## Girls’ Angle Bulletin, Volume 13, Number 2

This issue’s interview is with Prof. Raegan Higgins, associate professor of mathematics at Texas Tech University. Prof. Higgins went to college and graduate school with Prof. Christina Eubanks-Turner, who was our interviewee in the previous issue of the Bulletin. The two are the first two African-American women who achieved a doctoral degree in mathematics from the University of Nebraska Lincoln. We consider ourselves extremely fortunate to have had interviews with both of these remarkable women and to be able to present them to you in back-to-back issues.

Deanna Needell returns with a fascinating installment of The Needell in the Haystack which introduces neural nets and deep learning. Today, algorithms are capable of creating made-up human faces that are quite convincingly real. (Check out the faces at Generated Photos and see if you can tell which ones are fake.) Prof. Needell indicates how this is done.

Next, comes a clever self-referential True/False quiz by Michelle Chen. Self-referential tests are logic puzzles where there is a unique way to answer all the questions and have all the answers be correct. You don’t have to know any trivia because the statements refer to themselves, hence the name “self-referential.” It’s not that easy to come up with an interesting self-referential test that has a unique correct answer. If you like these, also check out the one by GhostInthehouseHolAnnherKatKatnis Everdeen, and Shark Inthepool, on pages 20-21 of Volume 11, Number 2 of this Bulletin. Can you solve Michelle’s?

Emily and Jasmine are giving themselves a thorough understanding of the areas of the shapes created by a double zigzag pattern across a rectangle. In this issue, they are able to determine all triangles of “type T” (as they call them) in such patterns by using a clever counting argument that spares them from a lot of computation.

Some members at Girls’ Angle have been thinking about and making perspective drawings. In Perspective On Perspective Drawing, Addie Summer takes a step back to explain the reason for mathematics in this subject. If you haven’t thought carefully about perspective drawing, the mathematics is actually rather subtle and quite interesting. (For example, the harmonic mean appears in a natural way in perspective drawing. See Math In Your World: Art and the Harmonic Mean on page 19 of  Volume 10, Number 4 of this Bulletin.) It’s already a challenge to produce a perspective drawing of cubes (see the cover).

If you like tennis, you’ve probably been thrilled with the relatively new Laver Cup tournament, which takes place two weeks after the US Open. In Laver Cup Scenarios we analyze how the very design of the tournament works to generate excitement.

Finally, we conclude with some Notes from the Club, which are authored by our Head Mentor Grace Work. In this one, you’ll find a few problems from our traditional end-of-session Math Collaboration which was designed and created by Girls’ Angle mentors Jenny Kaufmann and Laura Pierson.

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!

Also, the Girls’ Angle Bulletin is a venue for students who wish to showcase their mathematical achievements that go above and beyond the curriculum. If you’re a student and have discovered something nifty in math, considering submitting it to the Bulletin.

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 13, Number 1

We open with an interview with Loyola Marymount Associate Professor of Mathematics Christina Eubanks-Turner. Prof. Eubanks-Turner is a graduate of Xavier University of Louisiana and received her doctoral degree in mathematics at the University of Nebraska-Lincoln under the supervision of Sylvia Wiegand. She is an expert in commutative algebra and is also actively involved with mathematics outreach. Our interview with Prof. Eubanks-Turner was conducted by Wellesley College undergraduate Melissa Carleton.

Next, we have a delightful story by King’s College Professor of Mathematics Konstanze Rietsch who also served as illustrator.  You could say that the story is about a mathematically-inclined architect, or it’s about a nasty queen and her spoiled children, or it’s about a Diophantine equation, which is an equation to be solved in integers. And if Diophantine equations are your thing, you can also try your hand at solving Diophantine equations related to the Pythagorean equation in Another Diophantine Equation on page 25.

In between, Emily and Jasmine make steady progress at understand the pattern created by two zigzags across a rectangle, and there’s a Learn by Doing on using complex numbers to study plane geometry. Included in this Learn by Doing is a very brief introduction Möbius transformations, the topic that inspired this issue’s pumpkin cover.

We close with Notes from the Club, which are now being written by our recently hired Head Mentor, Grace Work. The club continues to be abuzz with mathematical activity, and we’re pretty confident that we’ll be showcasing member works in the Bulletin pretty soon.

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!

## Why the Girls’ Angle Bulletin?

Running Girls’ Angle, like most nonprofits, is a ton of work. There’s a club to run, outreach activities such as SUMIT to create, organize, and operate, and there’s fundraising and all the other aspects of maintaining a nonprofit.

So why, on top of that, do we produce a math magazine?

The answer is that the Bulletin is a critical component of Girls’ Angle’s math educational strategy for multiple reasons. I’d like to detail one of the more important of these reasons:  to provide more venues to showcase student achievement in mathematics.

Today, the math competition dominates extracurricular math, so much so that many consider winning a math competition to be the only way to show high achievement in math. Some go further and think that without stellar contest performance, they have no future in math. This is unfortunate because math competitions are an imperfect measure of mathematical ability. Just to list a few causes of this imperfection, math competitions

• place too much weight on computational accuracy and speed
• are generally confined to a limited bit of mathematical knowledge
• feature canned problems designed to be solvable within certain time constraints
• favor the ability to apply results over understanding them
• do not test for the ability to come up with good questions

Mathematics is about unraveling the mysteries of the unsolved. And it doesn’t matter how long it takes to do that. In fact, if you’re conditioned to always look for a quick, nifty solution, you’re likely to become frustrated with serious mathematical research.

If contests are the only venue to showcase mathematical ability, many mathematical talents will be forever hidden. Math educators must furnish alternative ways for students to show their mathematical achievement.

Enter the Girls’ Angle Bulletin. Students who have explored and come to a good understanding of some piece of mathematics can write up their observations and publish them in the Bulletin.

Perhaps you question the need for such a magazine since there are many math journals out there already. But the vast majority of those math journals are for professional mathematicians, and it is not reasonable to expect K12 students to produce mathematics of sufficient interest to professional mathematicians to warrant publication in those journals. It does happen, but it is rarer than qualifying for the USAMO.

Note that this absolutely does not mean that the Bulletin will only contain expository material. K12 students are fully capable of discovering new mathematics. What we can’t expect is that the math that a K12 student discovers will be something that a professional mathematician would find sufficiently interesting to justify publication in a professional math journal. (Though, as mentioned, it can happen, and I think there are some things like that already in the Bulletin.)

Academics have also recognized this problem in the founding of the journal Involve, which is about “bridging the gap between the extremes of purely undergraduate-research journals and mainstream research journals,” and “provides a venue to mathematicians wishing to encourage the creative involvement of students.” Though one difference between Involve and the Girls’ Angle Bulletin is that Involve involves undergrads whereas the Bulletin targets K12. (Note: MIT math professor Bjorn Poonen is both on the editorial board of Involve and the advisory board of Girls’ Angle.)

There are already several examples of student written articles in the Bulletin. Just to cite one, Milena Harned and Miriam Rittenberg wrote up their discoveries about equilateral hexagons inscribed in triangles. (See page 12 of Volume 11, Number 2.) To the best of our knowledge, their results are new. They showed that there’s a one-parameter family of equilateral hexagons inscribed in any triangle with the property that each of the three sides of the triangle are flush with at least one of the sides of the equilateral hexagon. For a professional mathematician, this result may be amusing to learn, but doesn’t shed light on the deep conundrums that keep mathematicians up at night. On the other hand, it’s definitely something that demonstrates above average mathematical creativity and ability, especially when you bear in mind that Milena and Miriam not only proved the result, but discovered it as well. (That is, they were not handed a conjecture and asked to prove it. They had to create the conjecture too.)

So, K12 students! If you discovered or did something nifty in mathematics, consider writing it up and submitting for publication in the Bulletin. We’d love to hear from you!

## 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!