N queens (Py-Contributors#568)

* Add file n_queens_math.py

* Add a description of the problem

* Add depth_first_search function

* Add n_queens_solution function and pass parameters to depth_first_search

* Add the main function

* Flake8 corrections

Author: DavidBanda

Python/Algorithms/BackTrackingAlgorithms/n_queens_math.py

@@ -0,0 +1,70 @@
+r"""
+Problem:
+The n queens problem is of placing N queens on a N * N chess board such that
+no queen can attack any other queens placed on that chess board.  This means
+that one queen cannot have any other queen on its horizontal, vertical and
+diagonal lines.
+"""
+
+from typing import List
+
+
+def depth_first_search(
+    possible_board: List[int],
+    diagonal_right_collisions: List[int],
+    diagonal_left_collisions: List[int],
+    boards: List[List[str]],
+    n: int,
+) -> None:
+    """
+    >>> boards = []
+    >>> depth_first_search([], [], [], boards, 4)
+    >>> for board in boards:
+    ...     print(board)
+    ['. Q . . ', '. . . Q ', 'Q . . . ', '. . Q . ']
+    ['. . Q . ', 'Q . . . ', '. . . Q ', '. Q . . ']
+    """
+
+    row = len(possible_board)
+
+    if row == n:
+        possible_board = [". " * i + "Q " + ". " * (n - 1 - i)
+                          for i in possible_board]
+        boards.append(possible_board)
+        return
+
+    for col in range(n):
+
+        if (
+            col in possible_board
+            or row - col in diagonal_right_collisions
+            or row + col in diagonal_left_collisions
+        ):
+            continue
+
+        depth_first_search(
+            possible_board + [col],
+            diagonal_right_collisions + [row - col],
+            diagonal_left_collisions + [row + col],
+            boards,
+            n,
+        )
+
+
+def n_queens_solution(n: int) -> None:
+    boards = []
+    depth_first_search([], [], [], boards, n)
+
+    """ Print all the boards """
+    for board in boards:
+        for column in board:
+            print(column)
+        print("")
+
+    print(len(boards), "solutions were found.")
+
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()
+    n_queens_solution(4)