Doing Math: Wrestling with the Super Magic Square

I started my magic square by putting in my birthday:

8

27

19

92

I then tried entering values into the next row by just adding or subtracting a few numbers from the initial date. It seemed to work, row-wise, so I tried the next few rows. I realized this strategy works for getting the proper sum in each row, but it will leave me with too many low numbers on the left column, and too many high numbers on the right. So I tried accounting for that by starting with just my initial row, and working on one quadrant at a time (top/bottom left 4 cells, top/bottom right 4 cells). I went from the top left to the bottom right, adjusting previous quadrants to make the columns and rows turn out as necessary. This strategy took a while, but it proved victorious: I ended up with a square that had all of the rows, columns, and top and bottom quadrants each adding up to 146. However, in order for it to be a Ramunujan square--a “super” magic square—I had to figure out how to get the center 4 numbers, the diagonals, and the side 4 squares to add up to 146 as well.

I tried messing around with the numbers for a while, without much concrete success. As I was manipulating the numbers, virtually strategy-less, I realized that since the top row has to stay as it is, I already know how some of the other combinations of rows/columns will have to add up. For example, the two center cells of the second row must add up to 100, since the top center 4 cells must add to 146, and 27+19=46. I extended this to other cell values, and found that some various “required” values for combinations of cells are:

Using this new information, I tried making the middle-side 4 cells equal to 146 as well. After a a while spent fiddling, this was successful:

Feeling victorious, I decided to try and tackle the center 4 next. This was much harder than I expected, and in order to obtain the center 4, I sacrificed each side’s center 4.

After trying to wrestle with my magic square for another while longer, I decided that my magic square and I had come to an impasse. I decided that I would stop here, at a truce with my square: I let it retain the magic that it didn’t want to expose to me, and it gave slight concessions by almost conforming.

            Patterns and puzzles are what first drew me to math, so making a “super” magic square was very appealing to me. I enjoy knowing that these mind games are indeed math—while making this magic square I used basic algebraic skills, but I also found myself making linear combinations and trying to intuitively distribute the numbers amongst the cells. Even with the technology that makes adding instantaneous, it still took me a long time to get to the partial “super” magic squares that I did. Struggling to make my a magic square that has all of Ramanugan’s square’s properties really put Ramanujan’s genius in perspective. His genius wasn’t just that he was able to try who-knows-how-many different squares before he found one that worked for his birthday without modern-day technology, but also that he saw the value and interest in making a magic square with these extra properties.

"Nature of Math"-- Mathematics is a language the world speaks that humans can understand

           Throughout the semester we have studied numerous mathematicians in MTH 495. We have learned about mathematicians from ancient Greece to China, and have been led through the centuries by great mathematical discoveries. As we learned about these great thinkers, it struck me how much “math” encompasses, and the contrast of this vastness to the small scope the term evokes in the minds of most people today.

            By nature, humans want to understand the world around them. Mathematics is a language the world speaks that humans can understand. Math is the tool that humans use to investigate the nature of the world, though it is rarely taught as so in today’s Western society.

            To many people today, math is seen as numbers, a calculator, and equations. It is seen as a requirement, a monotonous lecture, and as an overrated skill. Math has been parsed off from what it initially was, and, in a sense, what we consider “math” today is more confined that it ever has been.

            Archimedes was a mathematician. He was also a physicist, an engineer, an inventor, and an astronomer. Omar Khayyam was a mathematician, a philosopher, an astronomer, and a poet. Brahmagupta is considered a mathematician and an astronomer, but his texts were composed in elliptic verse, which shows a poetic intellect in the man as well. Rene Descartes was a mathematician, a great philosopher, and a writer. These great men are considered mathematicians, and in that overarching term lay discoveries in optics (Decartes), weaponry (Archimedes), a calendar that is more accurate than the one used today (Omar Khayyam), hydrostatics (Archimedes), and so much more.

            In their day, what were the labels put on these men? Were they identified as contributing to each of the disciplines mentioned above, were all of their discoveries considered mathematical discoveries, or were there few labels used at all? Why do we limit math as it is today, so early on? In grade school and even high school, math is taught separate from astronomy. Separate from the other sciences as well, in most cases. Math might be brought up in a physics class, but rarely is it explained why the formulas work the way they do, why we can put numbers and equations to a natural phenomenon. Perhaps more tragically, in high school I can’t remember more than a handful of times in a math class where any other discipline was brought up to help me understand and appreciate why I was learning what was taught. Students lose interest in math without even gaining the slightest understanding of its importance in the world.

             All of the men I mentioned were mathematicians, but they were also big thinkers who wanted to understand the world. They did this through math, but also though philosophy, physics, and astronomy. These disciplines are ones that frequently come to people’s minds when the question of “What discipline seeks to understand the world, and human’s place in it?” It is a tragedy that math, the foundation of physics and astronomy, and a sister to philosophy, would rarely be unveiled as an answer to many today.

“Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.”-Carl Friedrich Gauss

"Doing Math": Biz's Bracketology

I will preface this post by saying that the last time I watched a basketball game for more than ten minutes was when I was 11, with my grandma. I have never been interested in basketball (my family is of short, Polish stock), and have only the faintest idea of what March Madness is. But, I have a competitive nature, and I love applying my math and stats skills and my intuition. So, this year I decided to jump on the bracket bandwagon.

            To determine my bracket I used various sources:

·         FiveThirtyEight’s NCAA Tournament Predictions (accessible at http://fivethirtyeight.com/interactives/march-madness-predictions/)

I relied heavily on this tool, more so than seed, because of the breadth that it covers (from the pre-season rankings, various professionals’ estimates, the power rankings, player injuries, and geography, among many other factors).

·         The Washington Post’s list that contains the number of upsets per round each year (accessible at http://www.washingtonpost.com/wp-srv/special/sports/ncaa-march-madness-bracket-guide/)

This helped me determine how many upsets I should include in my bracket. Some upsets were already statistically determined by FiveThirtyEight’s tool, but others I decided for myself.

·         The tournament results from the past five years

·         Seed

·         My own (slight) biases

Mostly based on living in Michigan, my love for Oregon, and the graduate schools I applied to.

From the past five years I noticed that, usually, around one team from the previous year’s Final Four makes it to the current year’s Final Four. In three out of the last five instances, that team won. In the following table, the underlined teams were in the Final Four the year before:

2009	2010	2011	2012	2013
1 North Carolina (won)	1 Duke (won)	3 Connecticut (won)	1 Kentucky (won)	1 Louisville (won)
2 Michigan State	5 Michigan State	11 Virginia Commonwealth	4 Louisville	4 Michigan
1 Connecticut	5 Butler	8 Butler	2 Ohio State	4 Syracuse
3 Villanova	2 West Virginia	4 Kentucky	2 Kansas	9 Wichita State

This year Louisville, Michigan, and Wichita State are all in the same region, so only one of them can go to the Final Four. Louisville has, from what I’ve read, been mis-seeded because of their easy schedule during the year, and according to FiveThirtyEight they have a 38% chance at getting to the Final Four (against Wichita State and Michigan’s 14%), so I decided to include them. I have Ohio State beating Syracuse in the second round as one of my second round upsets, thus they can’t make it to the Final Four.

            Additionally, as you can see from the table, each time a team has shown up at the Final Four in two consecutive years, it does as well or better than the second year than the first. Thus, although Michigan State and Louisville are seeded the same, and Michigan State does have Tom Izzo, I decided to have Louisville going to the Semi-finals (also, on FiveThirtyEight it shows that Louisville has a much better chance at going to the Semi-finals than State).

            That is as far as I took the insights that I gained from the last fine year’s Final Four history—instead, I took into account Florida’s place as a #1 seed as well as the fact that no team has won the title two years in a row since the 1980s, and gave Louisville a respectable home in second place.

            I used the number of upsets per round over the past few years to guide the number of upsets I would have per round.

Round	1	2	3	4	5
Number of upsets	7	2	3	2	1

            Since 2009, the number of upsets in round 1 has been 10 four times and 7 once. When FiveThirtyEight projected that a lower seed would beat a higher seed, I would put that in as one of my upsets. My biases accounted for my putting NC State (12) beating Saint Louis (5), even though FiveThirtyEight projects otherwise (I’ll be attending NC State in the fall to obtain a masters in Data Analytics), Harvard ahead of Cincinnati (Harvard has done well the past few years, I looked into applying at Harvard’s statistics program, and FiveThirtyEight has Harvard winning a 42% chance and Cincinnati a 58%--relatively close), George Washington beating Memphis (FTE estimates 55/45 Memphis—but I applied to GW’s Data Analytics program and was accepted), and Stanford over New Mexico (Stanford’s stats program is ranked #1 in the US).

            The number of upsets in round 2 has been between 1 and 6 over the past five years, though mostly the number of upsets is in the higher range. I chose Ohio State over Syracuse almost arbitrarily, but more because I’ve heard of Ohio State and I haven’t heard much about Syracuse. Also, I needed an upset, and FTE estimated that Ohio State has a 40% chance of winning that round verses Syracuse’s 50%--not too much of a difference. So, Ohio State is my upset for this round. Oregon is my other upset, mostly because my family bought a house there this year. I visited Oregon over winter break with my family, and fell in love. I don’t have high hopes for them, but I thought that maybe I could send enough positive vibes their way for them to beat number 2-seeded Wisconsin.

            In round 3 I have a few more upsets than I normally would like—I have three, while the range over the past five years has been between 2 and 3, with the mode being 2. I already explained why I think Louisville will go to the finals—they are one of my upsets. FTE accounted for my putting Duke to beat Michigan, and my home-state bias (as well as Izzo) helped me choose Michigan State as beating Virginia.

            For the Final Four game I have 1 upset—the only time there has been an upset in the last 5 years was in 2009. But I see Louisville as a strong team, and FTE has them at a 1% chance of getting to the finals over Arizona, so I chose to have them beating Arizona.   

            Overall, I have enjoyed making my bracket. I have never thought of myself as wanting to go into sports statistics, but now I can see how people are drawn to it. I can definitely see myself getting into watching sports more often if I view the opportunity as a competition or for me to use my statistical abilities.

Book Response 1: Journey through Genius

            In Journey through Genius, William Dunham introduced some of the most influential mathematicians in history alongside explanations for one or two of their most profound discoveries in math. Journey through Genius gives a historical context for the mathematicians and their discoveries, and tries to convince the reader of the greatness of the discovery—the worthiness, as it may be, for why that piece of mathematical history has made it into this book. While the content provided in Journey through Genius is accessible to those with only a high-school level mathematics education, the means to coming to the results (via the proofs included in the book) are still so genius that one with a more in-depth understanding can appreciate the proofs given. If the purpose of the book was to inform the reader of some of the great mathematical theorems and those who discovered them, then this book mostly achieved its goal—its main fault is the short-sighted nature of its scope.

            There are twelve chapters in Journey through Genius, each one (more or less) dedicated to a main theorem and to the life of the man who discovered it. Each chapter introduced context of that area of math at the time of the discovery, gave a proof outlining the main theorem, and concluded with an explanation of how the theorem influenced math and sometimes summarized the rest of the discoverer’s life. Not only was each of the twelve chapters focused on someone (or, in Chapter 11’s case, some people) from the Western world, but even introductions and conclusion rarely mention anyone outside of the Western culture. Dunham tries to explain this obvious exclusion in one of his chapters, stating, amongst other things, that “Because the Islamic mathematicians put less emphasis on proving their results in complete generality, no great theorem appears here” (131). This weak excuse for omitting some of the great thinkers of the Islamic world is more consolation than the void of an excuse for why discoveries from China, India, and other realms of the world aren’t explored.

Even if the excuse Dunham supplies is true, it doesn’t explain why more of the Islamic mathematicians’ work isn’t explored in the introduction or conclusion of the chapters. Throughout the introductions and conclusions of the chapters, Dunham does thorough exploration of other theorems that aren’t the chapter’s central topic. He does this time and time again, and in a few of the chapters he goes so far as to mention that there was some progress made in the area by non-Western mathematicians, yet no short proof was supplied, no exploration of any non-Western “discovered” mathematics was ever explained by Dunham. In total, there are less than five pages dedicated to non-Western mathematical discoveries (out of 283), which, in my opinion, is Journey through Genius’s greatest downfall.

The information that Dunham did include about the mathematicians and the discoveries that he did highlight were both intellectually and emotionally satisfying. I appreciated learning how each great mathematician’s discovery was proven and about the background for each great mathematician; hearing how each embodied genius, handled his achievements, and went about his life. The background given for each mathematician increased the impact felt when being led through how each mathematician proved his theorem, giving the already extraordinary proofs another layer of meaning and understanding. I learned much about the state of mathematics from Journey through Genius —from the discipline’s history, to the construction of proofs of some truly great results, to the realities of the Western-centric mindset of some of the mathematical community that still pervade today.

History: Brahmagupta

As we learn about history’s great mathematicians, I am frequently struck by how often their pursuits are intertwined with other disciplines. We now have concrete job titles for what a person does—they are a mathematician, an astronomer, an engineer, a poet, a physicist—and it is implied that one will chose a profession and stick to it. I wonder if “mathematician” once encompassed all of these titles, and that to be a mathematician would have implied the various applications, or if we have now forced titles onto the work that people do, thereby corralling their ambitions to one topic.

One individual that was heavily involved in two disciplines—astronomy and mathematics—was Brahmagupta. It has been estimated that Brahmagupta was born in 598 in Ujjain, India, and that he passed in 670. During his lifetime he wrote (at least) two influential books, as well as held the position as the head of the astronomical observatory at Ujjain, “which was the foremost mathematical centre (sic) of ancient India at this time.” (O’Connor and Roberts). A portion of his work has been preserved in his book Brahmasphutasiddhanta, which displayed his insights in both astronomy and mathematics. The first ten chapters of the book described astronomical phenomena. The final fifteen further explored astronomical phenomena, but it also delved into algebra and geometry (O’Connor and Roberts).

Brahmagupta uncovered great mathematical truths in both algebra and geometry, but perhaps even more impressively, he is attributed to defining zero as a number and negative numbers. As early as 200 A.D., a closed circle-symbol or the word “kha” would be used by those who wanted to indicate an absence of a number or an empty place (“Zero as a Number”). Brahmagupta defined zero as a number and dug deeper, thereby unearthing negative numbers. In his book Brahmasphutasiddhanta he outlines rules for math using zero and negative numbers, where he calls an arbitrary negative number a “debt”, an arbitrary positive number a “fortune”, and zero as zero (Mastin).

Not only did Brahmagupta provide us with some of the building blocks of our modern number system, he concluded that quadratic equations could have two possible solutions (and one could be negative), he solved quadratic equations with two unknowns (which wasn’t considered in the West for another 1000 years), and gave a formula for the area of a cyclic quadrilateral as well as a formula for its length in relation to its sides (Mastin; Hayashi). In exploring these concepts, Brahmagupta even began to dabble in how we now view algebra by using the initials of the names of colors to represent the unknowns in his equations (Mastin).

Brahmagupta’s work explored the concretes of our solar system as well as the abstract concepts of mathematics. In describing zero and negative numbers, Brahmagupta unequivocally changed mathematics, and his other contributions further confirm the genius he possessed.

 

Works Cited

Hayashi, Takao. “Brahmagupta.” Encyclopedia Britannica, n.d. Web. 19 Feb. 2014.
Mastin, Luke. “Indian Mathemtatics-Brahmagupta.” The Story of Mathematics, n.d. Web. 19 Feb. 2014.
O’Connor, JJ, and E. F. Roberts. “Brahmagupta.” School of Mathematics and Statistics, University of
        St. Andrews, Scotland, n.d. Web. 19 Feb. 2014.
“Zero as a Number—Brahmagupta Period.” Wayne State University, n.d. Web. 19 Feb. 2014.

"Doing Math": Tesselation (s)

Using my novice Geogebra skills, I created the following tesselation:

I also made a few attempts previous to this one for my "base" shape, but the final one I used (above) is my favorite. Here is another "base" that I made, though I didn't find as aesthetically pleasing:

And here is another, more "full" tesselation that I was able to flesh out more.

Bimonthly number 1. "Communicating Math": Plain Language Euclid

                                                                         "Communicating Math": Plain Language Euclid

What is math?

My first reaction to the prompt, "What is math?" is a line of numbers running across my mind in a horizontal style not unlike the one in the Matrix movies. Once the number's dance is done, images of shells with the Fibonacci sequence outlined, the unit circle, and a triangle with the Pythagorean theorem flash into view.  For me, math has been a source of intrigue, education, and frustration. In the past I have focused on the puzzles that math offers rather than the history behind it, but there are a few names and times that have stuck with me in math's history.

The discoveries/milestones that strike me as "Top 5" worthy in the history of mathematics (that I can remember) are:

1. Use of numbers
2. "Discovery" of zero
3. The Fibonacci Sequence
4. Pi
5. The Fundamental Theorem of Calculus

I'll admit that I don't have the knowledge of when all of these discoveries/milestones hit. I recall learning that numbers first started being recorded for trade in the sense of "1 cow for 10 chickens." To count, add, and subtract numbers, and to figure out a way to write it down, may seem trite, but it was (in my opinion) the first great leap that started our current concept of mathematics.

Similarly, the "discovery" of zero by the Hindus and Mayans (independently) added another key component to today's mathematics. Recognizing that zero isn't just a numbers absence, but is a number in itself, influences much of how I look at math.

The Fibonacci Sequence has always fascinated me because it has an elegance to it and because it is found everywhere in nature--and that it has evolved in nature because of it's natural efficiency.

I think that pi deserves to be in the top 5 because it is the foundation of the circle, and the Fundamental Theorem of Calculus (which I remember was cemented in history by Issac Newton) because it has led to many breakthroughs in modern (and ancient) mathematics, and because calculus is in so many of our actions on a daily basis.