The goal of this assignment is to work on control structures and functions in Python.
You will be doing your work in a Jupyter notebook for this
assignment. You may choose to work on this assignment on a hosted
environment (e.g. tiger)
or on your own local installation of Jupyter and Python. You should use
Python 3.10 for your work, although 3.8 and 3.9 should work. To use
tiger, use the credentials you received. If you work remotely, make sure
to download the .ipynb file to turn in. If you choose to work locally,
Anaconda is the easiest
way to install and manage Python. If you work locally, you may launch
Jupyter Lab either from the Navigator application or via the
command-line as jupyter lab.
In this assignment, we will be working with sequences of numbers related to the Collatz Conjecture, sometimes called the \(3n+1\) problem. It states that for any positive integer, repeatedly applying the following piecewise function will eventually reach 1: \[\begin{equation} f(n) = \begin{cases} n/2 \text{ if $n$ is even} \\ 3n + 1 \text{ if $n$ is odd} \end{cases} \end{equation}\] Note that while this seems like a very simple problem, mathematicians have spent a lot of time working on it without a proof. Terry Tao, who proved one of the most recent results related to Collatz, has a nice presentation on the topic. Shizuo Kakutani has said that “A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”
In 2018, David Barina noted the \(7x\pm1\) problem seems to be a close relative of the Collatz Conjecture. It is also conjectured that for any positive integer, repeatedly applying the following piecewise function will eventually reach 1: \[\begin{equation} f(n) = \begin{cases} n/2 \text{ if $n$ is even} \\ 7n + 1 \text{ if } n \equiv 1 \mod 4 \\ 7n - 1 \text{ if } n \equiv 3 \mod 4 \end{cases} \end{equation}\] The orbit is the sequence of numbers generated from a starting number to 1, and the steps, the number of steps until reaching one, is also interesting to consider. In this assignment, we will write some functions to compute orbits and their stopping times.
The assignment is due at 11:59pm on Thursday, February 9.
You should submit the completed notebook file required for this
assignment on Blackboard. The
filename of the notebook should be a2.ipynb.
Please make sure to follow instructions to receive full credit. Use a markdown cell to Label each part of the assignment with the number of the section you are completing. You may put the code for each part into one or more cells.
The first cell of your notebook should be a markdown cell with a line for your name and a line for your Z-ID. If you wish to add other information (the assignment name, a description of the assignment), you may do so after these two lines. Because we are concentrating on control flow and functions, do not use lists, other collections, or comprehensions for this assignment.
Write a function orbit that for any positive integer
n, computes and prints the orbit of the \(7x\pm1\) function documented by Barina,
stopping at 1. For example, orbit(5) would produce the
result 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1. Print the
numbers with commas in between them, but no comma at the end. Test your
method by calling it on the inputs from 1 to 20. Sample output (scrolls
horizontally):
1
2, 1
3, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
4, 2, 1
5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
6, 3, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
7, 48, 24, 12, 6, 3, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
8, 4, 2, 1
9, 64, 32, 16, 8, 4, 2, 1
10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
11, 76, 38, 19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
12, 6, 3, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
14, 7, 48, 24, 12, 6, 3, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
16, 8, 4, 2, 1
17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
18, 9, 64, 32, 16, 8, 4, 2, 1
19, 132, 66, 33, 232, 116, 58, 29, 204, 102, 51, 356, 178, 89, 624, 312, 156, 78, 39, 272, 136, 68, 34, 17, 120, 60, 30, 15, 104, 52, 26, 13, 92, 46, 23, 160, 80, 40, 20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1
20, 10, 5, 36, 18, 9, 64, 32, 16, 8, 4, 2, 1Now, write a function steps that for any positive
integer n returns the number of steps in
the orbit produced by the \(7x\pm1\)
function. Recall that this is the number of steps required to get to 1.
For example, steps(5) would return 10. Do not use the print
method in the steps method. Test your method by calling it
on the inputs from 1 to 20. Write the results next to the input:
1 -> 0
2 -> 1
3 -> 13
4 -> 2
5 -> 10
6 -> 14
7 -> 18
8 -> 3
9 -> 7
10 -> 11
11 -> 53
12 -> 15
13 -> 19
14 -> 19
15 -> 23
16 -> 4
17 -> 27
18 -> 8
19 -> 50
20 -> 12Using the steps method, write a new method that, given
an input number k, returns two results:
the largest number which has the maximum stopping time of all numbers
between 1 and k and its stopping time. You must use the walrus
operator (:=) in a meaningful way in this method. For
example, calling this function on 100 returns 70, 326. Use
this function to compute and print the results of this longest
progression calculation for the inputs 10, 20, 30, 40, 50, 60, 70, 80,
90, 100.
range
method will be useful here; check all of its parameters.