Pressing Toward the Prize

I am afraid this update isn’t much of an update, in that it really doesn’t carry much by way of new revelation. I am continuing to ready Chaotic Elections, but it is going more slowly than I had hoped. My biggest challenge is carving out time to read, and then when I do, it is not the kind of material I can just breeze through and “get it.” The subject is very interesting, and I am learning a lot and enjoying it immensely. I am concerned, however, that I will have trouble keeping up with the Capstone timeline… the outline is due soon. Pesky things keep getting in the way – things like other classes, if you can imagine that!!

My goal is still the same, to continue reading the book and refining my outline.

On Wednesday I met with my faculty liaison, Dr. Edgar, about my Capstone project. He encouraged me to continue reading Chaotic Elections with an eye for linear algebra. Donald Saari uses vectors, specifically what he calls voting vectors, to analyze the various voting methods, do manipulations of outcomes, and support his argument that the Borda Count is the superior voting method whenever three or more candidates or issues are on the ballot. My goal for this week, then, is to continue reading and refining my understanding.

Although I did not get to read Chaotic Elections as much as I had hoped (I did read a little), I have begun to formulate ideas for my Capstone outline. These are my first thoughts:

1. Define voting paradox and explain election conditions that may be problematic, i.e., there are three or more candidates or issues on the ballot, and the winning candidate or issue does not receive a majority of the votes.

2. Define various voting methods: plurality, Borda Count, Condorcet.

3. Offer brief historical background: Borda, de Condorcet, Arrow, Saari.

4. Give simple example to demonstrate the three voting methods and how different methods can produce different results when applied to the same election.

5. Explain some of the voting biases inherent to each of the methods, with examples, and the kinds of fallacies that can result.

6. Discuss how Donald Saari used mathematics to identify specific kinds of voting paradoxes, to analyze each for the causes of the various voting biases, and to determine the most reasonable voting method, i.e., the one most immune from voting method biases.

7. Offer possible reasons why, in light of existing evidence, wholesale changes have not yet been made to elections to insure that outcomes accurately reflect the will of the people

I will need much more research before I am ready to finalize my outline, and continuing research will be my plan for the week. I will refine the outline as I go, using this rough draft as a  starting place.

It’s hard to believe that I just finished my first week of spring semester at Pacific Lutheran University. When fall semester ended in December, I had such high hopes for working on my Capstone project during the break, but life took an unexpected turn. My house sold after being on the market for only a week, so I spent Christmas break moving, spending time with my dad, and getting settled in my new home as best I could. Then J-term started, and that was a wild four-week ride through Tudor England, and a noble attempt at learning ballroom dance!! To top it off, I had a family crisis during the first week of J-term, and it is continuing to impact my life. The week between J-term and spring semester was filled with doctor appointments, the application process for grad school, the GRE, and more unpacking and organizing. I only had time to read a little in Donald Saari’s book, Chaotic Elections, and that was a far cry from what I had hoped to accomplish.

So here I am, back at it. This is my final semester at PLU, at least that’s the plan, and my Capstone project is due in May. Although I find myself a bit under the gun and nowhere near where I had hoped to be at this point, I will continue moving forward. My immediate goal is to refresh my LaTex skills by completing the assignment we were given this week. I hope to also continue reading Chaotic Elections and begin formulating my project outline. I’ll check in next week and report how it goes… wish me luck!!

Here we are at the end of the fall semester, and that means we are halfway through the Capstone process. This semester has been quite a learning experience for me. I remember being terrified at the prospect of doing a blog for all the world to see, because I have never done anything like that before. I’m not a texter, and I don’t follow Twitter or FaceBook, so this is a whole new realm for me. Why would anyone care what I have to say? I still don’t know the answer to that, but the good news is that I survived, and I found it wasn’t as bad as I thought it would be.

This semester I also learned how to use the mathematical typesetting program, LaTex, as well as the Beamer package that creates presentation slides. Our professor gave us awesome templates to use, so that was very helpful. We also practiced reading and “understanding” mathematics articles, and it was no surprise that I’m not very good at it. It takes a while to digest the information, if one can even follow what is being said in the first place. Those are the times when a translator would come in handy!! I have enjoyed being exposed to some of the math that is being studied by way of faculty, students, and literature, and if I had time, there are many topics I would love to pursue. But for now, I will be focusing my attention on my own Capstone project, The Complexities of Voting, and the linear algebra I will need in order to understand the articles and books on the subject!!

We had the final round of Capstone Proposal presentations today, and once again, a wide array of interests were represented. I am always humbled by others’ presentations, be it student or faculty, because they are a graphic reminder to me of how little I actually know! I listen and watch and try to understand… but mostly, I have no idea what they are talking about!! Some of the topics that totally lost me were Panel Data Modeling, the Riemann Hypothesis, and Applications of Fuzzy Set Theory and Finite State Automata (say that three times fast!!). I did a little better job of following the presentations about rankings and the Bowl Championship Series, the push for Proofs to be taught in high school, Digital Image Interpolation, and Error Detection and Correction in Data Transmissions. I was pretty excited that I understood some of the presentation on Markov Chains, since I can still recall a bit about stochastic processes from Linear Algebra. One of the presentations was about the Mozart Dice Game, which was apparently quite popular in Europe in the 18th Century. There is a 12 x 16 table, composition rules, and one uses the roll of the dice to randomly create a minuet. I don’t understand exactly how it works, but we were able to listen to a minuet that the student had created, and it was beautiful. She is working on a project that will connect art and math, and it should be quite interesting. I was also one of the presenters today, and all I can say about that is… I am glad it’s over!!

I recently came upon an article in Math Horizons magazine (Nov. 2010) that caught my eye: The Quest for God’s Number. Now I never knew God had a number, so I thought I better stop and check it out. The first thing I discovered is that God’s number is linked to God’s algorithm… who knew?

What this really has to do with is solving a Rubik’s Cube in the fewest moves, and hence the shortest amount of time. According to the article by Rik van Grol, “God’s algorithm is the procedure to bring back Rubik’s Cube from any random position to its solved state in the minimum number of steps. The maximum of all minimally needed number of steps is referred to as God’s number. This number can be defined in several ways. The most common is in terms of the number of face turns required, but it can also be measured as the number of quarter turns… Earlier this year, after decades of gradual progress, it was determined that God’s number is 20 face turns. Thus, if God’s algorithm were used to solve the cube, no starting position would ever require more than 20 face turns.” The article chronicles the process used to determine that God’s number is 20, and as one might imagine, it involves looking at an astronomically large number of possible pattern combinations, more than 4.3 x 10^19.

It was interesting to find that one of the mathematicians working on lowering the upper bound in search of God’s number used group theory in his calculations. He used the cube group, similar to the shape groups we studied in Abstract Algebra. His group had 6 moves on a cube fixed in space, Left, Right, Front, Back, Up, and Down, and the group operation was concatenation. He then divided the cube group into a nested chain of subgroups, and created an algorithm to eliminate some of the moves early on in the process. Using this method, he was able to show that the Rubik’s Cube could be solved in at most 85 moves. This was in 1979. By January of 1980 he had reduced it to 80, and by the end of that year, to 52. In 1995, another mathematician brought the upper bound to 29 face turns, but it was not until July of this year that the upper bound met the lower bound at 20.

Now that they have found God’s number for the classic 3x3x3 cube, they have turned their attention to the 4x4x4, 5x5x5, 6x6x6, and 7x7x7 cubes, all of which are currently on the market. There might be a hitch in this, though, since the 7x7x7 cube has 2.0 x 10^160 possible positions. Good luck with that!

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  • gramsonjanessa: I can't wait to listen to your capstone presentation in the spring! Your proposal was really interesting and I'm interested to see how the linear alge
  • dewittda: This is impressive! I thought I was good because I solved a rubik’s cube once in an hour. I served with a guy in the Air Force who could solve a r
  • ZeroSum Ruler: The Euclidean algorithm should me the mainstream way we teach students how to find the GCF. Why isn't it? A mystery.