

"""simple docstring"""






from __future__ import annotations

from collections import Counter
from random import random







class       snake_case__  :







   def __init__(			self			: Any  ):





        '''simple docstring'''



        UpperCAmelCase			:   List[Any]      =				{}







   def 			__lowerCAmelCase				(			self			: List[Any]			,  lowercase			: str  ):





        '''simple docstring'''



        UpperCAmelCase			:   Optional[int]      =				{}







   def 			__lowerCAmelCase				(			self			: Optional[int]			,  lowercase			: str			,  lowercase			: str			,  lowercase			: float  ):





        '''simple docstring'''



        if nodea not in self.connections:
             self.add_node(lowercase  )
        if nodea not in self.connections:
             self.add_node(lowercase  )
        UpperCAmelCase			:   int      =				probability







   def 			__lowerCAmelCase				(			self			: List[Any]  ):





        '''simple docstring'''



        return list(self.connections  )







   def 			__lowerCAmelCase				(			self			: int			,  lowercase			: str  ):





        '''simple docstring'''



        UpperCAmelCase			:   int      =				0
        UpperCAmelCase			:   Union[str, Any]      =				random()

        for dest in self.connections[node]:
             current_probability += self.connections[node][dest]
             if current_probability > random_value:
                  return dest
        return ""



def   lowercase_	(    _lowercase       :		str							,      _lowercase       :		list[tuple[str, str, float]]							,      _lowercase       :		int	):




     '''simple docstring'''






     UpperCAmelCase			:   int      =				MarkovChainGraphUndirectedUnweighted()

     for nodea, nodea, probability in transitions:
          graph.add_transition_probability(_lowercase							,      _lowercase							,      _lowercase	)

     UpperCAmelCase			:   List[Any]      =				Counter(graph.get_nodes()	)
     UpperCAmelCase			:   List[Any]      =				start

     for _ in range(_lowercase	):
          UpperCAmelCase			:   Optional[Any]      =				graph.transition(_lowercase	)
          visited[node] += 1

     return visited


if __name__ == "__main__":
 import doctest

 doctest.testmod()





"""simple docstring"""






from __future__ import annotations

snake_case_   :   str        =       list[list[int]]

# assigning initial values to the grid
snake_case_   :   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
snake_case_   :   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   lowercase_	(    _lowercase       :		Matrix							,      _lowercase       :		int							,      _lowercase       :		int							,      _lowercase       :		int	):




					'''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   lowercase_	(    _lowercase       :		Matrix	):




					'''simple docstring'''






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



def   lowercase_	(    _lowercase       :		Matrix	):




					'''simple docstring'''






					if location := find_empty_location(_lowercase	):
										UpperCAmelCase      ,							UpperCAmelCase			:   List[str]      =				location
					else:
										# If the location is ``None``, then the grid is solved.
										return grid

					for digit in range(1							,      10	):
										if is_safe(_lowercase							,      _lowercase							,      _lowercase							,      _lowercase	):
															UpperCAmelCase			:   int      =				digit

															if sudoku(_lowercase	) is not None:
																				return grid

															UpperCAmelCase			:   Optional[Any]      =				0

					return None



def   lowercase_	(    _lowercase       :		Matrix	):




					'''simple docstring'''






					for row in grid:
										for cell in row:
															print(_lowercase							,      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""" + """=""" * 2_0)
		print_solution(example_grid)
		print("""\nExample grid solution:""")
		snake_case_   :   Dict        =       sudoku(example_grid)
		if solution is not None:
			print_solution(solution)
		else:
			print("""Cannot find a solution.""")


