N Queens Problem in DSA C# - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of N Queens Board Representation

What is the printed board state after placing 4 queens in the first valid solution of the 4-Queens problem?

DSA C#

static void PrintBoard(int[] positions) {
    for (int row = 0; row < positions.Length; row++) {
        for (int col = 0; col < positions.Length; col++) {
            if (positions[row] == col) Console.Write("Q ");
            else Console.Write(". ");
        }
        Console.WriteLine();
    }
}

int[] solution = {1, 3, 0, 2};
PrintBoard(solution);

A

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

B

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

C

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

D

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

💡 Hint

Remember the array index represents the row, and the value at that index is the column where the queen is placed.

The array {1, 3, 0, 2} means queens are placed at (row0,col1), (row1,col3), (row2,col0), and (row3,col2). Printing '.' for empty and 'Q' for queen gives option A.

🧠 Conceptual

intermediate

Number of Solutions for 1x1 and 2x2 Boards

How many valid solutions exist for the N Queens problem on a 1x1 board and a 2x2 board respectively?

A 0 solutions for 1x1, 0 solutions for 2x2

B 1 solution for 1x1, 0 solutions for 2x2

C 1 solution for 1x1, 1 solution for 2x2

D 0 solutions for 1x1, 1 solution for 2x2

💡 Hint

Think about how many queens can be placed without attacking each other on very small boards.

On a 1x1 board, placing one queen is trivially valid (1 solution). On a 2x2 board, no arrangement exists where 2 queens don't attack each other (0 solutions).

🔧 Debug

advanced

Identify the Bug in the Diagonal Check

Given this snippet from an N Queens solver, which option correctly identifies the bug causing incorrect diagonal attack detection? bool IsSafe(int row, int col, int[] positions) { for (int i = 0; i < row; i++) { if (positions[i] == col || Math.Abs(positions[i] - col) == Math.Abs(i - row)) { return false; } } return true; }

A The diagonal check uses Math.Abs(i - row) instead of Math.Abs(row - i), causing wrong calculations.

B The loop should run until i <= row instead of i < row to check all previous rows.

C The diagonal check condition is correct; the bug is in the column check positions[i] == col.

D No bug; the code correctly checks for column and diagonal conflicts.

💡 Hint

Check the logic of absolute differences for diagonal attacks carefully.

The diagonal check uses absolute differences correctly: Math.Abs(positions[i] - col) == Math.Abs(i - row) detects diagonal attacks. The loop correctly checks all previous rows with i < row. The column check is also correct. So no bug here.

🚀 Application

advanced

Count Solutions for 5-Queens Problem

What is the total number of distinct solutions for the 5-Queens problem?

A 12

B 8

C 10

D 14

💡 Hint

The 5-Queens problem has fewer solutions than 8-Queens but more than 4-Queens.

The 5-Queens problem has exactly 10 distinct solutions. The options are designed so that 10 is the correct answer for unique solutions excluding symmetrical duplicates.

❓ Predict Output

expert

Output of Recursive Call Count in N Queens Solver

Consider this C# code snippet that counts how many times the recursive function is called when solving the 4-Queens problem. What is the printed output? int count = 0; void Solve(int row, int[] positions) { count++; if (row == positions.Length) return; for (int col = 0; col < positions.Length; col++) { bool safe = true; for (int i = 0; i < row; i++) { if (positions[i] == col || Math.Abs(positions[i] - col) == Math.Abs(i - row)) { safe = false; break; } } if (safe) { positions[row] = col; Solve(row + 1, positions); } } } int[] pos = new int[4]; Solve(0, pos); Console.WriteLine(count);

A 24

B 10

C 40

D 15

💡 Hint

Count includes all recursive calls including dead ends.

The recursive function is called 24 times in total for the 4-Queens problem with this backtracking approach, counting all partial attempts including those that fail early.

Practice

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

easy

A. Place N queens on an NxN board so that no two queens attack each other

B. Place N queens on an NxN board so that all queens attack each other

C. Place N queens on an NxN board in a single row

D. Place N queens on an NxN board in a single column

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 board so that no two queens attack each other -> Option A

4. Quick Check:

   Goal = No two queens attack each other [OK]

Hint: Remember: Queens must not threaten each other [OK]

Common Mistakes:

• Thinking queens can share rows or columns
• Ignoring diagonal attacks
• Confusing with placing queens in one line

2. Which of the following C# code snippets correctly declares a 2D array to represent an 8x8 chessboard for the N Queens Problem?

easy

A. int[,] board = new int[8,8];

B. int[][] board = new int[8];

C. int board[8][8];

D. int board = new int[8,8];

Solution

1. Step 1: Identify correct 2D array declaration in C#

   In C#, a 2D array is declared as int[,] board = new int[8,8];.

2. Step 2: Check other options for syntax errors

   int[][] board = new int[8]; declares a jagged array but misses initialization of inner arrays. int board[8][8]; is invalid syntax in C#. int board = new int[8,8]; tries to assign an array to an int variable.

3. Final Answer:

   int[,] board = new int[8,8]; -> Option A

4. Quick Check:

   Correct 2D array syntax = int[,] board = new int[8,8]; [OK]

Hint: Use int[,] for 2D arrays in C# [OK]

Common Mistakes:

• Using jagged arrays without initializing inner arrays
• Using C++ style array declarations
• Assigning arrays to single int variables

3. Consider the following C# code snippet for checking if a queen can be placed at row 1, column 2 on a 4x4 board:

bool IsSafe(int[,] board, int row, int col) {
    for (int i = 0; i < col; i++) {
        if (board[row, i] == 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 < 4 && j >= 0; i++, j--) {
        if (board[i, j] == 1) return false;
    }
    return true;
}

int[,] board = new int[4,4];
board[0, 1] = 1;
bool result = IsSafe(board, 1, 2);
Console.WriteLine(result);

What will be the output?

medium

A. true

B. Compile-time error

C. false

D. Runtime exception

Solution

1. Step 1: Analyze the board state and IsSafe checks

   The board has a queen at (0,1). We check if placing at (1,2) is safe. The function checks left row, upper-left diagonal, and lower-left diagonal.

2. Step 2: Check conflicts for position (1,2)

   Left row check: no queen at (1,0) or (1,1). Upper-left diagonal: (0,1) has a queen, so returns false.

3. Final Answer:

   false -> Option C

4. Quick Check:

   Queen at (0,1) blocks (1,2) = false [OK]

Hint: Check diagonals carefully for conflicts [OK]

Common Mistakes:

• Ignoring diagonal checks
• Assuming no conflict if row is clear
• Misreading array indices

4. The following C# code attempts to solve the N Queens Problem but has a bug. What is the bug?

bool SolveNQueens(int[,] board, int col) {
    if (col >= board.GetLength(0)) return true;
    for (int i = 0; i < board.GetLength(0); i++) {
        if (IsSafe(board, i, col)) {
            board[i, col] = 1;
            if (SolveNQueens(board, col + 1)) return true;
            board[i, col] = 0;
        }
    }
    return false;
}

// IsSafe method is correct

medium

A. The function does not backtrack properly

B. No bug; code works correctly

C. The board array is not initialized

D. The base case should check row instead of column

Solution

1. Step 1: Review the recursive backtracking logic

   The function tries to place a queen in each row of the current column, checks safety, places queen, and recurses to next column.

2. Step 2: Verify base case and backtracking

   Base case returns true when all columns are processed. Backtracking resets board[i, col] to 0 if recursion fails. This is correct.

3. Final Answer:

   No bug; code works correctly -> Option B

4. Quick Check:

   Backtracking logic and base case are correct [OK]

Hint: Check base case and backtracking steps carefully [OK]

Common Mistakes:

• Confusing row and column in base case
• Forgetting to reset board on backtrack
• Assuming uninitialized board causes error

5. You want to modify the N Queens solver to count all possible solutions for N=4 instead of stopping at the first. Which change should you make to the recursive method?

hard

A. Add a global boolean flag to stop after first solution

B. Remove the IsSafe check to speed up counting

C. Change the board size to 5x5

D. Replace the return type with int and sum all recursive calls instead of returning on first true

Solution

1. Step 1: Understand current behavior

   The current solver returns true on finding the first solution, stopping further search.

2. Step 2: Modify to count all solutions

   Change return type to int, accumulate counts from recursive calls, and do not return early. This counts all valid placements.

3. Final Answer:

   Replace the return type with int and sum all recursive calls instead of returning on first true -> Option D

4. Quick Check:

   Counting solutions = sum recursive counts [OK]

Hint: Sum all recursive results to count solutions [OK]

Common Mistakes:

• Stopping at first solution
• Changing board size unnecessarily
• Removing safety checks causing wrong counts