






import math






def   UpperCamelCase  (       snake_case__     :   Optional[int]  ):




     '''simple docstring'''






     __snake_case			:Any          =						[True] * n
     __snake_case			:Dict          =						False
     __snake_case			:Tuple          =						False
     __snake_case			:Optional[int]          =						True

     for i in range(3	,int(n**0.5 + 1  )	,2  ):
          __snake_case			:Optional[Any]          =						i * 2
          while index < n:
               __snake_case			:int          =						False
               __snake_case			:List[Any]          =						index + i

     __snake_case			:Optional[Any]          =						[2]

     for i in range(3	,lowercase__	,2  ):
          if is_prime[i]:
               primes.append(lowercase__  )

     return primes






def   UpperCamelCase  (       snake_case__     :   Optional[Any] = 9999_6666_3333  ):




     '''simple docstring'''






     __snake_case			:str          =						math.floor(math.sqrt(lowercase__  )  ) + 100
     __snake_case			:List[str]          =						prime_sieve(lowercase__  )

     __snake_case			:Union[str, Any]          =						0
     __snake_case			:List[Any]          =						0
     __snake_case			:str          =						primes[prime_index]

     while (last_prime**2) <= limit:
          __snake_case			:str          =						primes[prime_index + 1]

          __snake_case			:Dict          =						last_prime**2
          __snake_case			:Any          =						next_prime**2

          # Get numbers divisible by lps(current)
          __snake_case			:Tuple          =						lower_bound + last_prime
          while upper_bound > current <= limit:
               matches_sum += current
               current += last_prime

          # Reset the upper_bound
          while (upper_bound - next_prime) > limit:
               upper_bound -= next_prime

          # Add the numbers divisible by ups(current)
          __snake_case			:str          =						upper_bound - next_prime
          while current > lower_bound:
               matches_sum += current
               current -= next_prime

          # Remove the numbers divisible by both ups and lps
          __snake_case			:Dict          =						0
          while upper_bound > current <= limit:
               if current <= lower_bound:
                    # Increment the current number
                    current += last_prime * next_prime
                    continue

               if current > limit:
                    break

               # Remove twice since it was added by both ups and lps
               matches_sum -= current * 2

               # Increment the current number
               current += last_prime * next_prime

          # Setup for next pair
          __snake_case			:Any          =						next_prime
          prime_index += 1

     return matches_sum


if __name__ == "__main__":
     print(solution())










from __future__ import annotations

lowerCamelCase__							=     list[list[int]]

# assigning initial values to the grid
lowerCamelCase__							=     [
    [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
lowerCamelCase__							=     [
    [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   UpperCamelCase  (       snake_case__     :   Matrix	,snake_case__     :   int	,snake_case__     :   int	,snake_case__     :   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   UpperCamelCase  (       snake_case__     :   Matrix  ):




     '''simple docstring'''






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






def   UpperCamelCase  (       snake_case__     :   Matrix  ):




     '''simple docstring'''






     if location := find_empty_location(snake_case__  ):
          __snake_case       , __snake_case			:Optional[int]          =						location
     else:
          # If the location is ``None``, then the grid is solved.
          return grid

     for digit in range(1	,10  ):
          if is_safe(snake_case__	,snake_case__	,snake_case__	,snake_case__  ):
               __snake_case			:Union[str, Any]          =						digit

               if sudoku(snake_case__  ) is not None:
                    return grid

               __snake_case			:Tuple          =						0

     return None






def   UpperCamelCase  (       snake_case__     :   Matrix  ):




     '''simple docstring'''






     for row in grid:
          for cell in row:
               print(snake_case__	,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:""")
          lowerCamelCase__							=     sudoku(example_grid)
          if solution is not None:
               print_solution(solution)
          else:
               print("""Cannot find a solution.""")


