**An Unconventional Path **by Sarah Fleming ’17

For most of her life, Pam Harris had no plans to become a mathematician. Brought to the United States from Mexico at the age of eight, she had no prospects of attending university after graduating high school because of her immigration status. As she still wanted to continue her education, Pam began taking classes at Milwaukee Area Technical College, a community college near her home. She originally set out thinking she would become an art teacher because she had had several influential art teachers throughout her education. To complete her degree, though, she needed to take a math class, so she enrolled in college algebra. To her surprise, she enjoyed it so much that she decided to use her remaining elective credits to take trigonometry and three courses in calculus. She soon graduated from community college with associate degrees in art and science, and she longed for an opportunity to take more math courses.

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*John Damstra is a senior mathematics major at Williams College.*

There are 12 problems on the Putnam; students have 6 hours, 3 in the morning, followed by a two hour break for lunch and to recharge, and then another 6 problems over 3 hours in the afternoon. Each is worth 10 points. Check out the problems and solutions from the 2015 compeition. The median score was a 0.

If you are interested in math puzzles and competitions e-mail Professor Palsson and come join us at the weekly math puzzle dinners on Wednesdays in Mission at 5:30 PM.

]]>Dimensions 8 and 24 are especially interesting and easy cases, because there are very symmetric, very efficient ways of packing the spheres together, so good that it makes it much easier to prove that you cannot do any better.

Read my column on dimension 8 at The Huffington Post.

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I am writing from my homeland Montenegro, a small country in the Balkans, and in this post I just wanted to give a perspective from someone who now lives far away from Williams.

I am class of 2008, and I decided to go to Williams after attending Li Po Chun UWC in Hong Kong. I always wanted to study math, the only question was whether I would choose a second major. After experimenting a bit with physics and computer science, and due to my wish to spend my junior year studying abroad, I decided to stick with “just” math. When I arrived at Williams, I didn’t quite understand the Liberal Arts concept – I couldn’t comprehend why I was not allowed to take more math courses immediately in the first year, nor did I understand the purpose of compulsory writing intensive courses given my major of choice. It took me a while to understand that those courses are essential for any career you choose, and especially for an academic.

Fast forward four years, and I made a decision to come back to Montenegro. I was at the crossroads of my life, and I thought that if I stayed in the US for graduate study as well, I would never go back. Montenegro just gained its independence in 2006, and I felt I was needed there, as well as all the other young, bright people that left to get a better education somewhere else. It took some time to adjust, but I never regretted my decision. I wanted to continue to study mathematics, so I enrolled at the Mathematics Department of the University of Belgrade in Serbia, where I was lucky to have a wonderful mentor Prof. Zoran Petrović. I initially wanted to continue with algebraic number theory, which I fell in love with during Prof. Pacelli’s course my senior year at Williams, but eventually I got into the field of combinatorial algebraic topology. The research took a couple of years, and after several conferences, publishing three papers, and giving birth to my daughter, in October 2015 I defended my doctoral thesis.

I live in the capital of Montenegro, Podgorica, where I teach at the University of Donja Gorica. I am happy here – the university shares many of the values I inherited at Williams, and gives me the freedom to make a difference. Experience from Williams influences the way I teach every day, as I try to incorporate what I learned from professors that I admired: during lectures I try to be clear and meticulous like Prof. Loepp, encouraging like Prof. Burger, friendly and accessible like Prof. Pacelli, and excited about math like Prof. Morgan. I am very proud to tell people where I got my undergraduate degree, and excited when someone has already heard about Williams.

I haven’t been back to any reunions, and in fact I haven’t been back in the US since I graduated. But I got to see several of my Williams friends during travels through Europe. I even took a couple of vacations with them—one at Lago Maggiore in Italy and one on Santorini island in Greece—and some of them visited me in Montenegro.

So, my dear Williams alumni, students, and professors, if you ever come to Montenegro, just drop me an email. I would love to meet you, show you my country, and talk about Williams.

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I’ve had a lot of fun with some cubing events recently in Williamstown. The first was an official cube event organized by local student Ric Donati. For results see

- https://www.worldcubeassociation.org/results/c.php?top3=Top+3&competitionId=WilliamsWinter2016
- http://cubecomps.com/live.php?cid=1377

The second was earlier today, a cube workshop run by myself and my son, Cameron, at the Milne Library in town. We hope to hold one on campus soon; if you’re interested please contact me at sjm1@williams.edu.

I was taught how to do the 3x3x3 by two former SMALL students of mine, Alan Chang (now at Chicago) and Umang Varma (now at Michigan). I figured as a math professor I should know how to do it. Interestingly, once you learn how to do the 4x4x4 and the centers of the 5x5x5 you know all you need to solve any sized standard cube (provided you’re patient enough!). I did compete in the 4x4x4, but sadly I never trained for speed, and was disqualified for being too slow (I solved my first in 5mins7sec, but you needed 1.30 or less on the 4).

Not surprisingly there’s a lot of great math tied up in the cube. Similar to chess, there’s notation to allow us to discuss the moves; once you master the notation you can follow along. Unfortunately I cannot upload any new images or files, so I’ve posted some quick notes on my homepage here. Here’s a few interesting tidbits.

- Every cube can be solved in at most 20 moves.
- There are lots of great websites to see how to solve different issues; for example, here’s a great one to see issues with the 4x4x4 cube (I’ll leave the 2x2x2 case to the notes I’ve posted, and give you the fun of searching for the 3x3x3).

In that post my daughter Kayla and I did a fuse bead picture of the Fibonacci spiral, and we talked about how it can be used to give a geometric proof to the sum of the squares of the first n Fibonacci numbers is the product of the n-th and (n+1)-st Fibonacci numbers.

Today we want to share our most recent project:

Hopefully you can recognize the Mandelbrot set, though it pains me that it wasn’t quite to scale. We were hanging out in Paresky selling girl scout cookies (email me at sjm1@williams.edu if you want to buy some cookies, or if you and your friends would like some fuse beads for a project), and this took from roughly 3pm to a bit before 7pm (when fortunately majors Alyssa Epstein and Sarah Fleming helped fill the final color; if others helped on this and not Kayla’s elephant please let me know so I can add thanks). I quickly eyeballed where things should be from the picture on the right; it’s at least more colorful than one of the original pictures.

Here are some great videos zooming in:

- https://www.youtube.com/watch?v=gEw8xpb1aRA
- https://www.youtube.com/watch?v=0jGaio87u3A
- https://vimeo.com/6035941 (my favorite, and my son Cam’s as well)

There are many websites you can visit to learn more about this set.

- http://math.bu.edu/eap/DYSYS/FRACGEOM/index.html
- http://aleph0.clarku.edu/~djoyce/julia/julia.html
- http://www.math.cornell.edu/~lipa/mec/lesson5.html

I’m currently working on a book celebrating the 100th anniversary of Pi Mu Epsilon with my colleague and friend, Stephan Garcia of Pomona College; a rough draft of the chapter involving the Mandelbrot set is here: pme100_BOOK_mandelbrot (comments welcome).

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“The Magic of Math” is a book on general mathematics which aims to reveal the underlying magic. What lies behind “The Magic of Math” is the same thing that lies behind magic in general: manipulation and redirection. These themes underlie the book, and Arthur Benjamin uses them well to explain the basic concepts.

One of Benjamin’s best examples of the magic in mathematics is infinite series, where evaluating a sum may lead to contradictory results. One example is the “proof” that the sum of all powers of 2 is -1.

*Gabriel Ngwe ’17 is a math major from Chicago, Illinois. He enjoys chess, fencing, and mathematical analysis.*

Numbers and basic computation appeared in Ancient Egypt as early as 2700 BCE. But you might not know that Ancient Egyptians demanded that every fraction have 1 in the numerator. They wanted to write any rational between 0 and 1 as a sum of such “unit” fractions. Such sums are called *Egyptian fractions*.

Figure 1: Egyptian fractions written on a papyrus scroll [1]

There are many ways to write 2/3 as an Egyptian fraction:

Continue reading EgyptianFractions

*Nam Nguyen ’19 plans to become a math major. He enjoys playing badminton.*

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