High Jinks!

              When we were told to make a review about Ian Stewart’s: Cabinet of Mathematical Curiosities, I was quite hesitant for I believe that this book would contain intense discussion about mathematics and I already had enough of it. But when I opened and read the first two pages or so of the book, I quickly changed my mind.

          The book opened with the author’s belief that the most interesting topics of mathematics are not taught in school but are usually experienced by the person in the least expected places and that inspired him to make this book. He filled the book with different types of mathematical games, puzzles, trick and even stories that would surely stir the readers mind and interest. I have recognized some of the tricks mentioned for our instructor gave us some of those. The answer to the games and puzzles were given at the latter part of book which makes it very convenient to the reader for they would be able to know if they got the problem right or not. This book unveiled the true beauty of logic, geometry and probability which is associated to different sorts of problem.

          All in all, I found the book quite interesting for the puzzles had truly stirred my interest. The books contained no moral or any of that sort but only delight and amazement. In his quiet way, Stewart has done more than his colleagues have done in the past fast years. The way he wrote the problem was truly amazing and the curiosities were well explained but still remained the problems real tough. This book will certainly challenge both non mathematicians and mathematicians.

An Atypical Tale

              When we were told to read and to make a review about the book: Certain Ambiguity by Gaurav Suri and Hartosh Singh Bal, I immediately turned on my laptop and searched for a PDF for this book for I, a self-declared bookworm, wanted to get my hands on that book and boy I was shocked when the result came out! A mathematical novel? For real? For the past 10 years that I’ve been reading and burying myself with books, not even once I encountered this kind of genre. Who would have thought that some people would even bother to make a mathematical novel? I mean, isn’t it enough that we read them almost every day for an hour or so because they are one of our major subjects?  But still, curiosity killed the cat and unfortunately, that cat was me. I wasn’t really forced to read the book since the outline caught my attention ans made me wonder if the book was really that good. I decided to read it since I believe that it wouldn’t hurt to try and give this book a chance.

          The book opened with a man named Ravi Kapoor who was reminiscing to the time his mathematician grandfather gave him a math problem to try on a calculator. This part indicates the close relationship that the man had with his grandfather. The next day, the grandfather died and which then implies to the reader that Ravi’s grandfather had a big impact on his life for he pursued what he’s grandfather wanted for him to do in the future. But without his grandfather by his side, a man who had a strong passion for mathematics, Ravi became apathetic with the said subject. Still, he strived and got accepted at Stanford University and studied economics due to his father’s urging. There, he met his first friend Peter Cage whose career is inclined with business. As each succeeded page pass by, we learned that Peter and Ravi signed for “Thinking about Infinity” class under Professor Nico who Ravi eventually befriended. Ravi soon found out that his late grandfather Vijay Sahni, was jailed in 1919 at New Jersey and was charged for blasphemy. At this point, there was an alternation of events from Nico’s lecture to Ravi who was searching for the reason behind his grandfather’s imprisonment which then he discovers transcripts of philosophical discussions and a skeptical judge challenging his grandfather to defend his belief. The math topics conferred in the story includes Zeno’s paradoxes, Godel’s Incompleteness, Paul Cohen’s Consistency theorem and most of all the Euclidean Geometry which was the focal point of the story. At the end of the story, we learned that Ravi eventually left economics and chose a profession inclined with mathematics, he also end up with Claire, a fellow student during his infinity class.

          I found the book fascinating and was totally worth it to read. Though I’m not an enthusiast of mathematics, I was able to appreciate the topics discussed in the book for it was new to me and was believable enough. The authors of the book did a great job in explaining faith in a mathematical point of view. All in all, this book was pleasing enough and you can obtain lot of information from it especially in applying basic math axioms in real life situations. If given the chance to read another novel with this kind of genre, I wouldn’t have any second thoughts again J

Another Line of Attack

              When we were task to read the book: Rock, Paper and Scissors by Len Fisher, and give a review about it, I was devastated. My subconscious kept asking why is it again associated with mathematics? Isn’t it enough that we had to live with it ever since we entered kindergarten? But then again, my inquisitiveness overpowered me and so I decided to read the book. Not because we were required to do but for the reason that I am curious and I want to stop this madness.

          In the book, the author Len Fisher generally talked about Game Theory. Game theory, according to Fisher tells us what is going on behind the confrontation, broken promises, and just plain cheating. Aside from the Game Theory he also talked about other complicating factors such Nash’s Trap which implies that the equilibrium always leads to a bad outcome, N-person Situations which is cooperation between two individuals, Perfect and Imperfect Information and lastly, the Simultaneous or Sequential Strategic Decisions.  Fisher also demonstrated the limits of game theory and used it to show how obliging behaviors have evolved to escape the traps created by selfishness and how teamwork persists even in the most challenging situations. The author also suggested employing the same strategy of game theory to solve territorial disputes between countries. He also talked about cooperation and how we can achieve trust, the seven deadly dilemmas which was clearly stated and explained.

              The book was pleasing enough and was fun especially in the end notes part where it is full of stories, jokes and random trivia which will help you in future. Truly, Fisher has a knack for jokes that made the reader that they were reading a book full of witticism. 

Geometry: Rocks, Papers and Toothpicks

 “Sticks and stones may not break my bones, but Geom might somewhat crush me!” Ever since math was introduced to me, geometry was the least of my favorite. I didn’t even know how I survived my high school geometry because solving angles or anything related to it I a brain teaser for me. To be honest, doing a discussion and a group report about geometry really gave me pressure to do good and to prove that geometry, in it hardest concepts, also has an easy side and that through this reporting, I may be able to convince myself that geometry is not just a mere collection of formulas, theorems and postulates, that what we learn from it has applications, too which can be beneficial to human and the society.

When we think of the word “geometry”, the first that we associate directly to it are shapes whether quadrilateral, triangle, pentagon and the like. Having the opportunity to introduce this field to our classmates, we started our first meeting with a form this shape game I called “Search and Form Me.”  Before our classmates entered the room, we already hid figures of different sizes and shapes. Per group, 7 shapes were each assigned to be formed into a bigger shape, the heart (since that was a Valentine’s Week). After all the instructions were given, teams were given an unlimited time to form the unifying shape until one group can form the said shape. It almost took 5 minutes for the first team, the Team Computers, to form the shape. They have been given the necessary points for winning the game and we proceeded to the next game.

The next game that we had is the toothpick game where figure analysis of each player was tested by moving or removing toothpicks to form the shape desired by the game master. Initially, they were given 24 toothpicks and formed it into shapes drawn on the board. The game master would then instruct “Remove 2 toothpicks to form three boxes.” After the go signal, the players then find way to remove toothpicks to form the shape instructed by the game master. Groups who got the right answer were given points. The game was divided into three categories, the easy round (2 points), the average round (3 points) and the difficult round (5 points) where the group having the highest accumulated points from this game plus the first game will win the so-called competition. The players were given 2-3 minutes (sometimes with extension) to finish the task. With this game, you can really see how those imaginative minds work since they were not allowed to move or remove the toothpicks unless they are sure of their answers. We also saw how each group brainstormed and united to beat the challenge. For the first 2 games, Group number Systems led the ranking.

We played the next game on our second meeting which is called the “Form-ulate me” game. Groups were given a mathematical formula corresponding to the formula of the volume, mass, circumference, diameter, or density of a certain shape. After hearing the formula, they have to form the shape through the use of their bodies. Unluckily, we considered the game void due to a problem or misunderstanding experienced by the class. We declared the group Number Systems as the over-all winner and then we presented our hilarious video presentation.

Our video presentation mostly talked about Euclidean Geometry and Non-Euclidean Geometry. We have been able to compare them and explained them through the shapes and lines seen in the buildings of the school. We also talked about the applications of geometry and even incorporated the concept of BBC: The Story of Maths to our video. The video took 8 minutes, full of facts.

Lastly, we had our debate or discussion about “What shape should buildings or any architectural designs have? Should it be a quadrilateral shape or a triangle?” The pros and the cons expressed their thoughts supporting their own stand on the issue. The pros (or the quadrilateral side) said that should be a quadrilateral since most of the buildings were already built that way and that it would save a space more than using a triangular-shaped building. They also claimed that quadrilateral shapes of buildings originated since the early times and mostly followed by a lot of people. The cons counterattacked the other group and pointed out that a triangular building is more stable because of its base and can stand better and stronger. The 2 groups exchanged views until the audience interacted and shared their knowledge about the topic. We had great time absorbing new information about geometry while talking about architecture. The debate ended with a stand that it is on the option of the person involved whether a quadrilateral or a triangular shape is better for buildings.

Overall, I felt great after all the presentations, games and debate were done because I learned something new and something better. It was really fortunate for me to change my views about a certain thing I disliked through my researches. It was also a great reward to hear the applause of other groups plus Ma’am Blasing’s “Job well done” words which gave us a sigh of relief. I realized that geometry really is fun and I would really try hard to discover more about this subject. Though there were some challenges met while we were preparing, all the efforts that we put through paid off, from shooting, video editing, to fact finding. Through rocks, papers and toothpicks, I realized how important geometry is to the lives of the people and how shapes and angles affect each one’s perspective about geometry.

Rocks, Papers, Scissors Book Review

Reading a math book can be a tragedy especially when the one reading it do not have a proper mathematical background. A book concerning math (specifically the game theory), which does not really explains its matters with lots of math, is really an interesting book to read. The Rocks, Papers, Scissors: Game Theory in Everyday Life of Len Fisher links the concepts of game theory to everyday life situation. The book welcomes you to the journey in search for answers to some of our most intriguing social questions.

Have you ever heard of the Prisoner’s Dilemma?  When John Nash discovered the hidden logical trap, it led us to a disturbing series of social dilemmas. In the Tragedy of the Commons, the best overall outcome is produced by cooperation, but self-interest could bring individuals to cheat on the cooperation. An example of what game theorists call the Prisoner’s Dilemma is this story:

There were two thieves caught by the police. The prosecutor has enough evidence to imprison them for two years due to carrying concealed weapon rather than accusing them of burglary with a maximum penalty of ten years. The two thieves plead guilty but the prosecutor has something in mind that could change their pleas. His first encounter was with thieve A. He said that if thieve B plead guilty and he doesn’t, thieve B will only have four years in prison and thieve A will be the one to receive the maximum penalty of ten years. So it is better if thieve A will also plead guilty to receive four years rather than ten. But the prosecutor offered him another deal that if thieve B will not plead guilty and thieve A does, thieve A can be free for turning over state’s evidence. The prosecutor had offered the same talk with thieve B and they both found pleading guilty better than any other choices. As a result, they both got the four-year sentence rather than the two years they might have received if both of them just remained silent.

The world is facing serious problems relating to this Prisoner’s Dilemma. An example is the arms race which activated in the 1950s. The goal is to minimize arms production to use the money for more constructive purposes. This could benefit everyone but no nation could afford to be disarmed so long as other countries continued to build up stocks of nuclear weapons. The global warming which is a menace to everyone is still not given the right act ion because it is still troubled by a contradicting logic. Many polluting countries do not see the benefit of controlling carbon emissions because there are also other nations which cause continuous pollution.

Social dilemmas were at last understood through the discovery of John Nash in 1949. He made his discovery of basic logical trap when he was just twenty-one and not yet having his mental illness, schizophrenia. Nash was a genius. He was just eighteen when he identified the logical trap now called as the Nash Equilibrium: “It is a position in which both sides have selected a strategy without ending up in a less desirable position.”

According to the author, communication and negotiation are the twin keys that can unlock social dilemmas. Through these keys, we are permitted to share information and our own ideas to others. In this way alliances can be formed wherein coordination and joint agreement can coexist. Also, according to him, effects of social dilemmas of game theory and the real world concerns trust issues. If we will be able to trust one another, dilemmas could suddenly disappear. When we trust each other, we are free to deliver our thoughts to others. We can negotiate about things which needed our actions and we can coordinate our strategies to formulate best solutions. We should not break agreements just because of our selfish acts that patronize personal interests.  We should be able to trust and be able to free our minds with the possibilities of being cheated. Free yourself from the act of retaliating. As what Oscar Wilde said, “always forgive your enemies, nothing annoys them so much.”

An Individual Report for Algebra

                On the first meeting of MST3 section B, the whole class was divided into 5 groups. Also, there were five topics in this course that would be taken up namely Number Systems, Algebra, Geometry, Statistics and Computer. We were asked by our teacher to select one among the five topics and we were tasked to have a debate and make a creative presentation and a game about that. Our group selected Algebra and we were scheduled to have our presentation and debate on the first week of February.

                First, I thought that presenting Algebra in a creative way would be easy and also having a debate about it won’t be so hard but when the week were we would perform came,  we started to panic because we didn’t know what to do and by that time, we were busy for our other major subjects.

                We didn’t have much time to prepare for our presentation because we haven’t planned anything so we decided to have our game first and we had a quiz bowl show. The whole quiz bowl duration, I was so nervous because there was some misunderstanding with the rules and regulation of our game but eventually, it went fine. Then there was a game where a puzzle was scattered at the school grounds then again, there was a little problem because a puzzle set was not done for the Computer group but one of my group mate decided to add points to all of the groups so that it wouldn’t be unfair to the Computer group. The next meeting, we had our presentation, it was impromptu and I was so nervous and ashamed because we had a live performance. It was about a girl who had problems in Algebra and her friends encouraged her by sharing what were the history and other important uses of Algebra in the society. In the live performance, I was the one who acted as the problematic girl and I was ashamed because I really don’t do something like that. After the life performance, we had our debate. It was about whether Algebra is essential to our everyday life or not. Of course, it was essential. You are not aware but you are already doing algebra everyday because it is an organized arithmetic.

                Presenting algebra, having games and debate about it is not that easy especially when your group mates came from a different bloc and even from a different course. It was hard because you have no same free time and it was in the midst of many examinations. But at least, we still have done our task and we are thankful that were not reprimanded by our professor. I have learned one thing in what we have done. When there is something to do, you should not do it the night before or even minutes before it is going to be submitted or performed. 

A Book, A Novel, A Fiction

From kindergarten to high school – and even in college, we learned mathematics through reading math books and textbooks. Line by line, we were to read the lessons in the hope that we learn the concepts and principles behind those mathematical ideas, axioms and theorems. Line by line we read. Line by line we studied. And line by line we memorized. But let us face it, we don’t remember most of what we read, studied and memorized. The lines of those mathematical books have been left in the past and have long been forgotten.

We may have read them, studied them and memorized them but we have failed to understand them. And if given a mathematical textbook, most students won’t even flip the page if their teachers won’t force them to. And this is most probably because we easily get bored reading mathematical books because it is written in way that is so formal, so straightforward and just simply so boring. To put things simply, we believe mathematics is boring.

Contrary to this belief, the book A Certain Ambiguity: A Mathematical Novel written by Indian authors Gaurav Suri and Hartosh Singh Bal shows us that reading mathematics too can be fun and fascinating.  A winner of the 2007 Award for Best Professional/Scholarly Book in Mathematics, this mathematical fiction talks about finding certainty in mathematics and philosophy. (Wikipedia, 2013)

This compelling novel talks about the journey of Ravi Kapoor, of Indian nationality who pursued his studies in Stanford University, America, where he discovers that he confronts the same mathematical and philosophical dilemmas that his mathematician grandfather had faced before – which had been a factor that landed him in jail. The grandfather, Vijay Sahni, was charged in 1919 under an obscure blasphemy law in a small town in New Jersey. The skeptical judge challenged him to defend his belief that the certainty of mathematics can be extended to all human knowledge – even religion. (press.princeton.edu, 2013)

In the novel, Kapoor and his grandfather struggled with the questions of “whether there can ever be absolute certainty in mathematics or life?” Along with contemplating these very philosophical question is the act of reconsidering their fundamental beliefs and perspectives. Their journey in queering these mathematical questions led them to explore the parallel development that exists in the study of geometry and infinity. (press.princeton.edu, 2013)

Beginning with a nostalgic flashback experienced by Ravi Kapoor to the time his mathematician grandfather gave him a math problem to try on a calculator, the story starts with the passing along of the grandfather’s interest on mathematics to his grandson Ravi. Upon Vijay’s death, Ravi inherits a large sum of money for the purpose of sending him to college in America. And soon Ravi gets accepted to Stanford where he majors in Economics. (Vestal, D., 2007)

During his senior year, Ravi decides to take a class called “Thinking About Infinity”, and in this class he learns more than what he expected to learn. (Vestal, D., 2007)

In his class, not only did he learn about mathematical principles and concepts on infinity but as well as the facts about his grandfather spending some time in America which landed him in jail for blasphemy. Wanting to dig deeper into it, he reads court documents to find out what exactly happened. From there he finds out that the American judge, John Taylor, who was trying to decide what to do with his grandfather was a Christian who looks to religion for certainty while his grandfather, who was an atheist, looks to mathematics for certainty. However Vijay’s faith in mathematics was shaken by the appearance of non-Euclidean geometry in astrophysics.  (Vestal, D., 2007)

The novel tells us a lot of mathematical concepts and philosophy mostly through the author’s act of incorporating them to the lectures that Ravi attends. (Vestal, D., 2007) Some of these math topics that were discussed in the book were elementary mathematics of infinity, set theory, Euclidean geometry, non-Euclidean geometry, Zeno’s paradoxes and infinitude of primes through Godel’s incompleteness and Paul Cohen’s Consistency Theorems. (Bogomolny, A., n.d.) The authors also presented the history of mathematical certainty, and they have done this through imagining things like letters and journal entries from Pythagoras, Nicole Oresme, Cantor and many more. (Vestal, D., 2007) And while doing this, they have presented mathematics as a human endeavor.  (Bogomolny, A., n.d.)  (The authors have stressed that these writings were purely fictional and have expounded the history in the notes section by the end of the book). (Vestal, D., 2007)

The book ends with Ravi choosing the career leading to mathematics instead of the path leading to economic (which could have led him to make more money). (Bogomolny, A., n.d.) 

This novel is unlike any other math books I have encountered. Unlike the usual math books and textbooks, this book is compelling in every aspect you look at it. It is fun and fascinating without jeopardizing its mathematical content.  This is unlike any other math books. This is a mathematical novel. This is a mathematical fiction. And most of all this is a mathematical book.
 
Mathematical Novel. Math books are usually written in a manner that is formal, straightforward and full of jargons. They may be understandable to those within the academe of mathematics but for those like me, we see a jumble of words which makes it hard for us to understand the concepts behind them. However, this book has shifted anew, grasping another style. From being a boring math textbook, it wanted to metamorph into something different, something that people would actually want to take from the shelf, something that people would actually want to flip the pages, and something that people would actually want to look forward to read during their leisure and free time. It became something. It became a novel. And we have to remember that novel means there is a story behind those pages, and people love stories. People don’t like reading explanations and boring definitions, they want to feel the thrill, the mystery and the excitement that the characters feel in the novel. And with this book being a novel instead of the usual textbook, people are going to be enticed to read it. And instead of just forgetting the concepts people read in mathematical textbooks, in a novel, we would remember it more because they can associate it to circumstances and stories within the novel.

Mathematical Fiction. Aside from being a novel, this book is under the genre, fiction. That it is make-believe. And this ignites desire from people to actually read this book. People are more inclined to read fiction because men are adept to imagining things. Aside from this, people want things to happen, that is why more often than not, we find ourselves day dreaming of a better life and of a better status. And aside from this, fictional stories allow us to think ourselves. It stretches our imagination. We want to know why it happened, what could happen and what could have happened? It even ignites the curiosity that is within us. And if we are curious we actually dig deep into things. This is what happens once you read this book, you get curious and you end up searching and digging deeper into things. And we actually get more attached to it. We might even start looking up on mathematical concepts that were mentioned in the book just to feed our curious minds. And thus we learn more from this thought provoking book.

Mathematical Book. But above all, this is a mathematical book. Mathematical ideas, concepts and principles were intertwined with the plot and the characters of the story to make it more interesting. From the start of the book, we have come across different mathematical topics already. Almost every page of the book is talking about mathematical concepts, ideas and principles. We encounter mathematics in the different scenes and situation in the story. We learn new concepts, theorems and principles through Ravi’s experience, intellect and analysis. The plot, the character, the genre, is all but a cover to the true essence of the book, which is to bring mathematics to its readers. And it has done it so successfully. What is meant here is that, this book is so exciting that we actually read and understand mathematical concepts, ideas and principles better because we are inclined to the excitement of what is going on in the story. Let us face it, people can’t finish one whole mathematical textbook and understand rather, we’ll just confuse ourselves with the jargons and confusing mathematical axioms. We might not even finish one whole chapter. But in this book we not only have a novel and a fiction but what is most important is we learn and understand the mathematical concepts and topics that the authors want to impart on us. We actually get acquainted with more mathematical concepts and topics as we read deeper into the story all because many mathematical topics are intertwined with the flow of the story.

This book has provided us with new perspective into looking at mathematics. Who would have thought that mathematics could be intertwined with novel? Who would have thought that mathematics could be this interesting? But most of all whoa would have thought that the beauty of mathematics could be reflected through this book being a novel, a fiction and a mathematical book.
Before when we use to read math textbook, line by line we read, line by line we study, and line by line we memorize, yet through those lines we failed to understand and remember.

Now, though this book being a novel and through the fictional story, we have come to appreciate math and its beauty.



REFERENCE:

Bogomolny, A. (n.d.) Reviewed by Alexander Bogomolny [Review of the book  A Certain Ambiguity: A Mathematical Novel]. Cut the Knot. Retrieved February 13, 2014 from http://www.cut-the-knot.org/books/Reviews/CertainAmbiguity.shtml

Vestal, D. (2007, September 26). Reviewed by Donald L. Vestal, on 09/26/2007 [Review of the book A Certain Ambiguity: A Mathematical Novel]. Mathematical Association of America. Retrieved February 13, 2014 from http://www.maa.org/publications/maa-reviews/a-certain-ambiguity-a-mathematical-novel

A Certain Ambiguity. (2013, December 12). In Wikipedia. Retrieved February 13, 2014 from http://en.wikipedia.org/wiki/A_Certain_Ambiguity

A Certain Ambiguity: A Mathematical Novel. (2014, January 14) In Princeton University Press. Retrieved February 13, 2014 from http://press.princeton.edu/titles/8479.html