




from __future__ import annotations



def 				_A  (		lowerCAmelCase_	:   list[int]  ,							lowerCAmelCase_	:   list[int]  ,							lowerCAmelCase_	:   list[int]  ,							lowerCAmelCase_	:   list[list[str]]  ,							lowerCAmelCase_	:   int  ,							):
     """simple docstring"""

     lowerCAmelCase__            =					len(lowerCAmelCase_     )

     # If row is equal to the size of the board it means there are a queen in each row in
     # the current board (possible_board)
     if row == n:
          # We convert the variable possible_board that looks like this: [1, 3, 0, 2] to
          # this: ['. Q . . ', '. . . Q ', 'Q . . . ', '. . Q . ']
          boards.append([". " * i + "Q " + ". " * (n - 1 - i) for i in possible_board]     )
          return

     # We iterate each column in the row to find all possible results in each row
     for col in range(lowerCAmelCase_     ):
          # We apply that we learned previously. First we check that in the current board
          # (possible_board) there are not other same value because if there is it means
          # that there are a collision in vertical. Then we apply the two formulas we
          # learned before:
          #
          # 45º: y - x = b or 45: row - col = b
          # 135º: y + x = b or row + col = b.
          #
          # And we verify if the results of this two formulas not exist in their variables
          # respectively.  (diagonal_right_collisions, diagonal_left_collisions)
          #
          # If any or these are True it means there is a collision so we continue to the
          # next value in the for loop.
          if (
              col in possible_board
              or row - col in diagonal_right_collisions
              or row + col in diagonal_left_collisions
          ):
               continue

          # If it is False we call dfs function again and we update the inputs
          depth_first_search(
              [*possible_board, col]  ,							[*diagonal_right_collisions, row - col]  ,							[*diagonal_left_collisions, row + col]  ,							lowerCAmelCase_  ,							lowerCAmelCase_  ,							)



def 				_A  (		lowerCAmelCase_	:   int     ):
     """simple docstring"""

     lowerCAmelCase__            =					[]
     depth_first_search([]  ,							[]  ,							[]  ,							lowerCAmelCase_  ,							lowerCAmelCase_     )

     # Print all the boards
     for board in boards:
          for column in board:
               print(lowerCAmelCase_     )
          print(""     )

     print(len(lowerCAmelCase_     )  ,							"solutions were found."     )


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     n_queens_solution(4)




'''simple docstring'''
def 					lowerCamelCase_ ( __UpperCamelCase					:       int		)					->	bool:







     """simple docstring"""
     return str(__UpperCamelCase		) == str(__UpperCamelCase		)[::-1]

def 					lowerCamelCase_ ( __UpperCamelCase					:       int		)					->	int:







     """simple docstring"""
     return int(__UpperCamelCase		) + int(str(__UpperCamelCase		)[::-1]		)

def 					lowerCamelCase_ ( __UpperCamelCase					:       int = 1_0_0_0_0		)					->	int:







     """simple docstring"""
     _A				=							[]
     for num in range(1							,    __UpperCamelCase		):
          _A				=							0
          _A				=							num
          while iterations < 5_0:
               _A				=							sum_reverse(__UpperCamelCase		)
               iterations += 1
               if is_palindrome(__UpperCamelCase		):
                    break
          else:
               lychrel_nums.append(__UpperCamelCase		)
     return len(__UpperCamelCase		)


if __name__ == "__main__":
    print(F'{solution() = }')





