Lab 3: Recursion, Tree Recursion(CS61A 2020)

在网上没有lab3相应的答案，作者也卡蛮久

作者可能就自己的卡住过的问题做一些总结，不能面面俱到，请见谅

（就此补充一下答案）（完整答案在最后）

Q2: WWPD: Journey to the Center of the Earth

Use Ok to test your knowledge with the following "What Would Python Display?" questions:
python3 ok -q sr-wwpd -u
For all WWPD questions, type Function if you believe the answer is <function...>, Error if it errors, and Nothing if nothing is displayed.

>>> def crust():
...     print("70km")
...     def mantle():
...          print("2900km")
...          def core():
...               print("5300km")
...               return mantle()
...          return core
...     return mantle
>>> drill = crust
>>> drill = drill()

>>70km

>>> drill = drill()

>>2900km

>>> drill = drill()

>>5300km
>>2900km

>>> drill()

>>5300km
>>2900km
>>Function

1.之前的drill=drill()只是调用，

但不返回其值或类型（因为是赋值行为）

2.最后一个drill()

可以理解为，

函数调用过程中打印了两个值，并返回drill()的类型-->function

Q6: Ping-pong

The ping-pong sequence counts up starting from 1 and is always either counting up or counting down. At element k, the direction switches if k is a multiple of 8 or contains the digit 8. The first 30 elements of the ping-pong sequence are listed below, with direction swaps marked using brackets at the 8th, 16th, 18th, 24th, and 28th elements:

Index	1	2	3	4	5	6	7	[8]	9	10	11	12	13	14	15	[16]	17	[18]	19	20	21	22	23
PingPong Value	1	2	3	4	5	6	7	[8]	7	6	5	4	3	2	1	[0]	1	[2]	1	0	-1	-2	-3

Index (cont.)	[24]	25	26	27	[28]	29	30
PingPong Value	[-4]	-3	-2	-1	[0]	-1	-2

Implement a function pingpong that returns the nth element of the ping-pong sequence without using any assignment statements.

You may use the function num_eights, which you defined in the previous question.

Use recursion - the tests will fail if you use any assignment statements.

Hint: If you're stuck, first try implementing pingpong using assignment statements and a while statement. Then, to convert this into a recursive solution, write a helper function that has a parameter for each variable that changes values in the body of the while loop.

分析：pingpong(n) 递归返回 pingpong(n-1)+direction(n)

1.direction()作用是确定第n个数值正负，初始值都是1

n<8时可以确定一定为正即 1

n>8时递归统计有多少个结点n 及结点的上一个点n+1

（这两点会转换正负）

（因为结点的上一个点n+1处与更大的结点之间的数正负性均相同

结点处同理如图）

def pingpong(n):
    """Return the nth element of the ping-pong sequence.

    >>> pingpong(8)
    8
    >>> pingpong(10)
    6
    >>> pingpong(15)
    1
    >>> pingpong(21)
    -1
    >>> pingpong(22)
    -2
    >>> pingpong(30)
    -2
    >>> pingpong(68)
    0
    >>> pingpong(69)
    -1
    >>> pingpong(80)
    0
    >>> pingpong(81)
    1
    >>> pingpong(82)
    0
    >>> pingpong(100)
    -6
    >>> from construct_check import check
    >>> # ban assignment statements
    >>> check(HW_SOURCE_FILE, 'pingpong', ['Assign', 'AugAssign'])
    True
    """
    "*** YOUR CODE HERE ***"
        if n <= 8:
        return n
    return direction(n) + pingpong(n-1)


def direction(n):
    if n < 8:
        return 1
    if (n-1) % 8 == 0 or num_eights(n-1):
        return -1 * direction(n-1)
    return direction(n-1)

完整答案代码：

HW_SOURCE_FILE=__file__


def pascal(row, column):
    """Returns a number corresponding to the value at that location
    in Pascal's Triangle.
    >>> pascal(0, 0)
    1
    >>> pascal(0, 5)	# Empty entry; outside of Pascal's Triangle
    0
    >>> pascal(3, 2)	# Row 4 (1 3 3 1), 3rd entry
    3
    """
    "*** YOUR CODE HERE ***"
    if row<0 :
        return 0
    if row==0 and column==0:
        return 1
    elif column>row or column<0:
        return 0
    else:
        return pascal(row-1,column)+pascal(row-1,column-1)


def compose1(f, g):
    """"Return a function h, such that h(x) = f(g(x))."""
    def h(x):
        return f(g(x))
    return h

def repeated(f, n):
    """Return the function that computes the nth application of func (recursively!).

    >>> add_three = repeated(lambda x: x + 1, 3)
    >>> add_three(5)
    8
    >>> square = lambda x: x ** 2
    >>> repeated(square, 2)(5) # square(square(5))
    625
    >>> repeated(square, 4)(5) # square(square(square(square(5))))
    152587890625
    >>> repeated(square, 0)(5)
    5
    >>> from construct_check import check
    >>> # ban iteration
    >>> check(HW_SOURCE_FILE, 'repeated',
    ...       ['For', 'While'])
    True
    """
    "*** YOUR CODE HERE ***"
    if n==0:
        return lambda  x:x
    if n==1:
        return f
    return compose1(f,repeated(f,n-1))


def num_eights(x):
    """Returns the number of times 8 appears as a digit of x.

    >>> num_eights(3)
    0
    >>> num_eights(8)
    1
    >>> num_eights(88888888)
    8
    >>> num_eights(2638)
    1
    >>> num_eights(86380)
    2
    >>> num_eights(12345)
    0
    >>> from construct_check import check
    >>> # ban all assignment statements
    >>> check(HW_SOURCE_FILE, 'num_eights',
    ...       ['Assign', 'AugAssign'])
    True
    """
    "*** YOUR CODE HERE ***"
    if x==0:
        return 0
    if x%10==8:
        return num_eights(x//10)+1
    else:
        return num_eights(x//10)


def pingpong(n):
    """Return the nth element of the ping-pong sequence.

    >>> pingpong(8)
    8
    >>> pingpong(10)
    6
    >>> pingpong(15)
    1
    >>> pingpong(21)
    -1
    >>> pingpong(22)
    -2
    >>> pingpong(30)
    -2
    >>> pingpong(68)
    0
    >>> pingpong(69)
    -1
    >>> pingpong(80)
    0
    >>> pingpong(81)
    1
    >>> pingpong(82)
    0
    >>> pingpong(100)
    -6
    >>> from construct_check import check
    >>> # ban assignment statements
    >>> check(HW_SOURCE_FILE, 'pingpong', ['Assign', 'AugAssign'])
    True
    """
    "*** YOUR CODE HERE ***"
    if n <= 8:
        return n
    return direction(n) + pingpong(n-1)


def direction(n):
    if n < 8:
        return 1
    if (n-1) % 8 == 0 or num_eights(n-1):
        return -1 * direction(n-1)
    return direction(n-1)