




from __future__ import annotations

__UpperCamelCase					:       Union[str, Any]    					=      list[list[int]]

# assigning initial values to the grid
__UpperCamelCase					:       Matrix    					=      [
    [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					:       Matrix    					=      [
    [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:
	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:
	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:
	if location := find_empty_location(__lowerCamelCase    ):
		a				, a										=							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    ):
			a										=							digit

			if sudoku(__lowerCamelCase    ) is not None:
				return grid

			a										=							0

	return None





def 		__A		(	__lowerCamelCase    )      ->       None:
	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					:       Any    					=      sudoku(example_grid)
				if solution is not None:
						print_solution(solution)
				else:
						print("Cannot find a solution.")







from math import pi, sqrt





def 		__A		(	__lowerCamelCase    )      ->       float:
	if num <= 0:
		raise ValueError("""math domain error"""    )
	if num > 171.5:
		raise OverflowError("""math range error"""    )
	elif num - int(__lowerCamelCase    ) not in (0, 0.5):
		raise NotImplementedError("""num must be an integer or a half-integer"""    )
	elif num == 0.5:
		return sqrt(__lowerCamelCase    )
	else:
		return 1.0 if num == 1 else (num - 1) * gamma(num - 1    )





def 		__A		(	)      ->       None:
	assert gamma(0.5    ) == sqrt(__lowerCamelCase    )
	assert gamma(1    ) == 1.0
	assert gamma(2    ) == 1.0


if __name__ == "__main__":
		from doctest import testmod

		testmod()
		__UpperCamelCase					:       str    					=      1.0
		while num:
				__UpperCamelCase					:       Dict    					=      float(input("Gamma of: "))
				print(F'gamma({num}) = {gamma(num)}')
				print("\nEnter 0 to exit...")

