The 3N Problem
Executive Summary:
1. The problem:
“SCENARIO: You and your noted mathematician colleagues convene in (virtual) Geneva to present brilliant theories pertaining to one of the world’s great mysteries, the elusive 3n Problem.” (Stager 2005)
For a detailed PDF of the project, please follow this link: omaet3nproblem
2. Discussion of relevance/meaning:
The meaning of this Learning Adventure for me was to break away from the relative “safety” of solvable or achievable projects and work on something completely theoretical that is based on solid mathematical principles. In this way I was able to examine my own learning process at a deeper level and reflect on how I approach a challenge that may not have a clear ending.
In a way this project reminded me of some of what I read in the book The Long Haul, by Myles Horton. While his concepts are related to his never-ending pursuit of the goals of freedom and self-governance, much of what he speaks about relates to the process by which one goes about working toward any long-term goal and learning along the way. Here are a couple of quotes that stood out for me:
“I think it’s important to understand that the quality of the process you use to get to a place determines the ends…” (Horton 1998)
“Goals are unattainable in the sense that they always grow.” (Horton 1998)
“The nature of my visions are to keep on growing beyond my conception. That is why I say it’s never completed. I think there always needs to be struggle.” (Horton 1998)
I stress that these quotes are in regard to much more significant social goals than my working on a theoretically unsolvable mathematical problem, however their is something to be learned by reflecting on the process of going about solving this problem and working toward high-value, ever-growing and changing goals for societal change.
3. Possible conclusions/solution:
My conclusion for this Learning Adventure is that it helped me to recognize the value in establishing a process and keeping careful records and notes of the process in order to identify patterns and begin to develop a solid hypothesis that can be tested and refined. While I have no formal solution to this problem, as it would seem to be unsolvable, I do have a hypothesis that I formed in working on the problem. The basic idea of my hypothesis is that this equation illustrates the concept that all things eventually return to their most basic form, and that even something very complex or large (like a large number) is still based on the same basic building block.
Others have also reflected similar hypotheses and one member of the Cadre tied this problem to the reading we did at the beginning of the class on Wolfram. I agree that I see echoes of his work when I begin to analyze the results of my experiments with this problem.
4. Supporting evidence – may include links, graphics, references, supportive arguments:
The Collatz Conjecture: http://en.wikipedia.org/wiki/Collatz_conjecture
Prime number testing tool: http://www.math.com/students/calculators/source/prime-number.htm
Types of prime numbers: http://en.wikipedia.org/wiki/List_of_prime_numbers#Lists_of_primes_by_type
Overlay Graph of the numbers 1 through 1,000,000: Overlay_Graph (Note: Adobe Flash Player Required – may take a moment to load.)
Reflection on the Process:
The process of this Learning Adventure was challenging for me as I worked to move from my original, somewhat random approach to a more structured and methodical approach. This was not easy for me. Partly I believe this was due to working on the problem in many chunks of time, rather than focusing only on the problem for one or two lengthy periods of time. As I have shared in my posting and response to the debrief questions, I would begin to follow a pattern, then see another interesting result and begin pursuing that direction. While I enjoyed these “side trips of discovery” they did not usually lead to a more specific hypothesis or result.
The interesting thing about this Learning Adventure in comparison to the other Learning Adventures with MicroWorlds EX is that there was really no concrete result of the work. A hypothesis could be formed and to some extent tested, however coming to a formal conclusion with this project was difficult. Even as I was writing up my thoughts and reflections on this project I thought of another way that I would like to use the tools we were provided to analyze the data. I ended up creating an overlay graph of the numbers 1 through 1,000,000 by multiples of 10. The results of that experiment can be seen in the supporting evidence section above. It was fun to try this new approach as it seemed to add a new dimension to the problem and provide me a way to analyze multiple sets of data at one time, in a visual way – which I like. I could see using 3D imaging software to create detailed images that could be superimposed and digitally manipulated to see relationships with various integers and the resulting generational data.
I find it interesting how a problem like this can sort of stick with you and I can see returning to it from time to time, even just to show others the way that it works and get them thinking in possible a new way in the process.
Initial Research:
Here is my first experiment with the problem, done this morning (evening in the US) as I returned to the hotel from my work site here in the Philippines. I took my environment as inspiration for my experiment.
First number: 9 – The number of students wearing white shirts on their way to school at the last stop.
Results: 17 generations in a fairly fast process time. I find it interesting that such as small number produces what seems like a lot of generations and also took a little longer than I anticipated, although not very much time.
Second number: 2 – The number of cabs currently in sight.
Results: 2 generations and was fast, but almost the same time (from my perspective) as the first number.
Third number: 7 – The number of people on the sidecar motorcycle, called a tricycle here.
Results: 14 generations, not too long of a process.
Fourth number: 8 – The number of bananas on the bunch at the stand we just passed on the street.
Results: 1 generation, not surprising based on the formula, but interesting as it is so close to 9, but so far away in terms of generations.
Fifth Number: 17 – The number of people in the jeepney (like a small open-air bus) next to my car.
Results: 10 generations, fairly fast results.
Sixth number – 30 minutes – time from my work site to the hotel.
Results: 16 generations, not too remarkable, other than a small number like 9 takes a generation more to reduce.
Seventh number – 7773.8 – The number of miles from Tucson to Cebu.
Results: 272 generations, took a bit more time and took time for the tool to count the generations, I would almost call it long, but still relatively fast. I am very intrigued that even decimals resolve to the 4 2 1 cycle.
Eighth number: 7774 – The number of miles from Tucson to Cebu rounded to the nearest whole number.
Results: 50 generations – but still pretty fast.
Ninth number: 12510.37 – The number of kilometers from Tucson to Cebu.
Results: 189 generations – pretty fast, but takes a while to count generations. As with the initial decimal entry the process included very large numbers with extended decimal places throughout the process to reduce to 4 2 1, such as 285024.991563.
Tenth number: 12510 – The number of kilometers from Tucson to Cebu rounded to the nearest whole number.
Results: 110 generations, only slightly faster than the number with decimals.
Final initial experiment number – 2,000,000,010 – For the heck of it since a couple billion numbers have been tested.
Results: The program does not like this number and returned the following output – “remainder does not like 2,000,000,010 as input in even?”
My initial hypothesis is that this equation is representative of the concept that history repeats itself. Even when starting from a unique position (number in this case) if the same process is applied (3N) the results will always be the same, it is just a matter of time (generations).
Learning Adventure #10: Debriefing Questions
What did you learn from this experience?
This experience shined a light on the way that I approach a new problem. I began with some loose experimentation to get my bearings and understand the lay of the land, so to speak. I see this approach quite often when I am faced with a challenging problem to solve. Once I began to feel comfortable with the structure of the problem and how to use the available tools to analyze the problem I started to become more methodical.
This move from random experimentation to a more focused approach, involving the forming and testing of small theories, was also influenced by interactions with my colleagues, both through the artifacts others shared and through synchronous discussion of the process. Although I ultimately began to work in a more methodical way, I still saw myself using a bit of a tinkerer or painter approach. As I became fascinated by the way a particular pattern behaved I would follow down that path for a while, only to discover another pattern or anomaly and follow that new, promising direction.
In addition to my meta learning about how I approach this type of problem I also learned that there are ‘types’ of prime numbers, something I do not recall learning in the past and by which I am very intrigued.
What did you observe about the learning style(s) of your collaborators?
The learning styles seemed to vary between a highly-structured, methodical approaches to a similar approach that I took to begin with by just sort of messing around with the problem to find a place to begin focusing my efforts. While there was learning gained by the random approach, including finding a starting pattern on which to focus, eventually all collaborators moved to a more methodical approach in order to work toward a hypothesis.
Which subject(s) does this project address?
By being creative, I believe that almost any subject would benefit from this project. It is a great way to get people thinking critically and analyzing… seeking to form a hypothesis and defend that position. It would be ideal in everything from mathematics and science courses, to debate and art.
I could see this being used as a way to get people thinking deeply about a problem before asking them to move on to working on other problems that seem unsolvable and need to be worked on with the same desire to find reason and seek out the patterns. Perhaps this problem could be used with groups that are in conflict with each other to begin to work toward common ground and ultimately a resolution. It could help them see that there often is no easy, one-time solution, that a solution must be continually worked toward with passion and perseverance, like what Myles Horton shares in his book The Long Haul.
What might a student learn from this project?
As mentioned above, students could learn about critical thinking, analysis, and constructing meaning from the process. They could also learn many mathematical concepts and apply them in any number of related projects, such as creating artwork inspired by the graphical representations of the equation.
For what age/grade is this project best suited?
While it has been shared that this has been successfully used with third graders (that is great Gary!), I believe that it would be best suited for slightly older children on up through adults. It is not that I do not think that third graders could gain valuable insights and experience from the project. I just believe that there may need to be some additional framing and support given for students in say third and fourth grade.
What would a student have to know before successfully engaging in this project?
Very basic mathematics such as addition and multiplication and how to manipulate a computer – keyboard/mouse. The use of MicroWorlds EX could quickly be learned for this project, if the student had a very basic understanding of how to use a computer.