## Girls’ Angle Bulletin, Volume 16, Number 3

We have been incredibly fortunate to have had an interview with a woman in mathematics in every single issue of the Bulletin but one. All of these women are remarkable and have fascinating stories to tell. They’re wonderful role models. In Volume 16, Number 3, we interview Wellesley Professor of Mathematics Megan Kerr. Aside from this interview, Prof. Kerr has been a valued source of advice and has been supportive and helpful to Girls’ Angle in vital ways. Prof. Kerr is also a Wellesley alumna.

The cover is created by Juliette Majid who brilliantly illustrates the mathematical structure that Emily and Jasmine have been unravelling in their latest math adventure. Juliette also designed the cover of Volume 14, Number 6. To understand the meaning of the cover, read the 3rd installment of Romping Through the Rationals where Emily and Jasmine crack the mystery of a sequence they learned about on the internet that supposedly lists every single nonnegative rational number exactly once.

The Emily and Jasmine series aims to show how math is done. By contrast, in school, we are taught what math has been done, and it can be easy to forget that all the math that has been done once did not even exist. Someone had to create it. Professional mathematicians spend much of their time trying to create new math. We can all be creators of new math. As you read Emily and Jasmine, ask yourself if there’s anything they do that you cannot picture yourself doing?

In the same spirit, Addie Summer tells us about her adventure counting domino tilings of a zigzag path in a park. Before reading the whole article, we urge you to read just as far as the statement of the problem she considers, then try to solve the problem yourself. There are many ways to go about solving this problem, and you’ll likely find something different from what Addie does. (And if you do, tell us about it!)

Robert Donley continues his series on counting and partially ordered sets, this time stumbling upon multinomial coefficients and the generalization of Chu-Vandermonde convolution to them.

We also give our take on the oft-asked question, “What is $0^0$?” Often, people will say that it is 1 and explain how this is because $\lim_{x \to 0^+} x^x = 1$. However, it’s important to understand that this is not a proof that $0^0 = 1$. (Likewise, observing that $\lim_{x \to 0^+} x/x = 1$ does not imply that $0/0 = 1$!) Never forget that math is a creation of the human mind. We are the creators of math, and you can be too!

We close with some Notes from the Club where you can find a little-known way to produce the graph of the cosine function by using a sheet of origami paper.

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 16, Number 2

Volume 16, Number 2 opens with an interview with Professor Ellen Goldstein. Ellen is an Assistant Professor of the Practice of Mathematics at Boston College. She is a graduate of Skidmore College where she double majored in math and dance. She continues to dance today, as a member of Dance Prism. Her interest in communicating mathematics led her to develop and teach a calculus course for underprepared STEM majors.

The cover illustrates exponential growth and shows all the solid pool balls plus a cue ball. Ball number $n$ appears $2^n$ times. What percentage of balls are 8-balls? For more on exponentials, see this issue’s Learn by Doing, which explores why exponentials were created, how they are defined, and how they are extended to all real exponents.

Anna Ma writes about a very topical subject: text generating AIs. It’s getting harder and harder to tell whether a text was written by a human or a computer. How do these AIs work?

Robert Donley continues his series on counting by introducing “up” operators.

Emily and Jasmine continue their romp through the rational numbers. While thinking about the sequence of rational numbers that they learned through social media, they became interested in finding a sequence of nonnegative integers with the property that every nonnegative rational numbers occurs as the ratio of consecutive terms of the sequence exactly once. More precisely, they seek a sequence of nonnegative integers $a_1, a_2, a_3, \dots$ such that for every rational number $r$, there exists a unique positive integer $n$ such that $r = a_n/a_{n+1}$. Can you dream up such a sequence?

We close with some Notes from the Club, which include a wild system of equations in 4 unknowns created by 3 of our members this fall: Beatrice, Eleanor, and Sadie. The three worked this fall with our student mentor Alina. The system is nonlinear and has a unique solution in real numbers. Can you solve it?

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 16, Number 1

Volume 16, Number 1 opens with an interview with Professor Ann Trenk. Professor Trenk is the Lewis Atterbury Stimson Professor of Mathematics at Wellesley College. We thought it appropriate to include her interview in this issue because she briefly touches on partially ordered sets, which is the topic of Robert Donley’s article Examples of Posets which appears on page 13.

High school sophomore, Emily Caputo presents us with a challenge: to figure out what image she is depicting by the solutions to 24 different sets of simultaneous equations and inequalities. (See page 7.) Ever since Descartes introduced coordinates, geometry and algebra have been inextricably linked!

Emily and Jasmine begin a new adventure. On social media, they read that one can list the nonnegative rational numbers by using the function $f(x) = 1/(1+[ x ] -\{ x \})$, where $[x]$ is the greatest integer less than or equal to $x$ and $\{ x \}$ is the fractional part of $x$, namely $x - [x]$. According to what they read, the sequence $0$, $f(0)$, $f(f(0))$, $f(f(f(0)))$, … is a list of all the nonnegative rational numbers, and they’ve set out to understand and prove this assertion. If you don’t see what all the excitement is all about, please take a look at Addie Summer’s Learn by Doing on page 24.

Finally, Anna Ma explains the perceptron algorithm for finding a linear separator. This algorithm is readily implementable on a computer in your favorite programming language and we encourage you to try it!

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 15, Number 6

In Volume 15, Number 4, we had the pleasure of interviewing Professor Gloria Marí Beffa of the University of Wisconsin-Madison. In that interview, Prof. Marí Beffa mentioned that she’d need a second interview to address a question pertaining to gender and math. In this issue, we are thrilled that Prof. Marí Beffa agreed to this second interview, entirely on the subject of gender and math.

At the club, we rarely discuss gender and math, preferring to focus squarely on mathematics. If it does arise, it’s only because one of our members brings up the topic. But the continued underrepresentation of women in the field of mathematics underscores the importance of the topic. Many different explanations for the underrepresentation of women in mathematics have been proposed. Our opinion is that the so-called leaky pipeline is not at all an indication of a gender disparity in mathematical ability. From our experiences operating a math club for girls, we have seen that girls are fabulously talented at mathematics. We believe instead that the status quo in math education inherently favors boys over girls and we work to find and implement math educational methods that rectify this situation.

Following the interview, Robert Donley extends his exploration of path counting and derives the Binet formula for the Fibonacci numbers. He also connects binomial coefficients to multiset counting. If you like path counting, try to come up with your own network of streets. Who knows, you might discover some nifty new sequence of numbers, as Girls’ Angle members Esmé Krom and Molly M. Roughan did (see Volume 13, Number 3 or read this summary of it), which led to this new sequence on the On-Line Encyclopedia of Integer Sequence.

Next, Anna Ma begins an exploration of classification algorithms, starting with a linear classier to sort Halloween candy loot.

We have seen a number of times a situation where a student is having difficulty not so much with the ideas, but with the notation. It never feels good to feel left out and unable to participate because one lacks fluency with the notation, especially because the notation is not the math, it is only meant to facilitate the communication of math. If mathematical notation has been a sore spot for you, we hope this issue’s Notation Station can be of help.

We conclude with this summer’s batch of Summer Fun solutions. In fact, the cover is the solution to one of the problems from Laura Pierson’s problem set on labeled trees and parking functions. It shows every labeled tree on 5 nodes, together with the associated Prüfer sequence.

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 15, Number 5

In this issue, we’re thrilled to present our interview with mathematician Zoi Rapti, a professor in the mathematics department at the University of Illinois, Urbana-Champaign. Prof. Rapti is the 60th woman in mathematics who we’ve had the pleasure of interviewing. One of the reasons we’re excited about our interview with Prof. Rapti is that she is also a professor at the Carl R. Woese Institute for Genomic Biology, and we know that many of our members are interested in studying biology as well as mathematics. One of the applications of math that Prof. Rapti discusses is in analyzing the color patterns of bees.

Next, Robert Donley continues to develop his exposition on counting, turning now to the Catalan numbers. Did you know that the entry for the Catalan numbers on the On-Line Encyclopedia of Integer Sequences is longer than the corresponding entry for the Fibonacci numbers (as of this writing)?

In her latest installment of Needell in the Haystack, Anna Ma discusses data visualization – ways to present data in a way that helps humans to understand the data. This is an important, active topic.

Next, we present this summer’s batch of Summer Fun problem sets. There are three, and the third includes a special raffle contest for a chance to win a modest prize. The first, by Laura Pierson, connects labeled trees to parking functions. The second, by Mandy Cheung, is a modification on the water bucket genre of puzzle to the task of making orange juice. And the third, by AnaMaria Perez and Josh Josephy-Zack, represents a bit of an exception to our standard fare of mathematics. The two present a series of Fermi questions – questions that are either impractical or impossible to answer exactly, so you have to give your best estimate. Most of these questions will involve having some general world knowledge. At the end of their problem set, you’ll find two bonus questions and whoever (within the United States) submits the best estimate will receive a choice of chocolate or a math book.

We conclude with Notes from the Club, presenting three problems from an amazing Math Collaboration that was created by four of our members – a first at Girls’ Angle!

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 15, Number 4

It’s hard to imagine getting very far in mathematics without stumbling upon Pascal’s triangle. According to Wikipedia, the concept occurred even as far back as the 2nd century BC in India. The cover is a graphical depiction of the triangle, and in this issue’s Mathematical Buffet, you can find more graphic depictions of various aspects of this famous triangle of numbers, including images created by Sofia Egan and Yancheng Zhao, two 8th graders at the Buckingham, Browne, and Nichols Middle School. The triangle features prominently in Robert Donley’s article Pascal’s Triangle, the Binomial Theorem, and Chu-Vandermonde Convolution, where he presents several ways of interpreting the entries in Pascal’s triangle. If you were to create an image of Pascal’s triangle, how would you depict it?

In the last issue, we concluded our 3-part interview with University of Wisconsin, Madison Associate Professor of Mathematics Tullia Dymarz. One of the reasons we interviewed Prof. Dymarz is because she runs a wonderful program for high school girls called Girls Night Out! However, in her interview, she mentioned that the program was actually founded by her colleague Prof. Gloria Marí Beffa. So in this issue, we are pleased to present an interview with Gloria Marí Beffa. In addition to asking about her motivations for creating Girls Night Out!, we also discuss problem solving strategies and her unique route into mathematics.

Anna Ma and her cat return for an installment of Needell in the Haystack about data collection, which has changed dramatically with the advent of modern technology. Emily and Jasmine further their understanding of magic grids, and 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!

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 15, Number 3

On the cover, a kite with three congruent acute angles is seen at last, thanks to Milena Harned, who discovered that the only convex quadrilaterals whose angle bisectors are also perimeter bisectors are the rhombi, and such kites. To prove it, she analyzed the envelopes of the perimeter bisectors of polygons. For details, read her article in the peer-reviewed International Journal of Geometry, Volume 10, Number 4. For more, see this issue’s Member’s Thoughts and check out this article about Milena in the Notices of the American Mathematical Society by Scott Hershberger. I have a feeling we’ll be learning more math from Milena! (The photograph is fitting: It is the view from the mathematics department at UC Berkeley, courtesy of UC Berkeley Math Department Chair Michael Hutchings.)

We conclude our 3-part interview with University of Wisconsin, Madison Associate Professor of Mathematics Tullia Dymarz. Thank you, Tullia, for this wonderful opportunity to learn so much from you. In these interviews, we learned about quasi-isometry, the lamplighter group, and, in this concluding part, a fantastic model for involving high school girls in mathematics embodied in the program Girls Night Out!, which continues the brainchild of Gloria Marí Beffa.

What do cats and matrices have to do with each other? Read about one such connection in Anna Ma’s latest installment of Needell in the Haystack. Actually, what do matrices have to do with almost anything and everything? The field of data science could almost be described as a the mathematics of matrix manipulation.

We welcome back Robert Donley (aka Math Doctor Bob), who lays the groundwork for lattice path counting in Shortcuts to Counting. In Magic Grids, Emily and Jasmine become intrigued by a problem in the most recent Math Prize for Girls contest, and Addie Summer demonstrates that math accommodates many ways of thinking by presenting three different proofs of the angle sum formula for sine in Different Angles.

We conclude 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!

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 15, Number 2

The cover shows all 35 ways to tile a 4 by 4 rectangle with 1 by 1 and 2 by 2 tiles, as organized according to a scheme dreamed up by a group of 8th graders at the Buckingham, Browne, and Nichols Middle School in Cambridge, MA. The number of such tilings of a 4 by N rectangle has been well-studied and a number of recursion formulas are known. But one recursion formula was noted by Schneider without proof or reference:

$S_n = 2S_{n-1} + 3S_{n-2} - 2S_{n-3}$,

where $S_n$ is the number of ways to tile a 4 by n rectangle with 1 by 1 and 2 by 2 tiles. These 8th graders succeeded in proving it, and in just a few clever lines! Details are in their article on page 8.

In Part 1 of our interview with Professor Tullia Dymarz, Tullia describes the lamplighter group and a pipe cleaner model she made of one of its Cayley graphs. If you were interested in trying your hand at building the pipe cleaner model but haven’t learned about Cayley graphs yet, check out this issue’s Learn by Doing to get going. In Part 2 of Tullia’s interview, she tells us more about her research process and describes one of her own discoveries which concerns which Cayley graphs of different lamplighter groups are quasi-isometric.

One point we try to emphasize in the Bulletin is that doing math is just a matter of being curious, coming up with questions that can be analyzed rationally and abstractly, and trying to answer them. The mathematical universe is so enormous with so much unexplored territory that if you start doing this, it will not be long before you come upon something new. It is an astonishing journey, full of surprises and awesome beauty. For example, there was the time when Emily and Jasmine stumbled upon their “Zigzag” theorem. (See pages 16-20 of Volume 13, Number 3 and pages 20-26 of Volume 13, Number 4, or read this earlier blog post. Although a fictionalized account, the story does closely describe how the theorem was actually born.) In our ongoing attempt to further this point, we follow Lightning Factorial’s thoughts on 2022 and point out the math that oozes out from thinking about that number’s properties in All That Math.

Anna Ma, who has taken over The Needell in the Haystack from originator Deanna Needell, addresses a super important question in data science: What do you do if your data is incomplete? It is an extremely common predicament to find oneself in. She begins the discussion, but it is an active area of research.

We conclude 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!

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 15, Number 1

Our 15th year of the Girls’ Angle Bulletin begins with Part 1 of a multi-part interview with University of Wisconsin, Madison Associate Professor of Mathematics Tullia Dymarz. This was our first interview conducted over Zoom. In this first part, Tullia beautifully explains the notion of quasi-isometry and one of her favorite objects of study, the Diestel-Leader graph as the Cayley graph of the lamplighter group.

For this interview, we benefitted greatly from a biographical sketch of Tullia written by Isa Barth, and Isa’s essay follows the interview, although we urge all readers to read Isa’s essay first, as our interview takes off from there.

If you’ve been looking for an application of pipe cleaners to mathematics, look no further… Tullia provides a super interesting one!

Next, University of Oregon Associate Professor of Mathematics Ellen Eischen presents a selection of images from a mathematical art show that she curated and organized called Creativity Counts and which was on display at the Jordan Schnitzer Museum of Art in Eugene, Oregon. The cover features a contribution by Ellen herself. As Ellen note, “Aesthetic aspects of number theory, an area illustrated in most of the pieces in the gallery, have enthralled mathematicians since antiquity.”

Many topics in mathematics lend themselves well to visual imagery. If you create mathematically inspired visual art, we’d love to see it!

Anna Ma shows us that the Kaczmarz algorithm is an instance of a much more general minimization algorithm called “gradient descent.” If you have a real-valued function defined on some n-dimensional real space, and it is differentiable, then at each point in the n-dimensional space, it will have a vector, called the gradient, which points in the direction that the function locally grows the fastest. Gradient descent algorithms attempt to find minima by moving in a direction opposite to where the gradient points.

Emily and Jasmine analyze the Valentine heart equation from the Valentine heart app that launched their Valentine heart journey. They identify the 2D cross section which corresponds to the heart shape and compare and contrast it with the equation that they concocted.

We conclude 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!

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 14, Number 6

Volume 14, Number 6 opens with an interview with Sarah Bryant, a lecturer at Gettysburg College. Sarah brings a valuable and unique perspective to the mathematics profession and to becoming a mathematician. She is involved with many activities that draw people into mathematics, such as by creating the Shippensburg Area Math Circle for 4th and 5th graders. She has also applied mathematics to the study of questions in biology, specifically, she studied nematocysts in jellyfish. Normally, we truncate the interview for the electronic version, but you’ll find the complete interview with Sarah online.

Following the interview, three Girls’ Angle members, Eva Arneman, Altea Catanzaro, and Saideh Danison, present a game they created at the Girls’ Angle club and beautifully explain their winning first player strategy when this game is played on the edges of a tetrahedron. There are many follow-up questions that one can ask about this game, such as, on what graphs does the first player have a winning strategy? We hope some readers will have as much fun thinking about the possibilities as these three! Our cover is inspired by the game and created by Juliette Majid.

Next, we welcome Anna Ma of UC Irving, who authors our latest installment of The Needell In The Haystack. Anna earned her PhD under the supervision of series creator Deanna Needell. The importance of Data Science just grows and grows as the world becomes more digitized. We consider ourselves very fortunate to have this ongoing series. Deanna earlier wrote about the Kaczmarz algorithm, and in this issue, Anna Ma gives her own take on it.

The concluding half of Jovana Andrejevic’s article on paper crumpling comes next. She compares and contrasts crumpling with deliberate paper folder (such as paper folding as practiced by origami enthusiasts). Although crumpling doesn’t enjoy certain precise theorems that origami folding does, there are hints that there is some unrecognized hidden geometric structure to paper crumpling. Perhaps you can find it?

We conclude with solutions to the Summer Fun problem sets by Laura Pierson (on Wythoff’s game) and AnaMaria Perez and Josh Josephy-Zack (on Fibonacci partitions). We’re delaying solutions to Fan Wei’s probability problem set to honor a reader request.

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!