





'''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 					_a						(       _lowerCamelCase    ,	_lowerCamelCase    ,	_lowerCamelCase    ,	_lowerCamelCase ) ->				bool:



			"""simple docstring"""


			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 					_a						(       _lowerCamelCase ) ->				tuple[int, int] | None:



			"""simple docstring"""


			for i in range(9 ):
						for j in range(9 ):
									if grid[i][j] == 0:
												return i, j
			return None







def 					_a						(       _lowerCamelCase ) ->				Matrix | None:



			"""simple docstring"""


			if location := find_empty_location(_lowerCamelCase ):
						__snake_case      ,     __snake_case					:      Any							      =     location
			else:
						# If the location is ``None``, then the grid is solved.
						return grid

			for digit in range(1    ,	10 ):
						if is_safe(_lowerCamelCase    ,	_lowerCamelCase    ,	_lowerCamelCase    ,	_lowerCamelCase ):
									__snake_case					:      Optional[int]							      =     digit

									if sudoku(_lowerCamelCase ) is not None:
												return grid

									__snake_case					:      List[str]							      =     0

			return None







def 					_a						(       _lowerCamelCase ) ->				None:



			"""simple docstring"""


			for row in grid:
						for cell in row:
									print(_lowerCamelCase    ,	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.")








def 		lowerCamelCase__       (						_A	,      _A				):







	'''simple docstring'''





	_validate_point(_A				)
	_validate_point(_A				)
	if len(_A				) != len(_A				):
		raise ValueError("Both points must be in the same n-dimensional space"				)

	return float(sum(abs(a - b				) for a, b in zip(_A	,      _A				)				)				)


def 		lowerCamelCase__       (						_A				):







	'''simple docstring'''





	if point:
		if isinstance(_A	,      _A				):
			for item in point:
				if not isinstance(_A	,      (int, float)				):
					snake_case_						       =	(
					    "Expected a list of numbers as input, found "
					    f"{type(_A				).__name__}"
					)
					raise TypeError(_A				)
		else:
			snake_case_						       =	f"Expected a list of numbers as input, found {type(_A				).__name__}"
			raise TypeError(_A				)
	else:
		raise ValueError("Missing an input"				)


def 		lowerCamelCase__       (						_A	,      _A				):







	'''simple docstring'''





	_validate_point(_A				)
	_validate_point(_A				)
	if len(_A				) != len(_A				):
		raise ValueError("Both points must be in the same n-dimensional space"				)

	return float(sum(abs(x - y				) for x, y in zip(_A	,      _A				)				)				)


if __name__ == "__main__":
		import doctest

		doctest.testmod()
