Why N Queens Problem in DSA Java?

The Big Idea

What if you could solve a tricky puzzle by smartly trying and undoing moves instead of guessing blindly?

The Scenario

Imagine you have a chessboard and you want to place queens so that none can attack each other. You try to do this by guessing positions one by one, checking if they attack each other manually.

The Problem

Manually checking every possible position and combination is slow and confusing. It's easy to make mistakes and miss valid solutions or waste time on impossible ones.

The Solution

The N Queens Problem uses a smart way to place queens step-by-step, checking only safe spots and backtracking when stuck. This method finds all solutions without guessing blindly.

Before vs After

✗ Before

for (int i = 0; i < board.length; i++) {
  for (int j = 0; j < board.length; j++) {
    if (safeToPlace(i, j)) {
      placeQueen(i, j);
    }
  }
}

✓ After

void solveNQueens(int row) {
  if (row == N) printBoard();
  else {
    for (int col = 0; col < N; col++) {
      if (isSafe(row, col)) {
        placeQueen(row, col);
        solveNQueens(row + 1);
        removeQueen(row, col);
      }
    }
  }
}

What It Enables

This approach lets us efficiently find all ways to place queens safely on the board, even for large sizes.

Real Life Example

It's like scheduling tasks so none overlap or conflict, ensuring smooth operation without clashes.

Key Takeaways

Manual guessing is slow and error-prone.

Backtracking checks safe spots step-by-step.

Efficiently finds all valid queen placements.

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