
def 				a						(							snake_case__:					int						,   snake_case__:					list					):


				'''simple docstring'''
				_enforce_args(snake_case__						,   snake_case__					)
				if n == 0:
								return 0
				lowercase_       =					float('''-inf'''					)
				for i in range(1						,   n + 1					):
								lowercase_       =					max(
								    snake_case__						,   prices[i - 1] + naive_cut_rod_recursive(n - i						,   snake_case__					)					)

				return max_revue




def 				a						(							snake_case__:					int						,   snake_case__:					list					):


				'''simple docstring'''
				_enforce_args(snake_case__						,   snake_case__					)
				lowercase_       =					[float('''-inf'''					) for _ in range(n + 1					)]
				return _top_down_cut_rod_recursive(snake_case__						,   snake_case__						,   snake_case__					)




def 				a						(							snake_case__:					int						,   snake_case__:					list						,   snake_case__:					list					):


				'''simple docstring'''
				if max_rev[n] >= 0:
								return max_rev[n]
				elif n == 0:
								return 0
				else:
								lowercase_       =					float('''-inf'''					)
								for i in range(1						,   n + 1					):
												lowercase_       =					max(
												    snake_case__						,   prices[i - 1] + _top_down_cut_rod_recursive(n - i						,   snake_case__						,   snake_case__					)						,   )

								lowercase_       =					max_revenue

				return max_rev[n]




def 				a						(							snake_case__:					int						,   snake_case__:					list					):


				'''simple docstring'''
				_enforce_args(snake_case__						,   snake_case__					)

				# length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
				# length 0.
				lowercase_       =					[float('''-inf'''					) for _ in range(n + 1					)]
				lowercase_       =					0

				for i in range(1						,   n + 1					):
								lowercase_       =					max_rev[i]
								for j in range(1						,   i + 1					):
												lowercase_       =					max(snake_case__						,   prices[j - 1] + max_rev[i - j]					)

								lowercase_       =					max_revenue_i

				return max_rev[n]




def 				a						(							snake_case__:					int						,   snake_case__:					list					):


				'''simple docstring'''
				if n < 0:
								lowercase_       =					F'''n must be greater than or equal to 0. Got n = {n}'''
								raise ValueError(snake_case__					)

				if n > len(snake_case__					):
								lowercase_       =					(
								    '''Each integral piece of rod must have a corresponding price. '''
								    F'''Got n = {n} but length of prices = {len(snake_case__					)}'''
								)
								raise ValueError(snake_case__					)




def 				a						(							):


				'''simple docstring'''
				lowercase_       =					[6, 10, 12, 15, 20, 23]
				lowercase_       =					len(snake_case__					)

				# the best revenue comes from cutting the rod into 6 pieces, each
				# of length 1 resulting in a revenue of 6 * 6 = 36.
				lowercase_       =					36

				lowercase_       =					top_down_cut_rod(snake_case__						,   snake_case__					)
				lowercase_       =					bottom_up_cut_rod(snake_case__						,   snake_case__					)
				lowercase_       =					naive_cut_rod_recursive(snake_case__						,   snake_case__					)

				assert expected_max_revenue == max_rev_top_down
				assert max_rev_top_down == max_rev_bottom_up
				assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
						main()











from __future__ import annotations


def        SCREAMING_SNAKE_CASE      (       _UpperCAmelCase						,	_UpperCAmelCase )			->     list[tuple[int, int]]:
    lowerCamelCase__ ,							lowerCamelCase__    :      Union[str, Any]					 =		position
    lowerCamelCase__    :      Optional[Any]					 =		[
        (y + 1, x + 2),
        (y - 1, x + 2),
        (y + 1, x - 2),
        (y - 1, x - 2),
        (y + 2, x + 1),
        (y + 2, x - 1),
        (y - 2, x + 1),
        (y - 2, x - 1),
    ]
    lowerCamelCase__    :      Dict					 =		[]

    for position in positions:
        lowerCamelCase__ ,							lowerCamelCase__    :      Optional[int]					 =		position
        if 0 <= y_test < n and 0 <= x_test < n:
            permissible_positions.append(_UpperCAmelCase )

    return permissible_positions


def        SCREAMING_SNAKE_CASE      (       _UpperCAmelCase )			->     bool:
    return not any(elem == 0 for row in board for elem in row )


def        SCREAMING_SNAKE_CASE      (       _UpperCAmelCase						,	_UpperCAmelCase						,	_UpperCAmelCase )			->     bool:

    if is_complete(_UpperCAmelCase ):
        return True

    for position in get_valid_pos(_UpperCAmelCase						,	len(_UpperCAmelCase ) ):
        lowerCamelCase__ ,							lowerCamelCase__    :      Optional[int]					 =		position

        if board[y][x] == 0:
            lowerCamelCase__    :      List[Any]					 =		curr + 1
            if open_knight_tour_helper(_UpperCAmelCase						,	_UpperCAmelCase						,	curr + 1 ):
                return True
            lowerCamelCase__    :      Optional[Any]					 =		0

    return False


def        SCREAMING_SNAKE_CASE      (       _UpperCAmelCase )			->     list[list[int]]:
    lowerCamelCase__    :      Any					 =		[[0 for i in range(_UpperCAmelCase )] for j in range(_UpperCAmelCase )]

    for i in range(_UpperCAmelCase ):
        for j in range(_UpperCAmelCase ):
            lowerCamelCase__    :      Optional[int]					 =		1
            if open_knight_tour_helper(_UpperCAmelCase						,	(i, j)						,	1 ):
                return board
            lowerCamelCase__    :      Optional[Any]					 =		0

    lowerCamelCase__    :      Any					 =		F"""Open Kight Tour cannot be performed on a board of size {n}"""
    raise ValueError(_UpperCAmelCase )


if __name__ == "__main__":
    import doctest

    doctest.testmod()





