## Girls’ Angle Bulletin, Volume 12, Number 6

From Day One, Girls’ Angle has wished to hire a woman with a doctoral degree in mathematics as Head Mentor at our club, where girls explore mathematics under the guidance of our stellar mentors. This wish is now a reality with the hiring of Grace Work as our new Head Mentor, and this issue’s interview is with her.

The dream is to create a professorial class whose teaching duties pertain to the K12 arena instead of college/grad. A main reason for this dream is the observation that many girls come to like math when they do mathematical research. At Girls’ Angle, many girls have come up with their own interesting math questions and have embarked on multi-month journeys as they sought answers, and that’s what math research is. Another reason is to have a Head Mentor who has a research mathematician’s understanding of mathematics and experience tackling unsolved problems.

Next, Deanna Needell gives her take on the stable marriage problem. This is Needell’s 10th installment of her column Needell in the Haystack.

Emily and Jasmine continue discovering beautiful facts about the pattern created by two zigzags running across a rectangle. They keep finding neat things and it sure feels like they’re going to stumble on a nice mathematical gem soon – stay tuned!

We close with the solutions to last issue’s Summer Fun problem sets, which include quite a bit on ordinals, including a result of Paul Erdős concerning the maximum number of different ordinals one can obtain by adding up ordinals in different orders.

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 12, Number 5

This issue’s interview subject is Tanya Leise, Professor in the Mathematics and Statistics Department of Amherst College.  Tanya uses math to study circadian rhythms. She also has a mathematically gifted daughter and in this interview, we ask her about best practices for raising a mathematically gifted child. One of the tools in Tanya’s mathematical toolkit is the wavelet. To enable readers to get an even better idea of what a wavelet is, she also contributed the first Summer Fun problem set in this summer’s batch of Summer Fun problem sets.

Next, four students explain some of their discoveries about what happens when you fold a rectangular strip of paper in half, over and over. You’ll create a model with several layers. Exactly how are these layers ordered? They give a comprehensive answer. Their work inspired this issue’s cover, which represents a rectangular stripped folded in half 6 times to create 64 layers. (By the way, it’s a myth that you can’t fold a paper in half more than a certain number of times. It depends on the thickness and length of the paper.)

Deanna Needell delves deeper into graph theory with her 8th installment of The Needell In The Haystack where she defines the chromatic number of a graph and establishes some basic bounds on its size.

In addition to Tanya’s Summer Fun problem set, Whitney Souery, Laura Pierson, and Matthew de Courcy-Ireland return to give us two more, one on sine and cosine and one on ordinals. Whitney’s Sine and Cosine is designed for anyone who has not yet learned about the sine and cosine function but would be interested in challenging themselves to learn about them by solving problems. Laura and Matthew introduce ordinals, then provide a series of problems that recover work of Paul Erdős in How High Can You Count?

We conclude with brief 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 12, Number 4

We open with the fourth and final part of our interview with mathematician Dr. Kristin Lauter, a professor at the University of Washington and a principal researcher at Microsoft Research. In this segment, Dr. Lauter addresses gender issues in mathematics and gives advice to students. We hope you enjoyed this 4-part interview with Dr. Lauter.  We certainly did! A huge Thank You to Dr. Lauter and to Ke Huang for conducting the interview, which took place in April, 2018 at the University of Washington.

Next up is another wonderful installment of The Needell in the Haystack by Deanna Needell, this one on P vs. NP, the traveling salesperson problem, and Hamiltonian paths.

Emily and Jasmine continue their mathematical adventures getting deeper into their exploration of zigzags across rectangles. They’re making steady progress, increasing their knowledge of the patterns produced.

Then we have a peculiar problem set designed to induce you to think more conceptually about mathematics. Each of the problems in our “Anti-Calculator” game can be solved with a minimum of computation. In fact, you might find that you can solve them all entirely in your head and would encourage you to try.

We have an installment of Learn by Doing on the standard form of a line. If you’re a veteran of lines, most of this will be familiar, but perhaps the last two problems will not. If you’d like to try the last problem without seeing the result (which is in the problem statement), find a formula for the area of a triangle bounded by the lines $A_1x+B_1y=C_1$, $A_2x+B_2y=C_2$, and $A_3x+B_3y=C_3$, assuming that no two of these lines are parallel.

We then show how to see that the area under $1/x$ gives the logarithm without using calculus.

We cover the floor and ceiling in Notation Station, and close with a few 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!

## A Student’s Perspective on a Math Collaboration

Isabella-Marie participated in a Math Collaboration created and run by Girls’ Angle at her school in November, 2018. She wrote about it for her school’s journal, The Spark. The article below is reprinted with permission from the Buckingham, Browne, and Nichols School and the author, Isabella-Marie Selden from the Winter 2018-2019 edition of The Spark, page 5. For more information on Girls’ Angle Math Collaborations, visit our website.

# Girls Math Collaborative

by Isabella-Maria Selden

In November, the Girl’s Math Collaborative took place once again. The Collaborative is an annual event where girls in the Middle School come together to solve a variety of math problems to unlock a prize.

I participated in my first math collaborative last year. When I was first introduced to the idea, it sounded intriguing. But math is not my strongest subject, and I was tentative to sign up because of this. Regardless, I ended up joining and having the best time.

First, we were greeted with loads of food ranging from cheese and crackers to Oreos and fruit. We then received multiple packets of math problems, and another sheet to write our final answers on. The answer sheet helped to arrange the answers so they corresponded to another puzzle, that would eventually give clues to the lock combination needed to unlock the prizes. Once we solved all the problems on the packet and put them on the answer sheet, we were beyond proud of each other – but another problem stood in our way: the lock itself di not have any numbers. We proceeded to understand how the answer sheet corresponded to the lock, and we eventually succeeded and opened the chest with many surprised inside.

In addition to the math, the biggest aspect of the collaborative is teamwork. Since many problems and puzzles are built off of previous ones, if one question is answered incorrectly, the whole solving process can lead in the wrong direction. As a result, the girls were constantly working with peers to check answers, or to split up into groups to tackle different sets of problems.

It was amazing to see the teamwork between the girls; some were writing on the board together, others were carefully checking their math, and when someone ran out of ideas, other people involved would help that person to think of a new solution. I was so glad to be participating in this experience, and who knew math could be so fun?

The next math collaborative will take place on February 27th. Although it is fully booked, there will be many opportunities to join (for the current seventh graders) next year!

## Girls’ Angle Bulletin, Volume 13, Number 3

Our cover features a tessellation by two nonconvex tiles designed by Katherine Dawson. More on her in a bit.

We open with the third part of a multi-part interview with mathematician Dr. Kristin Lauter, a professor at the University of Washington and a principal researcher at Microsoft Research. In this segment, Dr. Lauter discusses more recent develops in cryptography and how quantum cryptography might affect security. She also discusses the differences between academia and industry.

Next up, an article from student Marlie Kass who gives us the low down on the mathematics lurking behind a contest problem from the Mathematical Association of America’s AMC 12 series. In particular, she shows that the contest problem is an application of Bézout’s lemma. Bézout’s lemma is important because it tells us what numbers are invertible, modulo n.

This semester, one of our fifth grade members, Katherine Dawson, got really excited about tessellations and ended up designing an entire tessellating font, not just for the letters, but also for the digits. Her entire font is shown here as well as an indication of how each character can be used to tile the plane. Special thanks to Girls’ Angle mentors Amy Fang, Adeline Hillier, and Elise McCormack-Kuhman for very rapidly preparing computer graphic versions of Katherine Dawson‘s work for the Bulletin.

Next, we present two beautiful facts about Pascal’s triangle for you to meditate on and figure out. Both facts involve overlaying one Pascal’s triangle over another, hence the subtitle: Pascal on Pascal.

Emily and Jasmine further their understanding of partitions of rectangles created by regular zigzags.

Hyperia gets her pet ant to walk about the surface of a hypercube as we watch its shadow to help us improve our understanding of the fourth dimension, and this is followed by a Learn by Doing on 4D polytopes. If you’ve ever been curious about the 4D versions of the Platonic solids, but haven’t had a chance to get around to thinking about them, check out this problem set. You’ll find each regular 4D polytope or its dual here.

We close with a few 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 12, Number 2

We open with the second part of a multi-part interview with mathematician Dr. Kristin Lauter, a professor at the University of Washington and a principal researcher at Microsoft Research. In this segment, Dr. Lauter gets into the nitty-gritty details of the Diffie-Hellman protocol for public key exchange and begins to explain what elliptic curves have to do with cryptography.

Next, Prof. Needell teases us with some tantalizing probability questions to underscore just how subtle and surprising probability can be.

Alana Axelrod-Freed, Milena Harned, and Miriam Rittenberg show how they proved a nifty result pertaining to paper folding that they discovered last summer. They were exploring the so-called “stamp folding problem” which asks for the number of ways a row of n stamps can be folded up by folding along the creases between stamps. This is an unsolved problem. However, they restricted to counting a certain subset of the folds and were able to get an explicit answer.

Emily and Jasmine begin a new math adventure exploring the areas of regions obtained by drawing zigzags across the face of a rectangle.  The cover shows an example of such a dissection. Will they find any interesting patterns? What patterns will they find?

Next, we have an article that aims to help those who are struggling to understand a fourth spatial dimension. The strategy presented is a pair of parallel dialogues that is based on work I did with eighth graders at the Buckingham, Browne, and Nichols Middle School that seemed to be fairly effective at helping them move into the fourth dimension.

Milena and Miriam return with the second part of our article on Umbrellas. Here we complete the proof of a characterization of the locus of points reachable in n unit-length step from the origin, such that each step has a nonnegative vertical component.

We close with a few 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 12, Number 1

Through the years, Dr. Kristin Lauter and Microsoft Research have been major financial supporters of Girls’ Angle, especially, our Math Collaboration initiative. Dr. Lauter is a Principal Researcher at Microsoft Research specializing in number theory and cryptography. We are thrilled to embark on a multi-part interview with Dr. Lauter in this issue. The interview was conducted in person by Ke Huang, a graduate student in the department of applied mathematics at the University of Washington. The interview was transcribed by New England Transcription Services with further assistance from Harvard math graduate student Peter Park.

In the previous issue, Prof. Needell showed how unintuitive high-dimensional geometry can be. In this issue’s installment of Needell in the Haystack, she exploits this unusual geometry to explain how to solve a problem in compressed sensing. In particular, she uses the fact that a corner of a high-dimensional hyperoctahedron has smaller and smaller high-dimensional solid angle. If you read this article and have trouble seeing this fact about hyperoctahedra, have no fear, Anna investigates the question in Anna’s Math Journal

Next up is a geometric result that Milena Harned, Miriam Rittenberg, and I stumbled upon a little over a year ago. The question is: For a fixed positive integer n, what is the locus of points that you can reach in a plane in n steps, where a step is one unit long and must have nonnegative vertical displacement. In Umbrellas, Part 1, we explain what the locus is for all n and prove it for n = 1 and 2. In the sequel, we will provide a proof for all n.

Harvard undergraduate Michael Kleistra gives us a chance to learn all about cardinality in a step by step installment of Learn by Doing.

Emily and Jasmine wrap up their investigation into stacked circles, seeking an elusive container for a stack of circles whose radii form a harmonic progression.

Addie Summer does more math while waiting for a bus.

And we close with a few 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!

## Head Mentoring at Girls’ Angle

Thanks to a grant from the Mathenaeum Foundation, Girls’ Angle is now in the process of hiring a full-time Head Mentor for a minimum of two-years. We are looking for a mathematician who loves to work with K12 students, especially girls.

To give prospective applicants a better idea about the position, the following describes in more depth the educational philosophy of Girls’ Angle and explains the critical role of the Head Mentor.

Girls’ Angle’s approach to math education

We believe that everyone benefits by studying more math, not just those with a special interest in math. Mathematics is a fabulous vehicle for improving one’s ability to think and solve problems, for no other subject shows up errors in reasoning so well. And the best way to obtain these benefits of studying math is to do math.

Consequently, Girls’ Angle welcomes all girls, currently in grades 5-12, to our club. Our members have diverse relationships to math. Some joined because they love math and can’t get enough of it. Others joined because they feel weak in math but would like to improve. And, yes, there are members who hate math and attend only because their parents want them to. For all these members, we aim to provide a safe, friendly, comfortable environment where they can feel at ease and not self-conscious, so they can focus on math without distraction.

To ensure the best experience for such diversity, our mentors need to be flexible, because what works for one girl may fail miserably for another. Some girls are perfectly happy to be given a challenging math problem and then given lots of space to think on it. Other girls need more guidance. Some members are motivated by competition, others by collaboration. Some members gravitate toward the concrete, while others revel in abstractions. Members have diverse personalities and hold a variety of different interests. Consequently, the stumbling blocks each member naturally encounters in the process of studying math are unique and fascinating. It’s a rich tapestry that also changes with time, even for the same member.

The role of the Head Mentor

Our Head Mentor is responsible for sorting out all this diversity and figuring out what would be best for each member, as an individual, to grow in thinking ability, problem-solving ability, mathematical knowledge, and mathematical understanding as effectively as possible. It is a big job, but it is extremely rewarding, and the Head Mentor has a lot of tools at her disposal to accomplish this task.

The first and most important tool is our group of super mentors, the heart and soul of Girls’ Angle. From the beginning, Girls’ Angle has been blessed with fabulous mentors who are excellent role models. They range from undergraduates majoring in math and related fields, to graduate students in math and related fields, to postdocs in math – each of them possesses strong fundamentals for their respective level in academia. The Head Mentor recruits, coordinates, and works with the mentors to deliver the highest quality math education we can muster.

Second, there is the enormous breadth and depth of mathematics itself. In our view, what is more important than the specific math being studied is to study math in the first place. Rather than insist that a member learn a particular piece of math, we prefer to help a member find some aspects of or problems in mathematics that resonate with her. The beauty of this approach is that mathematics is highly interconnected so that if a person gets hooked on some nice piece of mathematics (and it could be something that is never taught in the standard curriculum), it won’t be long before they branch out and pick up all the standard material. So we have all these members each navigating a unique path through the world of mathematics under the guidance of our mentors, who are, in turn, all coordinated by our Head Mentor.

Should a member become serious about mathematics and begin to contemplate making mathematics a profession, then it does become important for her to develop the discipline to learn important material that may not immediately appeal. When and how to develop this discipline is another matter that the Head Mentor must sort out. Ideally, the student’s own desires provide sufficient motivation to put the nose to the grindstone, but there certainly can be a region of transition.

Why do we need a mathematician for the Head Mentor position?

This is an important question, and one that is not easy to answer completely in a blog post because there are multiple reasons.

Members are a diverse group and represent many different stages in mathematical development. While it is not necessarily immediately appropriate for all members, one of the ideals we aim for is to help members develop into independent and capable thinkers who can solve the yet unsolved. We hope that members who spend some years with us are equipped with the tools and attitude to go into this world and contribute to the solutions to hard problems that have so far stymied us. Solving the unsolved requires creativity, persistence, and an ability to handle psychologically trying conditions. Having tackled unsolved problems and succeeded in creating new mathematics, mathematicians possess these qualities, and because we aim to impart these qualities to our members, we need a Head Mentor who possesses them and knows how to convey them, as well as help and/or facilitate our mentors to do the same.

Often, members don’t yet possess the vocabulary or language skills needed to express their thoughts well, but they do have precious thoughts. Our mentors have to have a radar for member thoughts, however ill-formed they may be, and be able to encourage them to pursue those thoughts. It may begin with helping a member to sculpt her thoughts into something mathematically actionable, or helping them learn to break a question down into tractable pieces, etc. A mathematician is practiced in this art. What we do not want at Girls’ Angle is for a member to have the inkling of an idea, try to express it, but then somehow lose that thought in the wind.

In fact, this also explains why we need excellent mentors for all our members, whether they excel at math or are floundering. It often takes a great deal of mathematical insight to figure out what a struggling member is having trouble with and how to help the member find a more effective and productive path; and it is so important for struggling members to get fundamentally sound guidance. When a member is bewildered, it does not help for her to have to deal with added layers of confusion created by poor instruction. A mathematician has thought about math to an unusual depth, and with that depth comes greater perspective for the relative importance of various concepts, an understanding of which descriptions of ideas have more generality than others, a knowledge of ideas that may seem expedient but lead to long-term confusion, etc. Our Head Mentor marshals her understanding of math to help our mentors help struggling members in ways that give them the best chance of future success.

There are many other such examples. Math is alive at Girls’ Angle. Mathematics is not a fossilized subject. It is a creative, conceptual art, and our Head Mentor must not only be an ambassador for this art, she must know how to practice it herself and be able to pass this art on to the next generation in an effective way.

## Girls’ Angle Bulletin, Volume 11, Number 6

It’s always nice to hear from old friends. Katy Cook née Bold was the first author for Math In Your World, a column about applied math. She contributed 8 installments before turning over the reins to Katherine Sanden. The cover features a rug that Katy designed together with Sara Eizen. What’s its mathematical significance? If you’ve made a math-themed carpet, we’d love to see pics. If we get enough, we’ll devote an installment of Math Buffet to them.

Rachel Pries gives us a wonderful interview for this issue. Rachel is a Professor of Mathematics at Colorado State University and received her doctoral degree in mathematics from the University of Pennsylvania under the supervision of David Harbater. Also, she is an alumna of Cambridge Rindge and Latin, which is just down the road from Girls’ Angle.

Next up, the fourth installment in Deanna Needell‘s series The Needell in the Haystack, about big data algorithms. Big data generally means high dimensions. For example, a very small 100 pixel by 100 pixel image can be thought of as a vector in a 10,000-dimensional space. In this installment, Deanna explains an unintuitive feature of high-dimensional spheres.

Addie Summer gives us an example of how she entertains herself in idle moments with mathematics in Systematic Counting, Part 1.

Then, we follow Emily and Jasmine on their continuing exploration of stacked circles. This time, their on a quest for stacked circles whose radii form a harmonic progression, but they run into a massive snag.

We conclude with solutions to the four Summer Fun problem sets of the previous issue. The best way to learn math is to do math, and tackling math problems is a good way to start doing math. But also, tackling math problems is a good way to develop your problem-solving skills. There is so much more to math problem-solving than being familiar with mathematics. I have seen students who know all the math involved with a particular problem, yet fail to solve the problem because of organizational issues. There are also psychological factors to learn to deal with. I don’t know how many times I’ve seen, and been victim to, an unwillingness to explicitly work out the simplest case of something, thinking that it is too trivial to pay attention to, and have that turn out to be a key obstacle. Once the reluctance is overcome and the seemingly trivial example is carefully worked out, the solution to the rest suddenly appears.

Special thanks to our four 2018 Summer Fun contributors, Matthew de Courcy-Ireland, Laura Pierson, Whitney Souery, and Vicky Xu!

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!