






from __future__ import annotations

__UpperCamelCase   :   Optional[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 					snake_case	(  lowerCamelCase					,	lowerCamelCase					,	lowerCamelCase					,	lowerCamelCase       ):




	'''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 					snake_case	(  lowerCamelCase       ):




	'''simple docstring'''







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





def 					snake_case	(  lowerCamelCase       ):




	'''simple docstring'''







	if location := find_empty_location(lowerCamelCase       ):
		__lowercase							,    __lowercase   							=	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       ):
			__lowercase   							=	digit

			if sudoku(lowerCamelCase       ) is not None:
				return grid

			__lowercase   							=	0

	return None





def 					snake_case	(  lowerCamelCase       ):




	'''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   :   Optional[Any]      =    sudoku(example_grid)
												if solution is not None:
																		print_solution(solution)
												else:
																		print("""Cannot find a solution.""")







from __future__ import annotations

from decimal import Decimal

from numpy import array

def lowercase    (  SCREAMING_SNAKE_CASE )			->							list[list[float]]:


			'''simple docstring'''







			SCREAMING_SNAKE_CASE_								=						Decimal

			# Check if the provided matrix has 2 rows and 2 columns
			# since this implementation only works for 2x2 matrices
			if len(SCREAMING_SNAKE_CASE ) == 2 and len(matrix[0] ) == 2 and len(matrix[1] ) == 2:
						# Calculate the determinant of the matrix
						SCREAMING_SNAKE_CASE_								=						float(
						    d(matrix[0][0] ) * d(matrix[1][1] ) - d(matrix[1][0] ) * d(matrix[0][1] ) )
						if determinant == 0:
									raise ValueError('This matrix has no inverse.' )

						# Creates a copy of the matrix with swapped positions of the elements
						SCREAMING_SNAKE_CASE_								=						[[0.0, 0.0], [0.0, 0.0]]
						SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_								=						matrix[1][1], matrix[0][0]
						SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_								=						-matrix[1][0], -matrix[0][1]

						# Calculate the inverse of the matrix
						return [
						    [(float(d(SCREAMING_SNAKE_CASE ) ) / determinant) or 0.0 for n in row] for row in swapped_matrix
						]
			elif (
			    len(SCREAMING_SNAKE_CASE ) == 3
			    and len(matrix[0] ) == 3
			    and len(matrix[1] ) == 3
			    and len(matrix[2] ) == 3
			):
						# Calculate the determinant of the matrix using Sarrus rule
						SCREAMING_SNAKE_CASE_								=						float(
						    (
						        (d(matrix[0][0] ) * d(matrix[1][1] ) * d(matrix[2][2] ))
						        + (d(matrix[0][1] ) * d(matrix[1][2] ) * d(matrix[2][0] ))
						        + (d(matrix[0][2] ) * d(matrix[1][0] ) * d(matrix[2][1] ))
						    )
						    - (
						        (d(matrix[0][2] ) * d(matrix[1][1] ) * d(matrix[2][0] ))
						        + (d(matrix[0][1] ) * d(matrix[1][0] ) * d(matrix[2][2] ))
						        + (d(matrix[0][0] ) * d(matrix[1][2] ) * d(matrix[2][1] ))
						    ) )
						if determinant == 0:
									raise ValueError('This matrix has no inverse.' )

						# Creating cofactor matrix
						SCREAMING_SNAKE_CASE_								=						[
						    [d(0.0 ), d(0.0 ), d(0.0 )],
						    [d(0.0 ), d(0.0 ), d(0.0 )],
						    [d(0.0 ), d(0.0 ), d(0.0 )],
						]
						SCREAMING_SNAKE_CASE_								=						(d(matrix[1][1] ) * d(matrix[2][2] )) - (
						    d(matrix[1][2] ) * d(matrix[2][1] )
						)
						SCREAMING_SNAKE_CASE_								=						-(
						    (d(matrix[1][0] ) * d(matrix[2][2] )) - (d(matrix[1][2] ) * d(matrix[2][0] ))
						)
						SCREAMING_SNAKE_CASE_								=						(d(matrix[1][0] ) * d(matrix[2][1] )) - (
						    d(matrix[1][1] ) * d(matrix[2][0] )
						)
						SCREAMING_SNAKE_CASE_								=						-(
						    (d(matrix[0][1] ) * d(matrix[2][2] )) - (d(matrix[0][2] ) * d(matrix[2][1] ))
						)
						SCREAMING_SNAKE_CASE_								=						(d(matrix[0][0] ) * d(matrix[2][2] )) - (
						    d(matrix[0][2] ) * d(matrix[2][0] )
						)
						SCREAMING_SNAKE_CASE_								=						-(
						    (d(matrix[0][0] ) * d(matrix[2][1] )) - (d(matrix[0][1] ) * d(matrix[2][0] ))
						)
						SCREAMING_SNAKE_CASE_								=						(d(matrix[0][1] ) * d(matrix[1][2] )) - (
						    d(matrix[0][2] ) * d(matrix[1][1] )
						)
						SCREAMING_SNAKE_CASE_								=						-(
						    (d(matrix[0][0] ) * d(matrix[1][2] )) - (d(matrix[0][2] ) * d(matrix[1][0] ))
						)
						SCREAMING_SNAKE_CASE_								=						(d(matrix[0][0] ) * d(matrix[1][1] )) - (
						    d(matrix[0][1] ) * d(matrix[1][0] )
						)

						# Transpose the cofactor matrix (Adjoint matrix)
						SCREAMING_SNAKE_CASE_								=						array(SCREAMING_SNAKE_CASE )
						for i in range(3 ):
									for j in range(3 ):
												SCREAMING_SNAKE_CASE_								=						cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
						SCREAMING_SNAKE_CASE_								=						array(SCREAMING_SNAKE_CASE )
						for i in range(3 ):
									for j in range(3 ):
												inverse_matrix[i][j] /= d(SCREAMING_SNAKE_CASE )

        # Calculate the inverse of the matrix
						return [[float(d(SCREAMING_SNAKE_CASE ) ) or 0.0 for n in row] for row in inverse_matrix]
			raise ValueError('Please provide a matrix of size 2x2 or 3x3.' )
