## Girls’ Angle Bulletin, Volume 11, Number 5

This issues’ interview is with Duke University Assistant Research Professor Heekyoung Hahn. Prof. Hahn had a remarkable journey from a rural farm in South Korea to mathematician. For most of her student years she would walk an hour each way to school. She had no toys. Instead, she would spend hours playing with math. Today, not only does Prof. Hahn play with math as a professional researcher, she also created a summer program for high school girls called SWiM.

Deanna Needell’s third installment to her column The Needell in the Haystack is about classification algorithms. How does an ATM machine read the checks you deposit into it to determine the correct amount of the deposit? It must look at the often handwritten digits and decide which digit each is. That is an example of a classification algorithm. In this case, written symbols are classified as various numerical digits. She concludes by describing some of the latest work she’s done with her colleagues on the topic. These days, the importance of such algorithms cannot be overstated.

Anna solves the math problem she began investigating two issues ago: Reduce Pascal’s triangle modulo 3 and interpret each row as a ternary number. In this way, one obtains a sequence that begins 1, 4, 16, 28, 112, 448, etc. How many of the first 2018 of these numbers are odd? This was one of the problems at this year’s SUMIT 2018.

Emily and Jasmine figure out a nice shape into which circles with radii in arithmetic progression can neatly be stacked. For a hint as to what this shape is, take a look at this issue’s cover.

Our June/July issue also contains this year’s batch of Summer Fun problem sets. As Prof. Hahn said in her interview, “Math is not something to be afraid of. Rather, it is something that you should enjoy playing with.” The Summer Fun problem sets provide an opportunity for you to play with math. This year, we have contributions from Texas A&M undergraduate Whitney Souery who is currently enrolled in Harvard’s Neonatology Summer Student Research Program at Boston Children’s Hospital and Beth Israel Deaconess Medical Center, two Harvard undergraduates, Laura Pierson and Vicky Xu, and one from newly minted Princeton PhD Matthew de Courcy-Ireland. They provide a wealth of problems to start playing with math on topics ranging from generating functions, step functions, probability, and Markoff triples. As with all Summer Fun problem sets, all members and subscribers to the Bulletin are welcome to send in their solutions and comments!

Also in this issue, a Meditate to the Math on Napoleon’s theorem and some Notes from the Club featuring the second end-of-session Math Collaboration created by mentors at Girls’ Angle, this one made by Northeastern graduate student Jacqueline Garrahan and MIT undergraduate Elise McCormack.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## Girls’ Angle Bulletin, Volume 11, Number 4

Betsy Stovall, assistant professor of mathematics at the University of Wisconsin-Madison, kicks off Volume 11, Number 4 with a wonderful interview. Her story of how she became interested in math is inspiring. She also describes one of the nifty problems that sparked her interest in math when she was a student. If you haven’t seen it before, it’s well worth thinking about.

Deanna Needell adds a second installment to her column The Needell in the Haystack, this time about filling in missing entries of a matrix. She describes an application to the study of Lyme disease. To fully understand Prof. Needell’s articles, one does have to be familiar with matrices. For those of you who don’t know about matrices, we included a very brief introduction to matrices intended to help reader’s unacquainted with the concept at least feel good enough about them to read Prof. Needell’s Bulletin contributions. Her articles are very much worth reading even if you don’t know about matrices.

Emily and Jasmine embark on a new adventure resulting from a graphic design project themed on the circle. The assignment inspired Emily to create a design of radiating circles. To create the design, she discovered some neat math. This new journey was inspired (in a somewhat indirect way) by work of students at the Buckingham, Browne, and Nichols Middle School.

Anna makes progress on her current problem by characterizing those $m$ for which ${2m} \choose m$ is congruent to 1 modulo 3.

Lightning Factorial authors an installment of Math In Your World, analyzing how to intercept an incoming tennis shot as quickly as possible.

Next, Girls’ Angle member Allie presents her wonderful inductive proof of a conjecture she came up with at the Girls’ Angle club regarding the largest number expressible using N ones, addition, multiplication, and parentheses.

We round out the issue with some Notes from the Club.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## Girls’ Angle Bulletin, Volume 11, Number 3

Can you figure out what the cover represents?

Volume 11, Number 3 kicks off with an interview with Rhonda Hughes, Professor Emeritus of Mathematics at Bryn Mawr College. In it, Prof. Hughes describes one of the most interesting paths into mathematics, and she gives lots of good advice to students.

Next up is the concluding half of Partitions from Mars, by Pamela E. Harris, Alexander Pankhurst, Cielo Perez, and Aesha Siddiqui. They conclude with an open problem about partition functions. Can you solve it?

Last issue, we interviewed UCLA Professor of Mathematics Deanna Needell. In this issue, Prof. Needell returns with a nifty article on topic modeling, explaining the technique of non-negative integer matrix factoring. Every day, people produce something on the order of $10^{19}$ bytes of data every day, and this rate is accelerating. But all this data is useless if we don’t have the tool to extract information from it. In The Needell in the Haystack, Prof. Needell shares with us her expertise and teaches us the basics of large-scale data analysis.

Anna’s Math Journal returns after a brief break to tackle a question about numbers formed by interpreting rows of Pascal’s triangle, modulo 3, as ternary numbers. Specifically, in the resulting sequence, which numbers are odd? Computing how many of the first 2018 of these numbers are odd was a problem at SUMIT 2018, a big math event we ran at the Broad Institute earlier this month.

In Errorbusters! we discuss a powerful way to minimize numerical computation errors: use variables. The use of variables delivers several benefits besides just reducing the number of dicey numerical computations. We illustrate by tackling problem 15 from the 2017 AIME I contest.

I’ve now been working with students on math for decades, and sometimes, I meet students who treat all facts equally, as if each were another isolated item to learn about this world we live in. But what is amazing about math is that some facts follow logically from others. Building connections between facts through implication enables us to understand a lot with little knowledge and helps us to retain facts because we can then understand them from many angles. To progress in mathematics, it is crucial to understand this. So I tried to address it directly with the article Implications.

We conclude with a Learn by Doing problem set on Fibonacci numbers, a note on the notation used for functions, and some notes from the club.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## Girls’ Angle Bulletin, Volume 11, Number 2

You could call this issue an ode to the triangle because half the content pertains to that most basic of geometric shapes, including the cover.

However, we open with an interview with UCLA Professor of Mathematics Deanna Needell. One of Prof. Needell’s mathematical interests is about how to get accurate views of things with less information, such as getting useful MRI scans in less scan time.

Next up is a nifty way to introduce vectors via partition functions by a group from Williams College, led by Pamela E. Harris. They motivate the whole concept by imagining a fictitious martian currency which behaves just like our dollar, except that there’s more than one unit of currency and they cannot be interchanged. The setup inevitably leads them to begin computing what is known as Kostant’s partition function for $A_3$. In fact, the authors dedicate the article to the memory of Bertram Kostant who passed away earlier this year.

Inscribed equilateral hexagons in an isosceles right triangle.

Milena Harned and Miriam Rittenberg are back explaining some results they found and proved this past summer about inscribing equilateral hexagons in triangles in such a way that every side of the triangle is flush with at least one side of the hexagon. In fact, for every triangle, they found a continuous family of such equilateral hexagons and they “rotate” about inside the triangle. You can explore these inscribed equilateral hexagons using a JavaScript app on the Girls’ Angle website.

Girls’ Angle mentor Ashley Wang pays tribute to triangles through her Math Buffet.

After an installment of Meditate to the Math (which asks you to interpret Sierpinski’s triangle as a sequence of binary numbers and contemplate its patterns), we present a self-referential test created by four Girls’ Angle members: Ghost Inthehouse, HolAnnHerKat, Katnis Everdeen, and Shark Inthepool. Earlier this fall, they solved Jim Propp’s self-referential multiple choice test and enjoyed it so much, they wanted to create their own. Their work was primarily overseen by Girls’ Angle mentor Rachel Burns.

Emily and Jasmine’s epic search for nice triangles comes to a heady conclusion. They succeed in putting all the pieces of the puzzle together to get a fairly complete classification of triangles with various numbers of integer side lengths and angles that measure rational multiples of $\pi$. Their journey took them through some lattice combinatorics, rational parameterizations of curves, and some basic Galois theory and a study of the values of cyclotomic polynomials at the square root of -1. And it all began with a fake triangle.

We conclude with some notes from the club.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## Get ready for a brand new math adventure – SUMIT 2018!

SUMIT 2018 is a fun, fully collaborative, math adventure for girls who like math in grades 6-10. Registration is open now. We’re running the event twice on February 3 and 4 at the Broad Institute in Cambridge.

SUMIT 2018 is designed not only to be a super fun, mathematically-interesting challenge for participants, but also to give participants a great opportunity to meet other girls who like math, build lasting friendships, and develop leadership skills. We’re creating the stage, but it’s up to the participants to take control of their destiny!

## Girls’ Angle Bulletin, Volume 11, Number 1

We begin the second decade of the Bulletin with a fascinating image by Arnaud Chéritat, CNRS/Institut de Mathématiques de Toulouse, called Two mating polynomial Julia sets. For more about these images, visit his website. The two images by him (on the cover and inside) were included at the urging of Sarah Koch, this issue’s interview subject. Prof. Koch is Associate Professor of Mathematics at the University of Michigan. She studies complex analysis, Teichmüller theory, and complex dynamics. In her interview, she describes a game you can play called the “chaos game”. If you play the chaos game long enough, you will create fractals. We include some images of such fractals right after the interview.

Recently, Girls’ Angle member π has been learning about integrals and decided to set herself the task of computing the center of mass of a semicircle of uniform mass density. One question led to another, and before we knew it, we had stumbled upon Euler’s formula

$\displaystyle \frac{\sin x}{x} = \prod_{k=1}^{\infty} \cos(x/2^k)$.

We retrace our journey in Pac-Man Meets Euler.

In Volume 10, Number 3 of this Bulletin, Addie Summer explained how she found the quadratic formula. As it turns out, Lightning Factorial also figured out the quadratic formula without having to be taught it. She explains her method in The Quadratic Formula, Revisited.

In Anna’s Math Journal, Anna succeeds in finding a way to show that the number of special tilings of a rectangle are counted by the Catalan numbers without using generating functions. This represents the culmination of 6 installments’ investigation.

Next comes a special Math Buffet. We asked a number of mathematicians to contribute an excerpt from their scratch work to give us a window on what it looks like when they are in the act of creating mathematics. A huge and heartfelt Thank You to Timothy Chow, Brendan Creutz, Danijela Damjanović, Laura DeMarco, Ellen Eischen, Elisenda Grigsby, Kathryn Mann, Elizabeth Meckes, Maria Monks, Radmila Sazdanović, Marjorie Senechal, Bianca Viray, Fan Wei, Kirsten Wickelgren, Lauren Williams, and Helen Wong for allowing us a look into their personal process of doing math. Special thanks to Ashley Wang for doing the layout.

Emily and Jasmine are getting very close to resolving their long-standing search for nice triangles. In this installment, they succeed in computing the constant terms of all the minimum polynomials of cosines of rational multiples of π. By Vieta’s formulas, this is equivalent to computing

$\displaystyle \prod_{(k,n)=1, 0 < k \le n/2} \cos(2\pi\frac{k}{n})$

for all $n > 1$.

We conclude with some notes from the club.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!