



from __future__ import annotations






def  UpperCAmelCase__    (    lowercase__						,       lowercase__						,       lowercase__						,       lowercase__						,       lowercase__						,       ) ->      None:
						__lowercase					=				len(lowercase__   )

						# 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(lowercase__   ):
												# 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]						,       lowercase__						,       lowercase__						,       )






def  UpperCAmelCase__    (    lowercase__   ) ->      None:
						__lowercase					=				[]
						depth_first_search([]						,       []						,       []						,       lowercase__						,       lowercase__   )

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

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


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			n_queens_solution(4)










from __future__ import annotations

from collections.abc import Callable

UpperCamelCase__ =						list[list[float | int]]






def  UpperCAmelCase__    (    lowercase__						,       lowercase__   ) ->      Matrix:
						__lowercase					=				len(lowercase__   )
						__lowercase					=				[[0 for _ in range(size + 1   )] for _ in range(lowercase__   )]
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42

						for row in range(lowercase__   ):
												for col in range(lowercase__   ):
																		__lowercase					=				matrix[row][col]

												__lowercase					=				vector[row][0]

						__lowercase					=				0
						__lowercase					=				0
						while row < size and col < size:
												# pivoting
												__lowercase					=				max((abs(augmented[rowa][col]   ), rowa) for rowa in range(lowercase__						,       lowercase__   )   )[
												    1
												]
												if augmented[pivot_row][col] == 0:
																		col += 1
																		continue
												else:
																		__lowercase	,			__lowercase					=				augmented[pivot_row], augmented[row]

												for rowa in range(row + 1						,       lowercase__   ):
																		__lowercase					=				augmented[rowa][col] / augmented[row][col]
																		__lowercase					=				0
																		for cola in range(col + 1						,       size + 1   ):
																								augmented[rowa][cola] -= augmented[row][cola] * ratio

												row += 1
												col += 1

						# back substitution
						for col in range(1						,       lowercase__   ):
												for row in range(lowercase__   ):
																		__lowercase					=				augmented[row][col] / augmented[col][col]
																		for cola in range(lowercase__						,       size + 1   ):
																								augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
						return [
						    [round(augmented[row][size] / augmented[row][row]						,       10   )] for row in range(lowercase__   )
						]






def  UpperCAmelCase__    (    lowercase__   ) ->      Callable[[int], int]:
						__lowercase					=				len(lowercase__   )
						__lowercase					=				[[0 for _ in range(lowercase__   )] for _ in range(lowercase__   )]
						__lowercase					=				[[0] for _ in range(lowercase__   )]
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42
						__lowercase					=				42

						for x_val, y_val in enumerate(lowercase__   ):
												for col in range(lowercase__   ):
																		__lowercase					=				(x_val + 1) ** (size - col - 1)
												__lowercase					=				y_val

						__lowercase					=				solve(lowercase__						,       lowercase__   )

						def interpolated_func(lowercase__   ) -> int:
												return sum(
												    round(coeffs[x_val][0]   ) * (var ** (size - x_val - 1))
												    for x_val in range(lowercase__   )   )

						return interpolated_func






def  UpperCAmelCase__    (    lowercase__   ) ->      int:
						return (
						    1
						    - variable
						    + variable**2
						    - variable**3
						    + variable**4
						    - variable**5
						    + variable**6
						    - variable**7
						    + variable**8
						    - variable**9
						    + variable**10
						)






def  UpperCAmelCase__    (    lowercase__ = question_function						,       lowercase__ = 10   ) ->      int:
						__lowercase					=				[func(lowercase__   ) for x_val in range(1						,       order + 1   )]

						__lowercase					=				[
						    interpolate(data_points[:max_coeff]   ) for max_coeff in range(1						,       order + 1   )
						]

						__lowercase					=				0
						__lowercase					=				42
						__lowercase					=				42

						for poly in polynomials:
												__lowercase					=				1
												while func(lowercase__   ) == poly(lowercase__   ):
																		x_val += 1

												ret += poly(lowercase__   )

						return ret


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





