N Queens Problem in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the N Queens Problem?

To place N queens on an N×N chessboard so that no two queens threaten each other. This means no two queens share the same row, column, or diagonal.

beginner

Which data structure is commonly used to represent the chessboard in the N Queens Problem?

A 2D array or matrix of size N×N, where each cell can hold a value indicating if a queen is placed there or not.

intermediate

How does backtracking help solve the N Queens Problem?

Backtracking tries placing a queen in a row and moves to the next row recursively. If a conflict arises, it removes the queen (backtracks) and tries the next position until all queens are placed safely.

intermediate

What checks are necessary before placing a queen on the board?

Check if the column, the row, the left diagonal, and the right diagonal are free of other queens to avoid attacks.

intermediate

Why is the N Queens Problem considered a classic example of backtracking?

Because it systematically explores all possible placements of queens and abandons paths that lead to conflicts, demonstrating the power of recursive problem solving.

What does the N Queens Problem require to avoid between queens?

A Being adjacent horizontally only

B Being placed on the same color square

C Sharing the same row, column, or diagonal

D Being placed in the first row

Queens must not share the same row, column, or diagonal to avoid threatening each other.

Which algorithmic technique is primarily used to solve the N Queens Problem?

A Backtracking

B Greedy

C Dynamic Programming

D Divide and Conquer

Backtracking tries all possible placements and backtracks when conflicts occur.

In the N Queens Problem, what does backtracking do when it finds a conflict?

A Places a queen anyway

B Removes the last placed queen and tries another position

C Stops the algorithm immediately

D Skips to the last row

Backtracking removes the last queen and tries the next possible position to find a safe placement.

What size is the chessboard in the N Queens Problem?

A 8×8 only

B 1×N

C N×8

D N×N

The board size is N×N, where N is the number of queens to place.

Which of these is NOT a valid check before placing a queen?

A Checking the color of the square

B Checking the same row

C Checking the diagonals

D Checking the same column

The color of the square is irrelevant; only rows, columns, and diagonals matter.

Explain how backtracking is used to solve the N Queens Problem step-by-step.

Think about trying positions and undoing moves when stuck.

Describe the conditions that must be checked before placing a queen on the board.

Remember queens attack in straight lines and diagonals.

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