N Queens Problem in DSA C# - Build from Scratch

N Queens Problem

📖 Scenario: You are helping a chess enthusiast who wants to place n queens on an n x n chessboard. The goal is to place the queens so that no two queens can attack each other. This means no two queens share the same row, column, or diagonal.This classic puzzle is called the N Queens Problem.

🎯 Goal: You will build a program that finds one valid way to place n queens on the board. The program will print the board with 'Q' for queens and '.' for empty spaces.

📋 What You'll Learn

Create a 2D character array called board of size n x n filled with '.'

Create an integer variable called n to represent the size of the board

Write a method IsSafe that checks if placing a queen at a given row and column is safe

Write a recursive method SolveNQueens that tries to place queens row by row

Print the board after successfully placing all queens

💡 Why This Matters

🌍 Real World

The N Queens problem is a classic example of constraint satisfaction problems used in artificial intelligence and optimization.

💼 Career

Understanding backtracking algorithms and recursion is important for software engineers working on puzzles, game development, and complex problem solving.

1

Create the chessboard and size variable

Create an integer variable called n and set it to 4. Then create a 2D character array called board of size n x n filled with the character '.' to represent empty spaces.

DSA C#

// Create the integer variable n and set it to 4
// Create the 2D char array board filled with '.'
// Your code here

Hint

Use nested for loops to fill the 2D array with '.'.

2

Write the IsSafe method

Write a method called IsSafe that takes parameters char[,] board, int row, and int col. It should return true if placing a queen at board[row, col] is safe (no other queens in the same column, upper left diagonal, or upper right diagonal). Otherwise, return false.

DSA C#

int n = 4;
char[,] board = new char[n, n];
for (int i = 0; i < n; i++)
{
    for (int j = 0; j < n; j++)
    {
        board[i, j] = '.';
    }
}

bool IsSafe(char[,] board, int row, int col)
{
    // Your code here
}

Hint

Check the column above, the upper left diagonal, and the upper right diagonal for queens.

3

Write the SolveNQueens recursive method

Write a recursive method called SolveNQueens that takes parameters char[,] board and int row. It should try to place a queen in each column of the current row if it is safe. If placing a queen leads to a solution for the next rows, return true. If no column works, return false.

DSA C#

int n = 4;
char[,] board = new char[n, n];
for (int i = 0; i = 0 && j >= 0; i--, j--)
    {
        if (board[i, j] == 'Q')
            return false;
    }

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

    return true;
}

bool SolveNQueens(char[,] board, int row)
{
    // Your code here
}

Hint

Try placing a queen in each column of the current row. Use recursion to solve for the next row. Backtrack if needed.

4

Print the board after solving

Call SolveNQueens(board, 0) to solve the puzzle. Then print the board row by row. For each cell, print 'Q' if there is a queen, or '.' if empty. Separate cells with a space. Print each row on a new line.

DSA C#

int n = 4;
char[,] board = new char[n, n];
for (int i = 0; i = 0 && j >= 0; i--, j--)
    {
        if (board[i, j] == 'Q')
            return false;
    }

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

    return true;
}

bool SolveNQueens(char[,] board, int row)
{
    if (row == n)
        return true;

    for (int col = 0; col < n; col++)
    {
        if (IsSafe(board, row, col))
        {
            board[row, col] = 'Q';

            if (SolveNQueens(board, row + 1))
                return true;

            board[row, col] = '.'; // backtrack
        }
    }

    return false;
}

if (SolveNQueens(board, 0))
{
    // Your code here
}

Hint

Call SolveNQueens(board, 0) first. Then use nested loops to print the board with spaces between cells and new lines after each row.

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