






from __future__ import annotations

__magic_name__   :  Optional[int]        =							list[list[int]]

# assigning initial values to the grid
__magic_name__   :  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
__magic_name__   :  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__  (				_UpperCamelCase       ,							_UpperCamelCase       ,							_UpperCamelCase       ,							_UpperCamelCase) ->			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 		lowercase__  (				_UpperCamelCase) ->			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 		lowercase__  (				_UpperCamelCase) ->			Matrix | None:
      """simple docstring"""

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

      for digit in range(1       ,							10):
            if is_safe(_UpperCamelCase       ,							_UpperCamelCase       ,							_UpperCamelCase       ,							_UpperCamelCase):
                  UpperCamelCase					       =  digit

                  if sudoku(_UpperCamelCase) is not None:
                        return grid

                  UpperCamelCase					       =  0

      return None
def 		lowercase__  (				_UpperCamelCase) ->			None:
      """simple docstring"""

      for row in grid:
            for cell in row:
                  print(_UpperCamelCase       ,							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:''')
              __magic_name__   :  Any        =							sudoku(example_grid)
              if solution is not None:
                     print_solution(solution)
              else:
                     print('''Cannot find a solution.''')











from __future__ import annotations

from collections.abc import MutableSequence





class 					A__     :

     '''simple docstring'''






     def __init__( self      :    Dict   ,						_SCREAMING_SNAKE_CASE      :    int   ,						_SCREAMING_SNAKE_CASE      :    MutableSequence[float] ):






           """simple docstring"""




           if len(_SCREAMING_SNAKE_CASE ) != degree + 1:
                 raise ValueError(
                     'The number of coefficients should be equal to the degree + 1.' )

           UpperCamelCase					       =  list(_SCREAMING_SNAKE_CASE )
           UpperCamelCase					       =  degree







     def __add__( self      :    Any   ,						_SCREAMING_SNAKE_CASE      :    Polynomial ):






           """simple docstring"""




           if self.degree > polynomial_a.degree:
                 UpperCamelCase					       =  self.coefficients[:]
                 for i in range(polynomial_a.degree + 1 ):
                       coefficients[i] += polynomial_a.coefficients[i]
                 return Polynomial(self.degree   ,						_SCREAMING_SNAKE_CASE )
           else:
                 UpperCamelCase					       =  polynomial_a.coefficients[:]
                 for i in range(self.degree + 1 ):
                       coefficients[i] += self.coefficients[i]
                 return Polynomial(polynomial_a.degree   ,						_SCREAMING_SNAKE_CASE )







     def __sub__( self      :    Optional[int]   ,						_SCREAMING_SNAKE_CASE      :    Polynomial ):






           """simple docstring"""




           return self + polynomial_a * Polynomial(0   ,						[-1] )







     def __neg__( self      :    Any ):






           """simple docstring"""




           return Polynomial(self.degree   ,						[-c for c in self.coefficients] )







     def __mul__( self      :    Union[str, Any]   ,						_SCREAMING_SNAKE_CASE      :    Polynomial ):






           """simple docstring"""




           UpperCamelCase					       =  [0] * (self.degree + polynomial_a.degree + 1)
           for i in range(self.degree + 1 ):
                 for j in range(polynomial_a.degree + 1 ):
                       coefficients[i + j] += (
                           self.coefficients[i] * polynomial_a.coefficients[j]
                       )

           return Polynomial(self.degree + polynomial_a.degree   ,						_SCREAMING_SNAKE_CASE )







     def   _SCREAMING_SNAKE_CASE     ( self      :    Optional[int]   ,						_SCREAMING_SNAKE_CASE      :    int | float ):






           """simple docstring"""




           UpperCamelCase					       =  0
           for i in range(self.degree + 1 ):
                 result += self.coefficients[i] * (substitution**i)
           return result







     def __str__( self      :    Union[str, Any] ):






           """simple docstring"""




           UpperCamelCase					       =  ''
           for i in range(self.degree   ,						-1   ,						-1 ):
                 if self.coefficients[i] == 0:
                       continue
                 elif self.coefficients[i] > 0:
                       if polynomial:
                             polynomial += " + "
                 else:
                       polynomial += " - "

                 if i == 0:
                       polynomial += str(abs(self.coefficients[i] ) )
                 elif i == 1:
                       polynomial += str(abs(self.coefficients[i] ) ) + "x"
                 else:
                       polynomial += str(abs(self.coefficients[i] ) ) + "x^" + str(_SCREAMING_SNAKE_CASE )

           return polynomial







     def __repr__( self      :    str ):






           """simple docstring"""




           return self.__str__()







     def   _SCREAMING_SNAKE_CASE     ( self      :    Optional[Any] ):






           """simple docstring"""




           UpperCamelCase					       =  [0] * self.degree
           for i in range(self.degree ):
                 UpperCamelCase					       =  self.coefficients[i + 1] * (i + 1)
           return Polynomial(self.degree - 1   ,						_SCREAMING_SNAKE_CASE )







     def   _SCREAMING_SNAKE_CASE     ( self      :    Optional[int]   ,						_SCREAMING_SNAKE_CASE      :    int | float = 0 ):






           """simple docstring"""




           UpperCamelCase					       =  [0] * (self.degree + 2)
           UpperCamelCase					       =  constant
           for i in range(self.degree + 1 ):
                 UpperCamelCase					       =  self.coefficients[i] / (i + 1)
           return Polynomial(self.degree + 1   ,						_SCREAMING_SNAKE_CASE )







     def __eq__( self      :    Any   ,						_SCREAMING_SNAKE_CASE      :    object ):






           """simple docstring"""




           if not isinstance(_SCREAMING_SNAKE_CASE   ,						_SCREAMING_SNAKE_CASE ):
                 return False

           if self.degree != polynomial_a.degree:
                 return False

           for i in range(self.degree + 1 ):
                 if self.coefficients[i] != polynomial_a.coefficients[i]:
                       return False

           return True







     def __ne__( self      :    str   ,						_SCREAMING_SNAKE_CASE      :    object ):






           """simple docstring"""




           return not self.__eq__(_SCREAMING_SNAKE_CASE )



