[Hard] 51. N-Queens

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.

Examples

Example 1:

Input: n = 4
Output: [[".Q..","...Q","Q...","..Q."],["..Q.","Q...","...Q",".Q.."]]
Explanation: There exist two distinct solutions to the 4-queens puzzle as shown above.

Example 2:

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

Constraints

• 1 <= n <= 9

Thinking Process

1. Row-by-Row Placement: Eliminates row conflicts automatically
   • Diagonal: row - col + n (shifted to avoid negatives)
   • Anti-diagonal: row + col

• Build solution incrementally; undo (backtrack) when constraints fail.
• Prune branches early to avoid exploring invalid partial states.
• Sort input to skip duplicate combinations efficiently.

Common Approaches

Typical techniques for this pattern:

Approach	Time	Space	Notes
Choose / explore / unchoose (this problem)	O(2^n)	O(n)	Subsets, combinations
Constraint pruning	Reduced search	O(n)	Early exit on invalid partial
Sort + skip duplicates	O(2^n)	O(n)	Combination sum II style
Path recording	O(n!) worst	O(n)	Permutations

Solution

Solution: Backtracking with Optimized Constraint Checking

class Solution:
    def solveNQueens(self, n):
        self.size = n
        self.board = [["."] * n for _ in range(n)]

        self.col = [False] * n
        self.diag = [False] * (2 * n)
        self.anti = [False] * (2 * n)

        self.rtn = []

        self.dfs(0)

        return self.rtn

    def dfs(self, row):
        if row == self.size:
            self.rtn.append(["".join(r) for r in self.board])
            return

        for c in range(self.size):
            d = row - c + self.size
            a = row + c

            if self.col[c] or self.diag[d] or self.anti[a]:
                continue

            self.col[c] = self.diag[d] = self.anti[a] = True
            self.board[row][c] = "Q"

            self.dfs(row + 1)

            self.board[row][c] = "."
            self.col[c] = self.diag[d] = self.anti[a] = False

Solution Explanation

Approach: Choose / explore / unchoose (this problem)

Key idea: 1. Row-by-Row Placement: Eliminates row conflicts automatically

How the code works:

1. Row-by-Row Placement: Eliminates row conflicts automatically
   • Diagonal: row - col + n (shifted to avoid negatives)
   • Anti-diagonal: row + col
   • Build solution incrementally; undo (backtrack) when constraints fail.
   • Prune branches early to avoid exploring invalid partial states.
   • Sort input to skip duplicate combinations efficiently.

Walkthrough — input n = 4, expected output [[".Q..","...Q","Q...","..Q."],["..Q.","Q...","...Q",".Q.."]]:

There exist two distinct solutions to the 4-queens puzzle as shown above.

Algorithm Explanation:

1. Initialize (Lines 4-9):
   • size = n: Store board size
   • board: n × n grid initialized with '.'
   • col: Boolean array of size n to track used columns
   • diag: Boolean array of size 2 * n to track diagonals (top-left to bottom-right)
   • anti: Boolean array of size 2 * n to track anti-diagonals (top-right to bottom-left)
2. DFS Function (Lines 16-30):
   • Base Case (Lines 17-20): If row == size, all queens placed successfully
     • Add current board configuration to result
     • Return
   • Try Each Column (Lines 21-29):
     • Calculate indices:
       • d = row - c + size: Diagonal index (add size to avoid negative indices)
       • a = row + c: Anti-diagonal index
     • Check Constraints (Line 23): If column, diagonal, or anti-diagonal is occupied, skip
     • Place Queen (Lines 24-25): Mark constraints and place 'Q'
     • Recurse (Line 26): Try next row
     • Backtrack (Lines 27-28): Remove queen and unmark constraints

Why This Works:

• Row-by-Row: Placing one queen per row eliminates row conflicts automatically
• Column Tracking: col[c] ensures no two queens in same column
• Diagonal Tracking:
  • Diagonal \: row - col is constant (shifted by +size to avoid negatives)
  • Anti-diagonal /: row + col is constant
• Optimization: Boolean arrays provide O(1) constraint checking vs O(n) board scanning
• Backtracking: Undoing choices allows exploring all valid configurations

Diagonal Index Calculation:

For an n × n board:

• Diagonal (top-left to bottom-right): row - col ranges from -(n-1) to (n-1)
  • Add n to shift range to [1, 2n-1]
  • Use row - col + n as index
• Anti-diagonal (top-right to bottom-left): row + col ranges from 0 to 2(n-1)
  • Use row + col directly as index

Example for n = 4:

Diagonal indices (row - col + 4):
    0  1  2  3
0   4  5  6  7
1   3  4  5  6
2   2  3  4  5
3   1  2  3  4

Anti-diagonal indices (row + col):
    0  1  2  3
0   0  1  2  3
1   1  2  3  4
2   2  3  4  5
3   3  4  5  6

Example Walkthrough:

For n = 4:

Initial: row=0, board=[[".",".",".","."], ...], all constraints false

Row 0:
  Try col 0:
    d = 0 - 0 + 4 = 4, a = 0 + 0 = 0
    Check: col[0]=false, diag[4]=false, anti[0]=false ✓
    Place Q at (0,0), mark constraints
    Recurse to row 1

  Row 1:
    Try col 0: col[0]=true ✗
    Try col 1:
      d = 1 - 1 + 4 = 4, a = 1 + 1 = 2
      Check: col[1]=false, diag[4]=true ✗ (conflict with (0,0))
    Try col 2:
      d = 1 - 2 + 4 = 3, a = 1 + 2 = 3
      Check: col[2]=false, diag[3]=false, anti[3]=false ✓
      Place Q at (1,2), mark constraints
      Recurse to row 2

    Row 2:
      Try col 0: col[0]=true ✗
      Try col 1:
        d = 2 - 1 + 4 = 5, a = 2 + 1 = 3
        Check: col[1]=false, diag[5]=false, anti[3]=true ✗
      Try col 2: col[2]=true ✗
      Try col 3:
        d = 2 - 3 + 4 = 3, a = 2 + 3 = 5
        Check: col[3]=false, diag[3]=true ✗
      Backtrack: Remove Q from (1,2)

    Try col 3:
      d = 1 - 3 + 4 = 2, a = 1 + 3 = 4
      Check: col[3]=false, diag[2]=false, anti[4]=false ✓
      Place Q at (1,3), mark constraints
      Recurse to row 2
      ... (continue until solution found)

Final: When row=4, add board to result

Complexity Analysis:

• Time Complexity: O(n!)
  • For each row, we try at most n columns
  • With pruning, actual complexity is better but still exponential
  • In worst case: O(n!) (factorial)
• Space Complexity: O(n²)
  • board: O(n²) for storing board state
  • col, diag, anti: O(n) each
  • Recursion stack: O(n) depth
  • Result: O(n! × n²) for storing all solutions

    Common Mistakes

1. n = 1: Return [["Q"]]
2. n = 2, 3: No solutions (impossible to place n queens)
3. n = 4: 2 solutions

4. n = 8: 92 solutions (classic 8-queens problem)

5. Wrong diagonal indexing: Not shifting row - col correctly
6. Missing backtrack: Forgetting to unmark constraints after recursion
7. Array bounds: Diagonal array size should be 2 * n (not n)
8. Board initialization: Not initializing board with '.' characters
9. Constraint checking order: Should check all three constraints before placing

Related Problems

• LC 52: N-Queens II - Count number of solutions (same problem, just count)
• LC 37: Sudoku Solver - Similar constraint satisfaction
• LC 51: N-Queens - This problem

Key Takeaways

1. Row-by-Row Placement: Eliminates row conflicts automatically
2. Optimized Constraint Checking: Boolean arrays provide O(1) checking vs O(n) scanning
3. Diagonal Indexing:
   • Diagonal: row - col + n (shifted to avoid negatives)
   • Anti-diagonal: row + col
4. Backtracking Pattern: Place → Recurse → Undo

References

• LC 51: N-Queens on LeetCode
• LeetCode Discuss — LC 51: N-Queens
• LeetCode Editorial (may require premium)

Template Reference

• Array & Matrix