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Archive for **December 2010**

### Halfway There!

Posted December 10, 2010

on: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!!

### The Final Round

Posted December 9, 2010

on:- In: Capstone | Speakers
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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!!

### God’s Number?

Posted December 4, 2010

on: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!

### Topics Galore!

Posted December 4, 2010

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In Capstone Seminar this week, we were treated to presentations in which nine different students shared the plans for his or her Capstone project. It was quite interesting to see the wide range of topics represented. One student will be programming virtual manipulatives (hands-on, interactive math learning tools) for mobile devices, one will be studying stocks, another dimension, and yet another will be evaluating Standards Based Grading in mathematics for secondary students. A couple of the students will be pursuing statistical topics, one involving Major League Baseball players and another looking at case control studies. In addition, modular origami and the Monte Carlo method will be explored. All of these are worthy topics, but my personal favorite is the study of traffic jams. Who has not been driving along and come upon a slow-down or back-up with no apparent cause? How frustrating! One student will be studying these “jamitons” in the hopes of using mathematics to gain insight into, and possible solutions for, such phenomenon. For all our sakes, I wish him well in his endeavor!