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Monday, May 30, 2016

Problem of the Week

For Memorial Day, we have a slightly modified Problem of the Week- since our office is closed for the holiday, there's no solution video for this week's problem. We will be back in business next Monday, so stay tuned! Have a great Memorial Day from all of us at the Center of Math!
See one potential answer below the break.

Thursday, May 26, 2016

Advanced Knowledge Problem of the Week

Don't miss out on this week's Advanced Knowledge Problem of the Week!
Solution below the break.

How to Have a Math Filled Summer

Summer of Math

Summer is a time for warm weather and relaxation, but it doesn't mean education has to come to a grinding halt. Learning and increasing your knowledge comes in many different forms, and we created the #EverydayMath series to prove this! So far, we have outlined ways that mathematics can be present in books, movies, television shows, or games. The summer is a great time to explore these areas, where math can intersect with other academic categories. To jumpstart your summer plans, we will recap these posts and include new ideas that can help to fill the coming months with educationally enriching activities!

Many people use the Summer as a time to get away. Whether it's a few days or just a few hours, here are some attractions that will appeal to all math enthusiasts!

Math Attractions

1. National Museum of Mathematics
    The National Museum of Mathematics is a great place for any math lover to visit. Located in New York City, the museum features a number of interactive exhibits, a gallery of mathematical structures, pictures and art, as well as programs that explore the wonders of mathematics. This is the place to be for anyone who wants to spend a day of fun, immersed in a world of mathematics. 

2. MIT Museum
Twitter: @MITMuseum
     Right in our backyard (Cambridge, MA), MIT is a center of math and sciences. The MIT museum is meant to engage and inspire visitors about the possibilities and opportunities science and technology have to offer. The museum features interactive exhibits, public programs, and its own world-renown collections. Any math lover should visit the museum and see one of the most influential mathematical places in the world.

3. The Tech Museum of Innovation

Twitter: @TheTechMuseum
     Located in San Jose, in the the heart of Silicon Valley, the Tech Museum of Innovation is a great place for any math lover to visit. The museum features many different programs and exhibits designed around tech innovation. One such exhibit involves designing and building your own robot! This is a great place to see what the latest innovations in technology are, all involving math of some kind.

Here are ways to stay in your own backyard, but still put a math twist on your summer!

Everyday Math: Books
April 29, 2016

Imagine this combination: a comfortable beach chair, beautiful weather, and a good book. It's difficult to imagine anything that may top that, but what if that book was a math book? 
Here are 3 top selling books about mathematics.

Fermat's Enigma
by Simon Singh

This National Bestseller is not a biography of Fermat or an explanation of his theorem, as the title might suggest. Instead, Singh delves into the lives of people who dedicated their time and careers to proving Fermat's Last Theorem. The book creates a more personalized view of mathematicians. The heartbreak, mastery, and critical moments of these great minds are documented in a way that humanizes the subject.

Zero: The Biography of a Dangerous Idea
Charles Seife

This non-fiction book examines the idea of 0 and explains the controversiality surrounding the number at some points in history. It traces back to the Babylonian roots of 0 and takes a look at "one of the great paradoxes of human thinking". He writes the topic in a thrilling and interesting way, receiving great reviews from the Mathematical Association of America.

The Drunkard's Walk: How Randomness Rules Our Lives
Leonard Mlodinow

This choice is not as firmly rooted in mathematics as some others on this list, however it is still a great read that combines probability, statistics, and how they impact society. Mlodinow demonstrates how our lives are strongly determined by chance, even when there seems to be a clear system set in place. Ranging from political polls to corporate success, his examples are compelling and interesting. By combining psychology and statistics, The Drunkard's Walk takes an applicable approach to mathematics.

Everyday Math: Math Games and Logic Puzzles
April 1, 2016

Maybe you prefer keeping your mind on tract with puzzles or logic games. Here are some Sudoku alternatives that will keep you thinking!


Similar to Sudoku and Killer Sudoku, Calcudoku follows the same basic directions. Instead of having to fill in the numbers according to sums, like in Killer Sudoku, Calcudoku provides a number and a certain mathematical operation. The numbers in the smaller boxes must compute to the number given, using the function noted. Again, no numbers are filled in before you begin, so the mathematical functions are the only hint you have! If you enjoy mental math, this is probably the twist on Sudoku you would enjoy the most! Below is an example with some numbers already filled in.

Bongard Problem

Bongard problems differ from the previously mentioned Sudoku puzzles. Have you ever played spot the difference in a children's magazine or book? Well Russian Scientist M.M Bogart created a similar game in 1967. Bongard problems are based on visual pattern recognition. There are 6 shapes or figures on the left, and six figures on the right. The six shapes on the left all share a common characteristic with each other or follow the same rule. The shapes on the right also share a common trait with each other, but something separates them from the shapes on the left. The game is to figure out what the difference is between the two sides. In other words, it's your job to find the rule that each side follows or does not follow. Check out the example below, a medium level problem.

Everyday Math: Math in the Movies
March 11, 2016

Even the Summer has its rainy days, and sometimes it can be nice to spend a day inside. Don't worry! This can be a great excuse to watch a movie centered on mathematics!

The Story of An Underdog: Good Will Hunting
Known as a classic math movie, Good Will Hunting is a must-see. This underdog tale has a romantic twist and stars both Matt Damon and Robin Williams. Matt Damon's character, Will Hunting is a troubled young adult who's life path weaved in and out of foster homes and trouble with the law. While working as a janitor at MIT, he is able to solve two math problems that were created for graduate students. A professor at MIT took interest in Will, noticing his affinity towards mathematics and genius-level ability. Along his journey, Will faces his inner struggles with the help of a therapist (Robin Williams), and meets a romantic interest who helps to shape his life. The math problem that Will faces on the board is actually a real problem, although not as difficult as it is made out to be. The problem involves a feature of graph theory, homeomorphically irreducible trees. Pictured below is the problem that sent Will into the realm of academia. Interestingly enough, the math brains behind the movie actually appeared on screen as well. Patrick O'Donnell, who had a minor roll in the bar scene, actually ran the math department at University of Toronto at the time. O'Donnell and John Mighton, who plays the professor's assistant, chose the equations and theorems used in the movie. 

The Classic: A Beautiful Mind
A Beautiful Mind is based loosely on the real life story of John Nash, a Nobel Prize winner. Russell Crowe plays John Nash, a mathematical genius that specialized in game theory, differential geometry, and differential equations. Game theory can be utilized in fields such as economics and political science. In fact, Nash won his Nobel Prize in economics. In a famous scene, the film dramatizes Nash's discovery of the Nash equilibrium, a term used in economics and game theory. The film is said to take artistic interpretation of Nash's real life, but the mathematics in the movie are based on real theorems and theories. The director of A Beautiful Mind enlisted the help of a mathematics consultant, Dave Bayer of Columbia University, to ensure the mathematics were correct throughout the film. The movie contains some math jokes and facts that may only be clear to those well versed in mathematics. For an example, at the end of the film, a student wants to show Nash a proof exploring the idea that "finite Galois extensions are the same as covering spaces", which is actually a true statement. Along with his prowess in mathematics, A Beautiful Mind also demonstrates Nash's story of mental illness and schizophrenia. His schizophrenia initially impacts his career, but he is able to recover and take his place as one of the leading mathematical and economic minds of his time.

Friday, May 20, 2016

The Stages of Buying Textbooks: Told by Social Media Posts

The Stages of Buying Textbooks: Told by Social Media Posts

The speed and accessibility of the Internet makes it the perfect place to voice your opinions, share your thoughts, and argue your beliefs. This is particularly true of young people, who use social media platforms to draw attention to issues that impact their lives. The quick spreading nature of Facebook, Twitter, Imagr, and other popular sharing platforms make them ideal forums for students to express themselves. Sometimes, this expression takes the form of a persuasive editorial. Other times, a picture with a witty caption can suffice in addressing the issue at hand. The latter what is the Center of Math chose to focus on in this post. Students have taken to the Internet to discuss the extravagant price of college textbooks and learning materials. While these students decided to use humor to express their frustrations, spending upwards of $200 on one textbook is not something to laugh off. Speak to your professors or do your part to #AdoptAffordable and visit the Center of Math

Here are the stages of purchasing textbooks, as told by social media posts. 

First, there is the purchase. Students are full of disbelief and use sarcasm to mask their feelings.

Next, comes sadness. Lots of sadness.

This feeling of sadness is followed by anger. Sometimes, this anger is misplaced towards bookstore employees. Sorry?

Student discounts and buying used books rarely goes as planned. 

Once classes begin, some teachers don't even use the assigned texts!

After the semester comes to a close, students seek relief by attempting to sell books back to the bookstore. This rarely goes as planned.

Lastly, students who are unable to sell back their books find some creative uses for them. Discarded textbooks become the most expensive dorm room furniture.

Now, what if I told you that you could purchase your math textbooks digitally for less than $10? No more anger, sadness, or frustration! You may not be able to add to your textbook graveyard/TV stand at the end of the semester, but with the money saved on textbooks you could buy real furniture! There's also no need to stress about selling back textbooks with an affordable, digital copy. Interested on hearing more? Head to our website to learn more about the Worldwide Center of Math!

Thursday, May 19, 2016

Advanced Knowledge Problem of the Week

Don't miss out on this week's Advanced Knowledge Problem of the Week! Let us know how you do in the comments.
Solution below the break.

Tuesday, May 17, 2016

Problem of the Week

See what you can come up with for this week's irrationally cool Problem of the Week. Let us know how you do in the comments!
Solution below the break.

Monday, May 16, 2016

How Math Education is Changing

In a culture that focuses on fast paced innovations, various aspects in our lives are constantly changing. It seems like every few months the newest phone, car, appliance is released. It can be exhausting and expensive to keep up, however we are all a little guilty of wanting to be constantly updated. Along with technological innovations there is no denying that education techniques and standards are also advancing. The disparity in just the last generation alone is astonishing. Classrooms have transformed from blackboards to smart boards, paper to laptops and textbooks to online resources. Parents and teachers often find themselves wondering, whatever happened to using a good old-fashioned No. 2 pencil?

To focus specifically on math in the classroom, a lot more has changed than just the tools used. The expectations facing 21st century students are constantly rising, and as a result teachers are raising the bar. Heres a few ways teachers and professors have altered the way they teach to fit the new standard:

1) Higher academic standards

Academic milestones need to be reached at much younger ages. This essentially means students need to develop a more wholesome understanding of what they are learning. According to the Common Core State Standards, "the idea is that the higher academic standards will ensure students are adequately prepared for college and the careers of the 21st century."

2) Digital Math

Technology, especially computers/cell-phones are second nature to most students who in this generation are coined to be "digital natives." Teachers are turning to digital math resources and adaptive leaning systems to engage students and make math enjoyable.

The Center of Math is centered around accessible and affordable digital textbooks. As we are in the height of the digital age, several teachers and students have decided to make the switch from print to digital. To view our digital resources click here.

3) Emphasis on the "why" and the "how"

Many see math as just a pointless subject consisting solely on memorizing formulas. Now more than ever, teachers are focusing much more on the reasoning behind a specific theories/formulas. Students are expected to not only master mathematical concepts, but also understand the reasoning behind them.

4) College on the horizon

It was not always the case that 70 percent of high school graduates attended college. The norm for students no more than 50 years ago was to stop at a high school degree, a much smaller percentage attended college. Teachers now must focus on catering their curriculum to serve students past the 12th grade.

5) Real-world applications

Recently math education has made the shift from being considered an abstract concept filled with formulas and theories to a concrete subject focusing on skills that can be applied to the real world. Teachers are shifting from more traditional curriculums and are beginning to incorporate real-life applications into their math classrooms.

6) Allow students to explore

Learning is dynamic, rather than static. It used to be that children were asked to repeat math concepts until their responses were automatic, multiplication tables ring a bell? Now, students are getting pushed to limits where they are encouraged to explore ideas and experiment with different avenues of the subject.

7) Link between researches and educators 

Teachers and researches are now encouraged to collaborate to produce data that can help improve education. Math education is always changing. To ensure it is moving in the right direction a researchers' input is necessary to better math studies.

8) Changing perceptions of math

Research has shown that how children view their math capabilities has an effect on their success. Students willing to take higher-level math is directly correlated with their self-esteem as learners. Teachers are working to transform math from a "scary" subject into something that is approachable and useful for students, so they can develop the confidence they need to be successful.


Friday, May 13, 2016

Friday the 13th

Did you know today is Friday the 13th? 

Spanning across many cultures and religions, the number 13 tends to have unlucky connotations. Furthermore, it may be a gateway to the weekend for some, but Friday also has historical and religious connections with 'evil' or 'unluckiness'. This being said, it is not surprising that the two culminate in what has been deemed the most unlucky day, Friday the 13th. This suspicion has been furthered by horror movies and pop culture. The legend and superstition began, and psychologists have tried to pinpoint why. Many say that because of the predisposed feelings towards Friday the 13th , people become more aware of unlucky or negative events. Whatever your feelings or superstitions, we have gathered 13 facts about the number 13. Hopefully with a few fun facts in your pocket, your Friday the 13th will be as lucky as any other day!

13 facts about the number 13!

1. Thirteen is often referred to as a baker's dozen.

2. (13)²= 169 and (31)²=961

3. 13 is the  6th  prime number

4. Triskaidekaphobia is the fear of the number 13. People who suffered from triskaidekaphobia include Winston Churchill (who refused to sit in row 13 of an aircraft or theatre), Napoleon, and Christopher Columbus.

5. 13 is the 7th   Fibonacci number. 

6. (1/13)=0.076923076923076923076923...
                 (rational periodic infinite decimal)

7. A thirteen-sided polygon is called a tridecagon. The interior angle measure in a regular tridecagon is 152.3077 degrees.

8. 13 can also be called a Long Dozen. 

9. 13 is the smallest emirp, a prime number whose digit reversal forms a different prime number (31 is prime). Other emirp pairs include 17 & 71, 79 & 97, and 1193 & 3911.

10. Many office buildings and hotels do not have a Floor 13. This is also true for some city streets. 

11. Traditional 13th wedding anniversary gifts have a theme of Lace.

12 . 13 holds a special significance for the United States. 
      There were 13 original colonies. There are 13 stripes on the American flag.

      The US $1.00 has:
      13 levels of the pyramid
      13 stars
      13 arrows
      13 stripes
      13 leaves
      13 olives

      The Great Seal of the United States has:
      13 stars
      13 arrows
      13 berries
      13 stripes
      13 leaves

13. If Friday the 13th is something you dread, in China and Italy, 13 is considered a lucky number! The same is true of Colgate University, in New York. 

Thursday, May 12, 2016

Interviewing a Math Student

Do you want to major in mathematics?

As freshman students spill through the doors of their respective colleges each fall, they are faced with a whole new world of opportunity, excitement, and exploration. However, before even taking a step on campus, some students are asked to select their major or program of study. This can be a daunting task, as choosing a college major begins a path of education specialized towards a particular career. This decision revolves around a personal inventory of skills, interests, and goals. A heavy demand for employees in the STEM fields has caused many students to select a major rooted in the math or science department. Sometimes, students are unaware of the career options they may have, preventing them from launching themselves into a math or science program. Mathematics can fall into this category, as many students don’t fully understand the major or opportunities it may lead to. Even students who have an affinity towards math may shy away from it at the college level wondering, “What does a mathematician do, anyway?”. At the Center of Math, we want students to embrace mathematics. We thought, what better what to see what a college math major is like than to speak to someone in the program! We interviewed Northeastern student Chloe Weiers, an intern at the Center of Math, to field our questions about college mathematics. 

WCoM: So Chloe, tell me about yourself briefly.

Chloe: I'm a rising third year student at Northeastern, and I'm a Music and Math double major. I play the flute. Also, I'm from Minnesota.

WCoM: Music and Math? That’s a very interesting combination. Did you know before you started college that combination was possible?

Chloe: No, going into college I thought I was just going to be a music major. I had always liked math but never considered it as a major before college. I've always thought math was beautiful, but not in the same way as music. Then I got the chance to look at math and music under the same light, I was better able to appreciate the symmetries and beauty in both and how they complement one another.

WCoM: What exactly made you change your mind and add math to your major.

Chloe: Once I got to school I had a great class called The Algebra and Geometry of Music, and after the course finished was invited to do research with the professor of the class on a topic of my choice. I chose dynamical systems and mathematical modeling, and then ran with it. After than, I got really interested in math and added it as a second major.

WCoM: Okay so I’m going to gear in a little more on the math side of your major now. Do you feel that you were aware of the career possibilities that were possible for math majors when you entered college?

Chloe: No, I had no idea what the options were. Many people seem to know going in, "Oh, I want to do specifically this or that", or like, "Oh, I want to be an actuary or an accountant or a professor", but I had no idea.

WCoM: Did that make you nervous at all?

Chloe: No, because I never really thought about it in that way. It seemed natural that I should add it as a major, because it’s one of those things where the more you learn, the more really interesting stuff you realize you don’t know, and its kind of addicting. I couldn't imagine never taking another math class. I have to keep growing as a mathematician.

WCoM: You mentioned 3 careers so far– becoming a professor, actuary, or accountant. What would you say to someone like yourself, who isn’t necessarily interested in any of those specific fields?

Chloe: Well, it's important to consider all the options, and sometimes that involves taking things into your own hands. There are tons of jobs that use math, but they aren't necessarily going to come to you- sometimes you have to seek them out. But when you do, the magnitude of really cool options is very encouraging. Also, when I was a first-year undergrad, my math professors kept encouraging me to learn some mathematical computing skills, like MATLAB and Python, and even LaTeX for typesetting mathematics. I was so reluctant, because I never considered myself even the slightest bit competent with computers. But it's so necessary. Even just learning the basics of something will help you get your foot in the door and help bolster your confidence. You don't have to be some coding expert, but having the ability to tackle new challenges with confidence instead of shutting down is an excellent thing to have going for you. The modes of thinking involved with all of these programming languages is completely logical and can even help you streamline your thought processes in other ways. So it's a win all around.

WCoM: Have you had a moment yet that you realized, yes I definitely made the right decision?

Chloe: I would say it’s less a moment than a series of realizations, usually coming when I’m working on or solving problems. That feeling you get of accomplishment is a really powerful, albeit selfish, indicator for me that I’m doing the right thing. Also, when you get into courses like Real Analysis and Number Theory and start learning the derivations behind the mathematics you have known and accepted your entire life, and then turning it inside out with new axioms or rules just kind of blows your mind. It’s also always very challenging and very interesting, so you’re never idling. Mathematics even helps me think rationally and logically in other areas of my life.

WCoM: Wow, so it seems like you have no regrets about your decisions so far, academically speaking. Is that correct?

Chloe: Definitely.

WCoM: So there is heavy discussion about women in STEM, or lack thereof. Do you notice this in your classes?

Chloe: Yeah, definitely. I mean, most of the time less than 30 or 40% or the class is women, often even less. It’s not necessarily a lack of opportunity for women in college, but rather a lack of opportunity and encouragement leading into the university setting that I think prevents many women from even considering math as a potential career field.

WCoM: Obviously there isn’t necessarily an easy fix to this issue, but what do you personally think can be done to encourage women in the field?

Chloe: I think that it needs to start much earlier than college. If you are looking at trying to get college-aged women into STEM fields you’re already too late. It needs to start as early as elementary school. Girls should be encouraged and celebrated as active participants in their math and science classes, so they can build confidence with their own abilities and learn about their options early on.

WCoM: Building confidence from an early age definitely makes sense. Do you have any advice for students thinking about majoring in mathematics, male and female alike?

Chloe: Try to get as involved as you can with extracurriculars and such. When I say extracurriculars, I mean pretty much anything- robotics, math Olympiads, just recreational math at home or with friends, coding, whatever. It doesn’t have to be these high level competitions you see kids doing. Math competitions aren’t for everyone, and they’re certainly not for me, so don’t feel pressured to participate in math like a competitive sport. There are often cool summer programs that I wish I would have known about as a high schooler, and they’re offered all over the summer, so definitely look into those if you’re interested in really finding out what interests 

WCoM: It’s nice to hear you so encouraging to young mathematicians. So what have you been up to lately at the Center?

Chloe: I've been working with Ruairi, who works with digital media here at the Center, to create a video series on mathematical music theory. We're calling it Musimathics, and it will be available soon on our Youtube channel. The videos give an overview of some of the especially interesting areas of mathematical music theory, explained in a casual setting by myself. You don't have to be a super advanced mathematician or skilled musician to understand what's going on the the videos. They're meant for everyone! So stay tuned for more information about that.