Collatz Patterns

MATH 242 practice problems

Learning Goals: Use computation to explore mathematical patterns and formulate specific conclusions. Ask precise mathematical questions that can be investigated computationally.

Your task: Complete the problems below, which are reprinted from our text.

Due: Wednesday, October 8, at 5:00pm. Upload your solution as a Mathematica notebook to the Collatz Patterns assignment on Moodle.

Grading: If you answer each question below, you will earn a Satisfactory grade for each problem.

Need help? Review the notes from class (which are posed on the course web site), talk with your classmates, email Prof. Wright, or visit the help session.

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Problem 1: Exercise 3.3 in the text

Confirm that the Collatz trajectories for all starting values n from 1 to 10,000 eventually reach 1. This gives you computational evidence in support of the Collatz conjecture.

Problem 2: Practice 3.5 in the text

Suppose that a Collatz trajectory increases from n to Col(n). That, is suppose that n<Col(n). Explain why the trajectory must decrease next. That is, explain why Col(n) > Col(Col(n)).

Note that this problem requires logical reasoning, but not necessarily computation. You should write your answer in sentences. You may do some computations as examples that help you figure out the logical reasoning.

Problem 3: Exercise 3.14 in the text (modified)

In class we saw that the Collatz height plot appears to contain a line of points satisfying height(n)=3n+1. Implement a method to select the values of n that satisfy height(n)=3n+1, and answer the following five questions.
(a) What proportion of starting values n∈{1, 2, 3, ...,10000} satisfy height(n)=3n+1?
(b) What proportion of starting values n∈{1, 2, 3, ...,20000} satisfy height(n)=3n+1?
(c) What proportion of starting values n∈{1, 2, 3, ...,30000} satisfy height(n)=3n+1?
(d) What proportion of starting values n∈{1, 2, 3, ...,40000} satisfy height(n)=3n+1?
(e) Does the proportion of starting values satisfying  height(n)=3n+1 appear to be stable as n increases?

Problem 4: Exercise 3.26

Find 17 consecutive integers with the same stopping time. That is, find integers n,n+1,n+2,...,n+16 such that stop(n)=stop(n+1)=stop(n+2)=⋯=stop(n+16).