






from __future__ import annotations

SCREAMING_SNAKE_CASE 							=						list[list[int]]

# assigning initial values to the grid
SCREAMING_SNAKE_CASE 							=						[
    [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
SCREAMING_SNAKE_CASE 							=						[
    [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      _lowerCamelCase	(						__A    :							Tuple		,		__A    :							List[str]		,		__A    :							int		,		__A    :							Optional[Any] )      ->					List[str]:
    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      _lowerCamelCase	(						__A    :							Optional[int] )      ->					List[Any]:
    for i in range(9 ):
        for j in range(9 ):
            if grid[i][j] == 0:
                return i, j
    return None

def      _lowerCamelCase	(						__A    :							Optional[int] )      ->					List[Any]:
    if location := find_empty_location(lowerCAmelCase__ ):
        _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(lowerCAmelCase__		,		lowerCAmelCase__		,		lowerCAmelCase__		,		lowerCAmelCase__ ):
            _UpperCAmelCase				:      List[str]			  =				digit

            if sudoku(lowerCAmelCase__ ) is not None:
                return grid

            _UpperCAmelCase				:      List[str]			  =				0

    return None

def      _lowerCamelCase	(						__A    :							Tuple )      ->					Optional[Any]:
    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:')
    SCREAMING_SNAKE_CASE 							=						sudoku(example_grid)
    if solution is not None:
      print_solution(solution)
    else:
      print('Cannot find a solution.')












import math

def      _lowerCamelCase	(						__A    :							int )      ->					int:

    if not isinstance(__A		,		__A ):
        _UpperCAmelCase				:      List[Any]			  =				f'''Input value of [number={number}] must be an integer'''
        raise TypeError(__A )

    if number < 1:
        _UpperCAmelCase				:      Optional[Any]			  =				f'''Input value of [number={number}] must be > 0'''
        raise ValueError(__A )
    elif number == 1:
        return 3
    elif number == 2:
        return 5
    else:
        _UpperCAmelCase				:      Any			  =				int(math.log(number // 3		,		2 ) ) + 2

        _UpperCAmelCase				:      Dict			  =				[3, 5]
        _UpperCAmelCase				:      Tuple			  =				2
        _UpperCAmelCase				:      Dict			  =				3
        for block in range(1		,		__A ):
            for _ in range(__A ):
                proth_list.append(2 ** (block + 1) + proth_list[proth_index - 1] )
                proth_index += 1
            increment *= 2

    return proth_list[number - 1]


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  for number in range(11):
    SCREAMING_SNAKE_CASE 							=						0
    try:
      SCREAMING_SNAKE_CASE 							=						proth(number)
    except ValueError:
      print(F'ValueError: there is no {number}th Proth number')
      continue

    print(F'The {number}th Proth number: {value}')




