VERY IMPORTANT: In this discussion, don't use a Python interpreter to run code until you are sure you are correct. Figure things out and check your work by thinking about what your code will do. Not sure? Talk to your group!
To get help from a TA, send a message to the
discuss-queue channel with the @discuss tag and your discussion group number.
What everyone will need:
- A way to view this worksheet (on your phone is fine)
- A way to take notes (paper or a device)
What the group will need:
- Someone to join Discord.
Suggestion: After Midterm 1, some students are looking for more effective ways to study. One great option is to meet up with your discussion group outside of class to review practice problems together. Now is a great time to schedule a time and place for some extra group practice of old Midterm 1 questions. This is optional and not everyone needs to come, but if there are Midterm 1 topics that haven't totally clicked yet, next week is a perfect time to review them.
Everything in this course builds on prior topics, and it's going to be hard to keep up if you don't have a solid understanding of the foundational topics from Midterm 1.
Remember, it's ok if someone hasn't learned everything yet and needs more time to master the course material. The whole point of the course is for students to learn things they don't already know. Please support each other in the process.
Ok, time to discuss problems! Remember to work together. Everyone in the group should understand a solution before the group moves on.
Q1: Recursion Mystery
Given the function below, what will be the output of
def f(n): if n < 10: print(n) else: print(n % 10) f(n // 10) print(n % 10)
6 5 2 3 2 5 6
When you've all agreed on an answer, write your response in your group's channel's text chat so that the course staff can provide feedback.
Q2: Skip Factorial
Define the base case for the
def skip_factorial(n): """Return the product of positive integers n * (n - 2) * (n - 4) * ... >>> skip_factorial(5) # 5 * 3 * 1 15 >>> skip_factorial(8) # 8 * 6 * 4 * 2 384 """ if n <= 2: return n else: return n * skip_factorial(n - 2)
def skip_factorial(n): if n < 3: return n else: return n * skip_factorial(n - 2)
Q3: Is Prime
is_prime that takes an integer
n greater than 1. It returns
n is a prime number and
False otherwise. Try following the approach
below, but implement it recursively without using a
def is_prime(n): assert n > 1 i = 2 while i < n: if n % i == 0: return False i = i + 1 return True
You will need to define another "helper" function (a function that exists just
to help implement this one). Does it matter whether you define it within
is_prime or as a separate function in the global frame? Try to define it to
take as few arguments as possible.
def is_prime(n): """Returns True if n is a prime number and False otherwise. >>> is_prime(2) True >>> is_prime(16) False >>> is_prime(521) True """ def check_all(i): "Check whether no number from i up to n evenly divides n." if i == n: # could be replaced with i > (n ** 0.5) return True elif n % i == 0: return False return check_all(i + 1) return check_all(2)
Finally, write a docstring for the helper function that describes what it does.
Don't just write, "it helps implement
is_prime." Instead, describe its
behavior. When you're done, paste the text of that docstring in your group's
Q4: Recursive Hailstone
hailstone function from Homework 1.
First, pick a positive integer
n as the start. If
n is even, divide it by 2.
n is odd, multiply it by 3 and add 1. Repeat this process until
n is 1.
Complete this recursive version of
hailstone that prints out the values of the
sequence and returns the number of steps.
def hailstone(n): """Print out the hailstone sequence starting at n, and return the number of elements in the sequence. >>> a = hailstone(10) 10 5 16 8 4 2 1 >>> a 7 >>> b = hailstone(1) 1 >>> b 1 """ print(n) if n % 2 == 0: return even(n) else: return odd(n) def even(n): return 1 + hailstone(n // 2) def odd(n): if n == 1: return 1 else: return 1 + hailstone(3 * n + 1)
Once your group has converged on a solution, it's time to practice your ability
to describe your own code. Pick a presenter, then send a message to the
discuss-queue channel with the @discuss tag, your discussion group number, and
the message "Hailing all course staff!" and a member of the course staff will
join your voice channel to hear your description.
You'll need your whole discussion group for the next question. If anybody isn't caught up with you, spend some time helping them out.
The Game of Sevens: Players in a circle count up from 1 in the clockwise direction. (The starting player says 1, the player to their left says 2, etc.) If a number is divisible by 7 or contains a 7 (or both), switch directions. Numbers must be said on the beat at 60 beats per minute. If someone says a number when it's not their turn or someone misses the beat on their turn, the game ends.
For example, 5 people would count to 20 like this:
Player 1 says 1 Player 2 says 2 Player 3 says 3 Player 4 says 4 Player 5 says 5 Player 1 says 6 # All the way around the circle Player 2 says 7 # Switch to counterclockwise Player 1 says 8 Player 5 says 9 # Back around the circle counterclockwise Player 4 says 10 Player 3 says 11 Player 2 says 12 Player 1 says 13 Player 5 says 14 # Switch back to clockwise Player 1 says 15 Player 2 says 16 Player 3 says 17 # Switch back to counterclockwise Player 2 says 18 Player 1 says 19 Player 5 says 20
Play a few games. Post the highest number your group reached without making a mistake or missing a beat in your group's channel's text chat.
sevens which takes a positive integer
n and a number of
k. It returns which of the
k players says
n. You may call
def sevens(n, k): """Return the (clockwise) position of who says n among k players. >>> sevens(2, 5) 2 >>> sevens(6, 5) 1 >>> sevens(7, 5) 2 >>> sevens(8, 5) 1 >>> sevens(9, 5) 5 >>> sevens(18, 5) 2 """ def f(i, who, direction): if i == n: return who if i % 7 == 0 or has_seven(i): direction = -direction who = who + direction if who > k: who = 1 if who < 1: who = k return f(i + 1, who, direction) return f(1, 1, 1) def has_seven(n): if n == 0: return False elif n % 10 == 7: return True else: return has_seven(n // 10)
Get help! There's a lot to keep track of in this question. If your group is
unsure of how to proceed, don't just start trying things in the interpreter to
see what works; ask for help in the
discuss-queue channel. We'll also review
this question in Friday's lecture, so if you don't finish, that's ok.
Document the Occasion
Please all fill out the attendance form (one submission per person per week).
This last recursion puzzle is just for fun. Work on it as a team if you have
time. If you'd like a hint, send a message to the
discuss-queue channel with
the @discuss tag, your discussion group number, and the message "Hint me!"
Q6: Karel the Robot
starts in the corner of an
n square for some unknown
n. Karel responds to only four functions:
move()moves Karel one square forward if there is no wall in front of Karel and errors if there is.
turn_left()turns Karel 90 degrees to the left.
front_is_blocked()returns whether there is a wall in front of Karel.
front_is_clear()returns whether there is no wall in front of Karel.
main() function that will leave Karel stopped halfway in the
middle of the bottom row. For example, if the square is 7 x 7 and Karel starts
in position (1, 1), the bottom left, then Karel should end in position (1, 4)
(three steps from either side on the bottom row). Karel can be facing in any
direction at the end. If the bottom row length is even, Karel can stop in either
n // 2) or (1,
n // 2 + 1).
Important You can only write
else statements and function
calls in the body of
main(). You may not write assignment statements, def
statements, lambda expressions, or while/for statements.