

from math import isclose, sqrt



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							float	,       SCREAMING_SNAKE_CASE_:							float	,       SCREAMING_SNAKE_CASE_:							float      )			->      tuple[float, float, float]:




				'''simple docstring'''





				A__        =			point_y / 4 / point_x
				A__        =			2 * normal_gradient / (1 + normal_gradient * normal_gradient)
				A__        =			(1 - normal_gradient * normal_gradient) / (
				    1 + normal_gradient * normal_gradient
				)
				A__        =			(sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

				# to find the next point, solve the simultaeneous equations:
				# y^2 + 4x^2 = 100
				# y - b = m * (x - a)
				# ==> A x^2 + B x + C = 0
				A__        =			outgoing_gradient**2 + 4
				A__        =			2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
				A__        =			(point_y - outgoing_gradient * point_x) ** 2 - 1_0_0

				A__        =			(
				    -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term      )
				) / (2 * quadratic_term)
				A__        =			(
				    -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term      )
				) / (2 * quadratic_term)

				# two solutions, one of which is our input point
				A__        =			x_minus if isclose(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      ) else x_plus
				A__        =			point_y + outgoing_gradient * (next_x - point_x)

				return next_x, next_y, outgoing_gradient



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							float = 1.4	,       SCREAMING_SNAKE_CASE_:							float = -9.6      )			->      int:




				'''simple docstring'''





				A__        =			0
				A__        =			first_x_coord
				A__        =			first_y_coord
				A__        =			(10.1 - point_y) / (0.0 - point_x)

				while not (-0.01 <= point_x <= 0.01 and point_y > 0):
								A__    ,				A__    ,				A__        =			next_point(SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )
								num_reflections += 1

				return num_reflections


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




from __future__ import annotations



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							list[int]	,       SCREAMING_SNAKE_CASE_:							list[int]	,       SCREAMING_SNAKE_CASE_:							list[int]	,       SCREAMING_SNAKE_CASE_:							list[list[str]]	,       SCREAMING_SNAKE_CASE_:							int	,       )			->      None:




				'''simple docstring'''





				A__        =			len(SCREAMING_SNAKE_CASE_      )

				# 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(SCREAMING_SNAKE_CASE_      ):
								# 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]	,       SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_	,       )



def 					lowerCAmelCase__							(	SCREAMING_SNAKE_CASE_:							int      )			->      None:




				'''simple docstring'''





				A__        =			[]
				depth_first_search([]	,       []	,       []	,       SCREAMING_SNAKE_CASE_	,       SCREAMING_SNAKE_CASE_      )

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

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


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	n_queens_solution(4)

