The Question
Coding

N-Queens Puzzle Solver

The n-queens puzzle is the problem of placing n queens on an n x n chessboard such that no two queens attack each other. Given an integer n, return all distinct solutions to the n-queens puzzle. You may return the answer in any order. Each solution contains a distinct board configuration of the n-queens' placement, where 'Q' and '.' both indicate a queen and an empty space, respectively. Constraints: 1 <= n <= 12.
Java
Backtracking
Pruning
Questions & Insights

Clarifying Questions

What is the maximum value for $N$? (Usually, N \le 12 for standard backtracking, as the complexity is factorial).
What should the output format be? (Should it return a list of all distinct board configurations, or just the total count?)
How is a board configuration represented? (Standard representation is a list of strings where 'Q' represents a queen and '.' represents an empty space).
Are there time or memory constraints? (Backtracking naturally consumes significant time for larger N; memory is usually O(N) for the recursion stack).

Assumptions

N will be between 1 and 12.
The function should return all distinct solutions.
Efficiency is prioritized using boolean arrays for O(1) conflict checks rather than scanning the board.

Thinking Process

State Space Search: We approach the problem row by row. Since each row must contain exactly one queen, we iterate through every column in the current row and attempt to place a queen.
Constraint Satisfaction: A queen at (r, c) threatens others in the same column, the main diagonal (r - c), and the anti-diagonal (r + c). We maintain boolean arrays to track which columns and diagonals are currently "under attack" to prune the search tree immediately.
Backtracking: After placing a queen and recursively exploring subsequent rows, we must "undo" the placement (backtrack) to explore other possible configurations for the current row.
Base Case: If the current row index reaches N, it means we have successfully placed N queens without conflict. We then convert the current board state into the required string format and add it to our results list.
Implementation Breakdown

Problem Set

Functional Requirements:
Solve the N-Queens puzzle such that no two queens attack each other.
Return all distinct solutions.
Constraints:
Time Complexity: O(N!) as we have N choices for the first row, N-2 or fewer for the next, and so on.
Space Complexity: O(N) for the recursion stack and tracking arrays (excluding the output list).

Approach

Algorithm: Backtracking with Pruning.
Data Structure: Boolean arrays (for constant time lookup) and a char array (to build the board).
Complexity:
Time: O(N!)
Space: O(N) (auxiliary arrays and recursion depth).

Implementation

Wrap Up

Advanced Topics

Bitmask Optimization: For N \le 31, we can replace boolean arrays with integer bitmasks (int or long). This uses bitwise operations (&, |, <<) which are significantly faster and reduce memory footprint, often being the preferred choice in competitive programming.
Symmetry Reduction: The N-Queens problem has rotational and reflective symmetries. We can cut the search space nearly in half by only placing the first queen in the first N/2 columns and then reflecting the found solutions.
Parallelization: Since each branch of the first row's decisions is independent, we can use a Thread Pool or Fork-Join framework to evaluate different starting positions of the first queen in parallel.
Iterative Repair / Local Search: For very large N (e.g., N=1000), backtracking is impossible. In such cases, one would use heuristic-based local search algorithms like "Min-Conflicts" to find a single valid configuration in near-linear time.