


'''simple docstring'''






from __future__ import annotations


def    _snake_case  (	A					,       A					,       A					,       A					,       A					,       )		->       None:
							lowerCAmelCase__			  =      len(A			)

							# 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(A			):
														# 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]					,       A					,       A					,       )


def    _snake_case  (	A			)		->       None:
							lowerCAmelCase__			  =      []
							depth_first_search([]					,       []					,       []					,       A					,       A			)

							# Print all the boards
							for board in boards:
														for column in board:
																					print(A			)
														print(''''''			)

							print(len(A			)					,       '''solutions were found.'''			)


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		n_queens_solution(4)


'''simple docstring'''




def    lowerCamelCase__		(    SCREAMING_SNAKE_CASE   : int = 1000    ):
  UpperCAmelCase    				=						-1
  UpperCAmelCase    				=						0
  for a in range(1							, n // 3    ):
    # Solving the two equations a**2+b**2=c**2 and a+b+c=N eliminating c
    UpperCAmelCase    				=						(n * n - 2 * a * n) // (2 * n - 2 * a)
    UpperCAmelCase    				=						n - a - b
    if c * c == (a * a + b * b):
      UpperCAmelCase    				=						a * b * c
      if candidate >= product:
        UpperCAmelCase    				=						candidate
  return product


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


