Collatz Heights

Math 242 practice problems

Name: _________________

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 three problems printed below. The first two ask you to investigate properties of Collatz sequences; for this you may adapt code from class. The third problem asks you to write your own questions about Collatz sequences.

Due: Wednesday, March 16, at 5:00pm. Upload your solution as a Mathematica notebook to the Collatz Heights assignment (Moodle link).

Grading: For problems 1 and 2, a Satisfactory grade is earned by answering each question, with your answers supported by computational evidence. A Satisfactory grade for problem 3 is earned by stating two questions that exhibit the four stated characteristics.

Need help? Review the notes from class, which are posed on the course web site. If you still need help, email Prof. Wright.

1. In class, we observed that height(n)= 9232 occurs for 1579 different values of n. What is the smallest height that occurs for more than 1579 different starting values?

2. Some starting values of n are such that height(n)=3n+1. What proportion of values n<=1000 satisfy height(n)=3n+1? Is this proportion stable as N increases?

3. In class, we said that a good mathematical question satisfies four characteristics:

The question is interesting to you.

You don’t already know the answer to the question.

You haven’t already seen the question before, or at least not exactly.

You can begin to make computational investigations to shed light on the question.

(These characteristics are adapted from Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis.)

Write at least two good mathematical questions about Collatz sequences.