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Solve the N-Queens Problem

Certification: Advanced Level Accuracy: 100% Submissions: 10 Points: 15

Write a Python function that solves the N-Queens problem. The N-Queens problem is placing N chess queens on an N×N chessboard so that no two queens threaten each other. Thus, a solution requires that no two queens share the same row, column, or diagonal.

Example 1
    

  • Input: n = 4
    • 

  • Output: [[".Q..","...Q","Q...","..Q."],["..Q.","Q...","...Q",".Q.."]]
    • 

  • Explanation: 

      

    • Step 1: Create a 4×4 chessboard.
      • 

    • Step 2: Find all possible arrangements of 4 queens where none threaten each other.
      • 

    • Step 3: There are two valid solutions:
      • 

    • Solution 1:
      
  .Q..
      
  ...Q
      
  Q...
      
  ..Q.
      • 

    • Solution 2:
      
  ..Q.
      
  Q...
      
  ...Q
      
  .Q..
      • 

    • Step 4: Return both solutions as shown in the output format.
      • 

    

    •
Example 2
    

  • Input: n = 1
    • 

  • Output: [["Q"]]
    • 

  • Explanation: 

      

    • Step 1: Create a 1×1 chessboard.
      • 

    • Step 2: Place 1 queen on the board.
      • 

    • Step 3: There is only one possible arrangement: a single queen on the only square.
      • 

    • Step 4: Return the solution as [["Q"]].
      • 

    

    •
Constraints
    

  • 1 ≤ n ≤ 9
    • 

  • Time Complexity: O(N!), where N is the number of queens
    • 

  • Space Complexity: O(N²)
    •
SetAlgorithmsGoogleTCS (Tata Consultancy Services)
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Hint
Solution

Solution Hints

    
  
  • Use backtracking to place queens row by row
    • 
  
  • Check if a position is safe by verifying no queen threatens it
    • 
  
  • Use sets to keep track of occupied columns and diagonals
    • 
  
  • For each valid solution, construct the board representation
    •

Steps to solve by this approach:

 Step 1: Create an empty n×n board represented as a 2D array.

 Step 2: Use sets to track occupied columns and diagonals.

 Step 3: Implement a recursive backtracking function starting from row 0.

 Step 4: For each position, check if placing a queen is safe (no attacks possible).

 Step 5: If safe, place the queen and mark columns and diagonals as occupied.

 Step 6: Recursively solve for the next row and backtrack if needed.

 Step 7: Add complete board configurations to the result when all rows are filled.

Submitted Code :