Collatz Creation¶

Project Goals¶

This engineering effort invites you to investigate how to apply all that you have learned about Python programming and discrete structures to implement a program, called collatzcreation, that can solve the Longest Collatz Sequence problem posed on Project Euler. The Collatz sequence is defined for the positive integers according to the rule that \(n\) becomes \(\frac{n}{2}\) when \(n\) is even and \(3n + 1\) when \(n\) is odd. To date, computer scientists and mathematicians do not know whether or not the Collatz sequence will terminate with the value of \(1\) when it is started with an arbitrary positive integer \(n\). However, for all of the values tried to date the sequence always yields a Collatz chain (i.e., the sequence values that arise from iteratively applying the rules) of a finite length. The Longest Collatz Sequence problem posed by Project Euler asks "Which starting number, under one million, produces the longest chain"? The collatzcreation program that you implement for this project should efficiently produce an answer to this question.

Project Access¶

If you are a student enrolled in a Computer Science class at Allegheny College, you can access this assignment by clicking the link provided to you in Discord. Once you click this link it will create a GitHub repository that you can clone to your computer by following the general-purpose instructions in the description of the technical skills. Specifically, you will need to use the git clone command to download the project from GitHub to your computer. Now you are ready to add source code and documentation to the project!

Expected Output¶

As part of this assignment, you are going to implement a collatzcreator program that takes as input a complete document stored in a text file and then performs an automated analysis of the document's contents. If you run the collatzcreator program with the command poetry run collatzcreator --minimum 1 --maximum 10 --display it will try the numbers 1 through 10 as the input number to the Collatz Sequence and then calculate the length of the Collatz chain before the sequence produces the value of 1. The collatzcreator program will also compute some summary statistics about the length of the Collatz chains that it constructed when using the inputs that start at the minimum and go up to the maximum. When the collatzcreator accepts the input flag of --display it will also produce a graph that will visualize the relationship between the value of the numerical input and the length of the Collatz chain.

🕵  Let's investigate the behavior of the Collatz sequence!

  The first input to try will be 1
  The last input to try will be 10

The inputs to the Collatz function:

[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

✨ What is the length of the Collatz chain before the function produces
the value of 1?

📏 The length of the resulting Collatz chain:

[1, 2, 8, 3, 6, 9, 17, 4, 20, 7]

✨ What is the summary information about the length of the Collatz chain?

  The minimum length of a Collatz chain is: 1
  The maximum length of a Collatz chain is: 20
  The mean of the length of a Collatz chain is: 7.70
  The median of the length of a Collatz chain is: 6.50
  The standard deviation of the length of a Collatz chain is: 5.97

🤷 Can you find a pattern between the input number and the length of the
Collatz chain?

📦 Check the file called 'graphs/collatz.pdf' to see a graph that
visualizes the results!

Adding Functionality¶

If you study the file collatzcreator/collatzcreator/main.py you will see that it contains a single TODO that reminds you to call the compute_collatz_chain function that takes as input a specific number and returns an Iterator[int] as its output. This means that the compute_collatz_chain function should use yield to incrementally produce the int values in the Collatz sequence. When you look at the collatzcreator/collatzcreator/collatz.py file you will notice that the TODO marker instructs you to provide a complete implementation of the aforementioned compute_collatz_chain function. Finally, a review of the collatzcreator/collatzcreator/summarize.py will show that you also need to implement the following functions that characterize the computed Collatz sequences:

• def compute_mean(numbers: List[int]) -> float:
• def compute_median(numbers: List[int]) -> float:
• def compute_difference(numbers: List[int]) -> List[float]:
• def compute_variance(numbers: List[int]) -> float:
• def compute_standard_deviation(numbers: List[int]) -> float:

The following source code segment provides a complete implementation of the compute_collatz_chain function. Line 3 of this function firsts yields the number since the first numerical value in the Collatz chain is always the initially provided number. Next, lines 4 through 9 iteratively compute the values in the Collatz sequence, continuing until the number takes on the value of 1. When number is even, lines 5 and 6 use the // operator to assign to number to the integer value of number / 2. When number is odd, line 8 assigns to number the value of 3 * number + 1. Ultimately, the compute_collatz_chain function follows the sequence's definition by which \(n\) becomes \(\frac{n}{2}\) when \(n\) is even and \(3n + 1\) when \(n\) is odd, only terminating when the value of \(n\) is \(1\).

1
2
3
4
5
6
7
8
9

def compute_collatz_chain(number: int) -> Iterator[int]:
    """Compute the numbers in the Collatz sequence for the starting number."""
    yield number
    while number != 1:
        if number % 2 == 0:
            number = number // 2
        else:
            number = 3 * number + 1
        yield number

Finally, don't forget that the Longest Collatz Sequence problem posed on Project Euler is "Which starting number, under one million, produces the longest chain"? This means that you will need to run the program with the following command-line arguments: poetry run collatzcreator --minimum 1 --maximum 1000000 --display. It is important to note that it is possible that running collatzcreator on your laptop with these command-line arguments may require a significant amount of computation time. This means that you will either have to wait a long time for collatzcreator to finish or implement a more efficient version of the compute_collatz_chain function! Which approach did you pick? Why?

Running Checks¶

As you continue to add and confirm the correctness of collatzcreator's functionality, you should study the source code in the pyproject.toml file. This file contains the specification of several tasks that will help you to easily run checks on your Python source code. Now, you can run commands like poetry run task lint to automatically run all of the linters designed to check the Python source code in your program and its test suite. You can also use the command poetry run task black to confirm that your source code adheres to the industry-standard format defined by the black tool. If it does not adhere to the standard then you can run the command poetry run fixformat and it will automatically reformat the source code. By following a tutorial, you can configure VS Code to use the black tool to automatically reformat the source code when you save a file.

Along with running tasks like poetry run task lint, you can leverage the relevant instructions in the technical skills to run the command gatorgrade --config config/gatorgrade.yml to check your work. If your work meets the baseline requirements and adheres to the best practices that proactive programmers adopt you will see that all the checks pass when you run gatorgrade. You can study the config/gatorgrade.yml file in your repository to learn how the GatorGrade program runs GatorGrader to automatically check your program and technical writing. You can also run the command poetry run task test to run the Pytest test suites provided in the files test_collatz.py and test_summarize.py. Did all of your tests pass?

Project Reflection¶

Once you have finished both of the previous technical tasks, you can use a text editor to answer all of the questions in the writing/reflection.md file. For instance, you should provide the output of the Python program in a fenced code block, explain the meaning of the Python source code segments that you implemented, and answer all of the other questions about your experiences in completing this project. One of the main goals of the reflection is for you to explain the trends that you see in relationship between the input number and the length of the Collatz chain. You should also discuss how the collatzcreator program uses discrete structures like the list or the tuple to store the Collatz chain and then visualize its characteristics with a scatterplot.

Project Assessment¶

Since this project is an engineering effort, it is aligned with the evaluating and creating levels of Bloom's taxonomy. You can learn more about how a proactive programming expert will assess your work by examining the assessment strategy. From the start to the end of this project you may make an unlimited number of reattempts at submitting source code and technical writing that meet every aspect of the project's specification.

Seeking Assistance¶

Emerging proactive programmers who have questions about this project are invited to ask them in either the GitHub discussions forum or the Proactive Programmers Discord server. Before you ask your question, please read the advice concerning how to best participate in the Proactive Programmers community. If you find a mistake in this project, please describe it and propose a solution by creating an issue in the GitHub Issue Tracker.