The N-Queens Problem

The N-Queens problem is a classic puzzle in computer science and mathematics that challenges you to place N chess queens on an N×N chessboard such that no two queens threaten each other. A queen can attack any piece in the same row, column, or diagonal. This problem is a fundamental example of a constraint satisfaction problem and is often used to illustrate backtracking algorithms.

Problem Description

The goal is to find all distinct solutions to the N-Queens puzzle for a given board size N. A solution is a configuration of N queens on an N×N chessboard where no two queens share the same row, column, or diagonal.

Requirements:

1. Placement: Exactly N queens must be placed on the N×N board.
2. No Attacks: No two queens can be placed in positions where they threaten each other. This means:
   • No two queens can be in the same row.
   • No two queens can be in the same column.
   • No two queens can be on the same diagonal. There are two types of diagonals:
     • Top-left to bottom-right diagonals (where row - col is constant).
     • Top-right to bottom-left diagonals (where row + col is constant).
3. Distinct Solutions: The output should list all unique arrangements of queens that satisfy the above conditions. The order of solutions or queens within a solution typically doesn't matter, but a consistent representation is preferred.

Expected Behavior:

Your program should accept an integer N representing the size of the chessboard. It should return a list of all valid N-Queens configurations. Each configuration can be represented as a list of strings, where each string represents a row of the board. For example, a '.' could represent an empty square, and a 'Q' could represent a queen.

Edge Cases:

• N = 1: A single queen on a 1x1 board.
• N = 2 or N = 3: Boards where no solution exists.

Examples

Example 1:

Input: N = 4
Output:
[
 [".Q..",  // Solution 1, Row 0
  "...Q",  // Solution 1, Row 1
  "Q...",  // Solution 1, Row 2
  "..Q."], // Solution 1, Row 3

 [ "..Q.",  // Solution 2, Row 0
   "Q...",  // Solution 2, Row 1
   "...Q",  // Solution 2, Row 2
   ".Q.."]  // Solution 2, Row 3
]

Explanation:

For N=4, there are two distinct ways to place 4 queens on a 4x4 board without them attacking each other. The output shows these two arrangements.

Example 2:

Input: N = 1
Output:
[
 ["Q"]
]

Explanation:

For N=1, there is one queen on a 1x1 board, and it trivially satisfies the conditions.

Example 3:

Input: N = 3
Output:
[]

Explanation:

For N=3, it is impossible to place 3 queens on a 3x3 board without them attacking each other. Therefore, the output is an empty list.

Constraints

• 1 <= N <= 14
• The input N will be an integer.
• The solution should be reasonably efficient for the given constraints, suggesting that an exponential time complexity solution is acceptable for N <= 14.

Notes

• This problem is an excellent candidate for a recursive backtracking approach.
• You'll need a way to efficiently check if placing a queen at a particular (row, col) position is valid given the queens already placed in previous rows.
• Consider how to represent the board and the placement of queens.
• Think about how to keep track of occupied columns and diagonals.