code logic

   The code counts all valid solutions for 4 queens on a 4x4 board using backtracking.

2. Step 2: Recall known result for 4 queens

   It is a well-known fact that there are 2 valid solutions for the 4 queens problem.

3. Final Answer:

   2 -> Option A

4. Quick Check:

   Number of solutions for N=4 = 2 [OK]

Hint: Remember: 4 queens have exactly 2 solutions [OK]

Common Mistakes:

• Assuming only 1 solution exists
• Confusing number of solutions with board size
• Ignoring backtracking resets

4. Identify the bug in this snippet of the N Queens solver:

static boolean isSafe(int row, int col, int[][] board, int n) {
    for (int i = 0; i < row; i++) {
        if (board[i][col] == 1) return false;
    }
    for (int i = row, j = col; i >= 0 && j >= 0; i--, j--) {
        if (board[i][j] == 1) return false;
    }
    for (int i = row, j = col; i >= 0 && j <= n; i--, j++) {
        if (board[i][j] == 1) return false;
    }
    return true;
}

medium

A. It incorrectly checks columns beyond the board size

B. It does not check the left diagonal upwards

C. It does not check the current row for other queens

D. It misses checking the lower diagonals for threats

Solution

1. Step 1: Analyze the safety checks

   The column check and upper-left diagonal are correct.

2. Step 2: Identify the bug in upper-right diagonal

   The loop condition 'i >= 0 && j <= n' allows j to reach n (e.g., when col = n-1, after first iteration j = n), causing ArrayIndexOutOfBoundsException on board[i][n].

3. Final Answer:

   It incorrectly checks columns beyond the board size -> Option A

4. Quick Check:

   Upper-right diagonal: j <= n causes out-of-bounds [OK]

Hint: Always verify loop bounds in diagonal checks to avoid index errors [OK]

Common Mistakes:

• Forgetting off-by-one errors in diagonal loop conditions
• Overlooking array bounds in upper-right diagonal traversal
• Confusing upper vs lower diagonal checks

5. You want to modify the N Queens solver to print all distinct solutions for N=4. Which approach is best to achieve this?

hard

A. Stop recursion as soon as the first solution is found to save time

B. Place all queens in the first row and check if they attack each other

C. Use a greedy algorithm to place queens in the first available safe spot in each row

D. Modify the recursive function to store each valid board configuration in a list instead of returning after first success

Solution

1. Step 1: Understand the requirement

   We want to find and print all distinct solutions, not just one.

2. Step 2: Choose the correct approach

   Modifying the recursive function to continue searching after finding a solution and storing each valid board is the correct approach.

3. Step 3: Eliminate wrong options

   Stopping after first solution won't find all solutions. Greedy algorithm to place queens in the first available safe spot in each row and placing all queens in the first row and check if they attack each other are incorrect strategies.

4. Final Answer:

   Modify the recursive function to store each valid board configuration in a list instead of returning after first success -> Option D

5. Quick Check:

   Store all solutions by continuing recursion [OK]

Hint: Keep searching after each solution to collect all boards [OK]

Common Mistakes:

• Stopping recursion after first solution
• Using greedy instead of backtracking
• Placing multiple queens in one row