N Queens Problem in DSA Python - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the N Queens Problem?

The N Queens Problem is a puzzle where you must place N chess 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

Why can't two queens be placed in the same row or column in the N Queens Problem?

Because queens attack along rows and columns, placing two queens in the same row or column means they can capture each other, which is not allowed.

intermediate

What technique is commonly used to solve the N Queens Problem?

Backtracking is commonly used. It tries placing queens one by one in different columns of a row and moves forward only if the placement is safe. If stuck, it backtracks to try other positions.

intermediate

How do you check if a queen placement is safe in the N Queens Problem?

Check if no other queen is in the same column, and no other queen is on the same diagonal (both left and right diagonals) above the current row.

advanced

What is the time complexity of the N Queens Problem using backtracking?

The worst-case time complexity is O(N!), because for each row, you try placing a queen in N columns and recursively check further rows.

In the N Queens Problem, what does a 'safe' position mean?

A No other queen is in the same row, column, or diagonal

B No other queen is in the same row only

C No other queen is in the same column only

D No other queen is in the same diagonal only

A safe position means no queen can attack the new queen, so no other queen is in the same row, column, or diagonal.

Which algorithmic technique is best suited for solving the N Queens Problem?

A Backtracking

B Greedy

C Dynamic Programming

D Divide and Conquer

Backtracking tries all possible placements and backtracks when a conflict is found, making it ideal for N Queens.

How many queens must be placed on the board in the N Queens Problem?

A 2N queens on an N×N board

B N queens on a 2N×2N board

C N/2 queens on an N×N board

D N queens on an N×N board

The problem requires placing N queens on an N×N board.

What is the main reason for backtracking in the N Queens Problem?

A To speed up the solution by skipping rows

B To try different queen placements when a conflict occurs

C To count the number of queens placed

D To sort the queens by position

Backtracking allows undoing a queen placement to try other positions when conflicts arise.

Which of these is NOT a direction a queen attacks in chess, relevant to the N Queens Problem?

A Horizontally

B Vertically

C Knight moves

D Diagonally

Queens do not attack in knight moves; only horizontally, vertically, and diagonally.

Explain how backtracking helps solve the N Queens Problem step-by-step.

Think about trying positions and undoing them when stuck.

Describe how to check if a queen placement is safe on the board.

Remember queens are placed row by row from top to bottom.

Practice

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

easy

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

B. Count the total number of queens on a chessboard

C. Move queens to capture all opponent pieces

D. Arrange queens in a single row on the chessboard

Solution

1. Step 1: Understand the problem statement

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

2. Step 2: Analyze the options

   Only Place N queens on an NxN chessboard so that no two queens attack each other correctly describes placing queens so none attack each other. Other options describe unrelated tasks.

3. Final Answer:

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

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 be in the same row or column
• Confusing the problem with counting queens
• Assuming queens can attack each other

2. Which of the following Python code snippets correctly initializes an empty chessboard for the N Queens Problem with size n?

easy

A. board = [[0 for _ in range(n)] for _ in range(n)]

B. board = [[0]*n]

C. board = [0]*n

D. board = [[0]*n]*n

Solution

1. Step 1: Understand board initialization

   We need a 2D list with independent rows to represent the chessboard.

2. Step 2: Analyze each option

   board = [[0]*n]*n creates rows referencing the same list, causing bugs. board = [[0 for _ in range(n)] for _ in range(n)] creates independent rows correctly. board = [0]*n creates a 1D list, not 2D. board = [[0]*n] creates a single row only.

3. Final Answer:

   board = [[0 for _ in range(n)] for _ in range(n)] -> Option A

4. Quick Check:

   Independent rows = board = [[0 for _ in range(n)] for _ in range(n)] [OK]

Hint: Use nested list comprehension for independent rows [OK]

Common Mistakes:

• Using [[0]*n]*n causing shared references
• Creating 1D list instead of 2D
• Forgetting to repeat for each row

3. What will be the output of the following Python code snippet for n = 4?

def is_safe(board, row, col):
    for i in range(col):
        if board[row][i] == 1:
            return False
    for i, j in zip(range(row, -1, -1), range(col, -1, -1)):
        if board[i][j] == 1:
            return False
    for i, j in zip(range(row, len(board), 1), range(col, -1, -1)):
        if board[i][j] == 1:
            return False
    return True

board = [[0]*4 for _ in range(4)]
board[1][0] = 1
print(is_safe(board, 1, 1))
print(is_safe(board, 0, 1))

medium

A. True\nFalse

B. False\nFalse

C. False\nTrue

D. True\nTrue

Solution

1. Step 1: Check is_safe for position (1,1)

   Column check: board[1][0] == 1, so returns False.

2. Step 2: Check is_safe for position (0,1)

   Row left: board[0][0] == 0. Upper diagonal: board[0][1] == 0. Lower diagonal: board[1][0] == 1, so returns False.

3. Final Answer:

   False\nFalse -> Option B

4. Quick Check:

   Check columns and diagonals = False, False [OK]

Hint: Check columns and diagonals left of position [OK]

Common Mistakes:

• Ignoring diagonal checks
• Confusing row and column indices
• Assuming safe if no queen in same row only

4. Identify the error in the following code snippet for solving the N Queens Problem:

def solve_n_queens(board, col):
    if col >= len(board):
        return True
    for row in range(len(board)):
        if is_safe(board, row, col):
            board[row][col] = 1
            if solve_n_queens(board, col + 1):
                return True
            board[row][col] = 0
    return False

board = [[0]*4 for _ in range(4)]
solve_n_queens(board, 0)
print(board)

medium

A. The base case condition is incorrect

B. The board is not reset properly after backtracking

C. The function returns True too early, missing all solutions

D. The is_safe function is not called

Solution

1. Step 1: Analyze the recursion and return statements

   The function returns True immediately after finding the first solution, so it stops searching further.

2. Step 2: Understand impact on solution finding

   This means it finds only one solution and does not explore all possible solutions.

3. Final Answer:

   The function returns True too early, missing all solutions -> Option C

4. Quick Check:

   Early return stops full search [OK]

Hint: Return True only after exploring all columns [OK]

Common Mistakes:

• Stopping recursion after first solution
• Not resetting board on backtrack
• Incorrect base case check

5. Given the N Queens Problem, which approach best improves efficiency by reducing unnecessary checks during queen placement?

hard

A. Place queens randomly and retry if conflicts occur

B. Check the entire board for conflicts every time before placing a queen

C. Use nested loops to try all possible board configurations without pruning

D. Use backtracking with arrays to track columns and diagonals already occupied

Solution

1. Step 1: Understand efficiency in N Queens

   Checking the entire board repeatedly wastes time; pruning reduces checks.

2. Step 2: Analyze options for pruning

   Using arrays to track columns and diagonals occupied avoids repeated scanning, improving speed.

3. Final Answer:

   Use backtracking with arrays to track columns and diagonals already occupied -> Option D

4. Quick Check:

   Pruning with tracking arrays = Use backtracking with arrays to track columns and diagonals already occupied [OK]

Hint: Track columns and diagonals to skip invalid spots fast [OK]

Common Mistakes:

• Checking full board every time
• Trying random placements without pruning
• Using brute force without backtracking