Collatz Extensions

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: Friday, October 10, at 5:00pm. Upload your solution as a Mathematica notebook to the Collatz Extensions 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 3.31 in the text

Let α(n) be the number of odd integers in the Collatz trajectory of n, not including 1. For example, α(3)=2 and α(7)=5. Rafael Ruggiero conjectured that the stopping time satisfies stop(n)=⌈(Subscript[log, 2])(n·(6^(α(n))))⌉. Investigate Ruggiero’s conjecture. Either produce computational evidence in support of the conjecture, or find a counterexample. Note that ⌈x⌉ denotes the least integer greater than or equal to x (also called the “ceiling” of x).

Problem 2: Collatz variant cycles

In class, we defined a variant of the Collatz function, ColVar, to be ColVar(n)=3n-1 if n is odd, and ColVar(n)=(n/2) if n is even. We then observed three distinct cycles that occur when iterating the ColVar function on positive integers. Consider the ColVar trajectories starting with integers n∈{1,2,3,...,1000}. How many times does each of the three cycles occur in these 1000 trajectories?

Hint: How many of these trajectories contain a 1? How many contain the number 5? How many contain 17?

Problem 3: Collatz variant heights

Define the height of a ColVar trajectory to be the greatest integer that occurs in the trajectory. What is the most common height in the first 1000 ColVar trajectories?