Master of Arts in Educational Technology (OMET) Pepperdine University  
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3n+1 Problem

Learning Adventure #10

Assignment: Dear Distinguished Mathematicans:For your penultimate Learning Adventure, I've pulled a classic out of the archives. It uses MicroWorlds EX, but only as a laboratory for running an experiment using tools I have created for you. You don't have to write a line of code, unless of course you wish to customize the environment. All of the instructions are in the attached PDF file.

What did you learn from this experience?

I learned the difference between math and mathematics. I have a greater appreciation for mathematicians and their love of the subject. Being in the Mathematics, Science and Technology program at WestEd, I have to interact with math people on a frequent basis. When they are met with a math challenge, their eyes light up and a smile breaks across their face – now I know why. Playing around with numbers can be fun…hard fun.

What did you observe about the learning style(s) of your collaborators?

That we have some deep thinkers amongst us. Some dove into the problem, wresting it to the ground offering a wealth of mathematical ideas, while others dabbled with the learning adventure. Many focused intently on the numbers themselves while others, like myself, saw
things from a more right-brain viewpoint, like what patterns formed on the graphs.

Which subject(s) does this project address?

Mathematics (statistics, algebra, algorithms and discrete mathematics), social & behavioral science, computer technology (finding patterns and solutions that would be extremely time consuming to do by hand).

What might a student learn from this project?

How to be a creative thinker, trying to come up ways to solve an unsolvable problem.

For what age/grade is this project best suited?

Since this conjecture remains unproven, I am not sure this is meant for third graders as mathematical geniuses have yet to solve the problem. Algebra is not even taught until middle school or high school.

What would a student have to know before successfully engaging in this project?

A basic understanding algebra and how to operate a computer program like MicroWorlds that does the 3x+1 computations for you.


2) Look at: 54, 55 --Do you see a pattern? Can you test it?

What I noticed about the numbers 54 and 55 is that a side-by-side comparison indicates that the numbers are different up to the seventh generation and then they are the same all the way through to the 112 [final generation]. I did the problems "long hand" to see if I could tell anything that was making the numbers arrive at 94 together at the seventh generation. The number 94 is what both 54 and 55 have in common. This is the point where they have a conjuncture. If worked backwards 188/2=94 and 31x3+1=94. So for all numbers to infinity, there is a set pattern that they will all eventually follow back to 4…2…1 because it recursive pattern. This 3n problem is basically multiple patterns of recursive numbers. I did use MicroWorlds EX to test the numbers, running over 100 numbers from 3 through 62728127. I also discovered another site (in German) that ran the numbers faster and easier http://did.mat.uni-bayreuth.de/personen/wassermann/fun/3np1_e.html.