


'''simple docstring'''






import doctest
from collections import deque

import numpy as np




class 						a__							:





							'''simple docstring'''






							def __init__(						self  )			->				None:
														lowerCAmelCase__			  =      [2, 1, 2, -1]
														lowerCAmelCase__			  =      [1, 2, 3, 4]







							def     __SCREAMING_SNAKE_CASE				(						self  )			->				list[float]:
														lowerCAmelCase__			  =      len(self.first_signal  )
														lowerCAmelCase__			  =      len(self.second_signal  )

														lowerCAmelCase__			  =      max(lowerCamelCase_   ,       lowerCamelCase_  )

														# create a zero matrix of max_length x max_length
														lowerCAmelCase__			  =      [[0] * max_length for i in range(lowerCamelCase_  )]

														# fills the smaller signal with zeros to make both signals of same length
														if length_first_signal < length_second_signal:
																					self.first_signal += [0] * (max_length - length_first_signal)
														elif length_first_signal > length_second_signal:
																					self.second_signal += [0] * (max_length - length_second_signal)
														for i in range(lowerCamelCase_  ):
																					lowerCAmelCase__			  =      deque(self.second_signal  )
																					rotated_signal.rotate(lowerCamelCase_  )
																					for j, item in enumerate(lowerCamelCase_  ):
																												matrix[i][j] += item

        # multiply the matrix with the first signal
														lowerCAmelCase__			  =      np.matmul(np.transpose(lowerCamelCase_  )   ,       np.transpose(self.first_signal  )  )

														# rounding-off to two decimal places
														return [round(lowerCamelCase_   ,       2  ) for i in final_signal]


if __name__ == "__main__":
		doctest.testmod()



'''simple docstring'''






from __future__ import annotations

__UpperCAmelCase       							=				list[list[int]]

# assigning initial values to the grid
__UpperCAmelCase       							=				[
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
__UpperCAmelCase       							=				[
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]


def    _snake_case  (	A					,       A					,       A					,       A			)		->       bool:
							for i in range(9			):
														if grid[row][i] == n or grid[i][column] == n:
																					return False

							for i in range(3			):
														for j in range(3			):
																					if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
																												return False

							return True


def    _snake_case  (	A			)		->       tuple[int, int] | None:
							for i in range(9			):
														for j in range(9			):
																					if grid[i][j] == 0:
																												return i, j
							return None


def    _snake_case  (	A			)		->       Matrix | None:
							if location := find_empty_location(A			):
														lowerCAmelCase__  ,   lowerCAmelCase__			  =      location
							else:
														# If the location is ``None``, then the grid is solved.
														return grid

							for digit in range(1					,       10			):
														if is_safe(A					,       A					,       A					,       A			):
																					lowerCAmelCase__			  =      digit

																					if sudoku(A			) is not None:
																												return grid

																					lowerCAmelCase__			  =      0

							return None


def    _snake_case  (	A			)		->       None:
							for row in grid:
														for cell in row:
																					print(A					,       end=''' '''			)
														print()


if __name__ == "__main__":
		# make a copy of grid so that you can compare with the unmodified grid
		for example_grid in (initial_grid, no_solution):
				print('''\nExample grid:\n''' + '''=''' * 20)
				print_solution(example_grid)
				print('''\nExample grid solution:''')
				__UpperCAmelCase       							=				sudoku(example_grid)
				if solution is not None:
						print_solution(solution)
				else:
						print('''Cannot find a solution.''')