Discussion 3: Recursion
Pick someone in your group to join Discord. It's fine if multiple people join, but one is enough.
Now switch to Pensieve:
- Everyone: Go to discuss.pensieve.co and log in with your @berkeley.edu email, then enter your group number. (Your group number is the number of your Discord channel.)
Once you're on Pensieve, you don't need to return to this page; Pensieve has all the same content (but more features). If for some reason Penseive doesn't work, return to this page and continue with the discussion.
Post in the #help channel on Discord if you have trouble.
Getting Started
Say your name and share a food that you really liked as a child. (It's ok if you still like that food now.)
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, this weekend 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 Midterm 1 material.
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.
Recursion
Ok, time to discuss problems! Remember to work together. Everyone in the group should understand a solution before the group moves on. Many students find this discussion challenging. Everything gets easier with practice.
VERY IMPORTANT: In this discussion, don't check your answers until your whole group is sure that the answer is right. Figure things out and check your work by thinking about what your code will do. Your goal should be to have all checks pass the first time you run them! If you need help, ask.
Q1: Swipe
Implement swipe, which prints the digits of argument n, one per line, first backward then forward. The left-most digit is printed only once. Do not use while or for or str. (Use recursion, of course!)
Run in 61A Code
First print the first line of the output, then make a recursive call, then print the last line of the output.
Q2: Skip Factorial
Define the base case for the skip_factorial function, which returns the product of every other positive integer, starting with n.
Run in 61A Code
If n is even, then the base case will be 2. If n is odd, then the base case will be 1. Try to write a condition that handles both possibilities.
Q3: Is Prime
Implement is_prime that takes an integer n greater than 1. It returns True if n is a prime number and False otherwise. Try following the approach below, but implement it recursively without using a while (or for) statement.
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.
Run in 61A Code
Define an inner function that checks whether some integer between i and n evenly divides n. Then you can call it starting with i=2:
def is_prime(n): def f(i): if n % i == 0: return ____ elif ____: return ____ else: return f(____) return f(2)
Come up with a one sentence 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 channel's text chat.
Q4: Recursive Hailstone
Recall the hailstone function from Homework 1. First, pick a positive integer n as the start. If n is even, divide it by 2. If 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.
Run in 61A Code
An even number is never a base case, so even always makes a recursive call to hailstone and returns one more than the length of the rest of the hailstone sequence.
An odd number might be 1 (the base case) or greater than one (the recursive case). Only the recursive case should call hailstone.
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.
Document the Occasion
Please all fill out the attendance form (one submission per person per week).
You'll need your whole discussion group for this question. At least try it out. You might have fun. We'll review the question in lecture on Friday.
Q5: Sevens
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 1Player 2 says 2Player 3 says 3Player 4 says 4Player 5 says 5Player 1 says 6 # All the way around the circlePlayer 2 says 7 # Switch to counterclockwisePlayer 1 says 8Player 5 says 9 # Back around the circle counterclockwisePlayer 4 says 10Player 3 says 11Player 2 says 12Player 1 says 13Player 5 says 14 # Switch back to clockwisePlayer 1 says 15Player 2 says 16Player 3 says 17 # Switch back to counterclockwisePlayer 2 says 18Player 1 says 19Player 5 says 20
Play a few games. Post the highest score your group reached on Discord.
Then, implement sevens which takes a positive integer n and a number of players k. It returns which of the k players says n. You may call has_seven.
An effective approach to this problem is to simulate the game, stopping on turn n. The implementation must keep track of the final number n, the current number i, the player who will say i, and the current direction that determines the next player (either increasing or decreasing). It works well to use integers to represent all of these, with direction switching between 1 (increase) and -1 (decreasing).
Run in 61A Code
First check if i is a multiple of 7 or contains a 7, and if so, switch directions. Then, add the direction to who and ensure that who has not become smaller than 1 or greater than k.
Q6: Karel the Robot
Karel the robot starts in the corner of an n by n square for some unknown number 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.
Implement a 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 position (1, n // 2) or (1, n // 2 + 1).
Important You can only write if or if/else statements and function calls in the body of main(). You may not write assignment statements, def statements, lambda expressions, or while/for statements.
For every two steps forward, take one step back to end up in the middle.