The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

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

.

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

for all .

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!

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The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

10 years of the Girls’ Angle Bulletin!

With Volume 10, Number 6, we’ve published 42 interviews with women in mathematics, dozens of Summer Fun problem sets, and some 1500 pages of math and math educational content, authored by professional mathematicians and scientists, graduate students, math teachers, undergraduates, and Girls’ Angle members. There have been galleries of math related art, comic strips, dialogues, articles, brain teasers, math challenges, math games, and much more. Thank you to the over 150 people have contributed to Bulletin content over the last decade! And Thank You to Mathworks, whose continuing support for the Girls’ Angle Bulletin has made so much of this possible.

Volume 10, Number 6 opens with an interview with Kathryn Mann, assistant professor in the Department of Mathematics at Brown University. Prof. Mann received her doctoral degree from the University of Chicago under the supervision of Benson Farb and was previously an assistant professor and NSF postdoctoral researcher at UC Berkeley. Her research interests include geometry, topology, and geometric group theory.

Next, Emily and Jasmine make steady progress on their quest to classify “nice triangles”. Their journey has taken them into the realm of algebra where they have been evaluating the cyclotomic polynomials at *i*. Hopefully, their perseverance will pay off, but even if it doesn’t, they’ve learned a lot of neat facts about the cyclotomic polynomials.

In this issue’s *Math In Your World*, we explain the rationale behind so-called “geometric probability.” The reason for this article, like so much content in the Bulletin, is because there are some current Girls’ Angle members who may be about to begin an investigation that requires understanding continuous probability distributions. (To all members: your content requests are taken very seriously and given a high priority!)

In Anna’s Math Journal, Anna gets an idea for looking at special tilings of 1 by rectangles that could potentially lead to a derivation of the formula for the number of such tilings that doesn’t use generating functions.

And, finally, we close with the solutions to this summer’s Summer Fun problem sets.

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!

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*The yellow pig has a 2 sentence solution!*

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The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

Volume 10, Number 5 opens with an interview with Ruth Charney, the Theodore and Evelyn Berenson Professor of Mathematics at Brandeis University. Prof. Charney studied geometric group theory and received her doctoral degree in mathematics from Princeton University. She was formerly a professor at Ohio State University.

Next comes the second half of the article by Milena Harned and Miriam Rittenberg on NIM Counting. They give a general formula for the number of ways a NIM game with 2 starting piles can be played out and they investigate the form of the formula.

Errorbusters! returns after a long absence! The column, which was originated by Cammie Smith-Barnes, has been revived by Hamilton College Assistant Professor of Mathematics Courtney Gibbons. In her first installment, she writes about “Errors of Apathy,” which include such dastardly errors as substituting for .

The cover is a homage to the 24th cyclotomic polynomial. Of late, Emily and Jasmine have been filling reams of scratch paper with computations involving cyclotomic polynomials as they continue their quest to classify all “nice triangles”. The computation is daunting and they don’t even know if the computation will prove useful in their quest, but they bravely press on! Such is math research…

Next up: This summer’s batch of Summer Fun problem sets. This year, we have problem sets on cyclotomic polynomials (as a kind of primer to Emily and Jasmine’s work), sets (by Debbie Seidell), the fourth dimension, and unsuspected appearances of geometry (by Matthew de Courcy-Ireland).

We conclude with a few problems from our traditional end-of-session math collaboration and a few chocolate Hasse diagrams from the mini-chocolate tasting of Meet 11.

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!

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The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

Volume 10, Number 4 opens with an interview with Nalini Joshi, Professor of Mathematics in the School of Mathematics and Statistics at the University of Sydney. Prof. Joshi received her doctoral degree from Princeton University. She studied differential equations. The cover represents the iterates of an integrable third-order difference equation that arose out of joint work between her and Dr. C.-M. Viallet of CNRS and Sorbonne Universités. Dr. Viallet created the cover image.

In the 10th installment of *In Search of Nice Triangles*, Emily and Jasmine apply the knowledge they learned from Prof. Alison Miller from Part 9 and succeed in finding the minimum polynomials of the cosines of “nice” angles, a big step toward their goal of classifying all triangles that have 3 “nice” angles and 2 sides of integer length.

In *Anna’s Math Journal*, Anna gives in to the temptation of the generating function and uses it to prove her conjecture about the number of “special” tilings of a 1 by rectangle.

Next up, Milena Harned and Miriam Rittenberg explain different ways of counting the number of ways a game of NIM can be carried out. In this first half, they find explicit formulas when the game begins with two piles, one of which has a small number of counters. In the final half, which will appear in June, they will examine the asymptotics of the general 2-pile case.

Recently at the Girls’ Angle club, some members have been studying perspective drawing and practicing the theory by making drawings of geometric objects such as a checkerboard, like the one in the floor of Vermeer’s *Art of Painting*. This issue’s *Math In Your World* shows how the harmonic mean lurks within this topic. In fact, there’s a lot of beautiful mathematics in perspective drawing.

Milena Harned decided to generalize an idea she found on the 2002 American Invitational Mathematics Exam I. Along the way, she found some polynomials that relate to the famous way of obtaining the Fibonacci numbers from Pascal’s triangle shown at left.

We hope you enjoy it!

*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!

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After you make your intuitive guess, compute to find out the truth!

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Volume 10, Number 3 opens with an interview with Sommer Gentry, professor of mathematics at the US Naval Academy. Prof. Gentry invented a system for optimizing kidney transplantation that positively. This interview was conducted by Girls’ Angle progam assistant Long Nguyen. As has been our practice, we truncate the interview in the electronic version. For the full version, please subscribe!

Next comes Villanova assistant professor of mathematics Beth Malmskog‘s concluding part of *Quilt-Doku!* Here she shows that a 5 by 5 row complete Latin square is impossible. Shown at right is a row complete Latin square of order 15. It remains an unsolved problem to determine if there are any row complete Latin squares of odd prime order. In fact, it isn’t even known that there isn’t one of order 13. Can you prove it?

Test your area computation skills in *Area Area*. Can you solve these area problems in your head?

Anna B. continues to search for a combinatorial proof that the conjectured formula she found for the number of special tilings of a 1 by rectangle is, in fact, valid. She remains stumped. Can you help her prove that the close-form formula she found is correct?

Emily and Jasmine continue their pursuit of triangles with 3 nice angles but only 2 sides of integer length at Cake Country where they run into Alison Miller, a Benamin Peirce and NSF Postdoctoral Fellow at Harvard University. Prof. Miller gives them a lot to chew on for next time! She also mentions Gauss’s Lemma, but doesn’t have time to prove it, so we provide an installment of *Learn by Doing* where you can prove the lemma for yourself.

The cover pertains to Addie Summer’s follow-up to her article on the quadratic formula. In this issue, she creates a graphical representation of monic quadratics and interprets properties of them geometrically. On the cover, the surface represents monic cubics with multiple roots. Specifically, it is the surface of points (*b*, *c*, *d*) in *bcd*-coordinate space such that the cubic has multiple real roots. (Note that if a cubic with real coefficients has multiple real roots, then all of its roots are real.) This graph was created using MATLAB, a powerful suite of math software created by MathWorks. MathWorks has been a valuable sponsor of the Girls’ Angle Bulletin for several years.

For fun, we offer a self-referential true/false quiz that was inspired by a self-referential multiple choice test created by Jim Propp.

Finally, we conclude with Notes from the Club, which contains a description of one of our more versatile and popular games: Describe that Drawing.

We hope you enjoy it!

*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!

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Volume 10, Number 2 opens with an interview with Doris Schattschneider, professor emerita of mathematics at Moravian College. Among many other things, Professor Schattschneider was the first woman to serve as editor of the MAA’s *Mathematics Magazine*. This interview was conducted by Girls’ Angle summer intern Sandy Pelkowsky. As has been our practice, we truncate the interview in the electronic version. For the full version, please subscribe!

The cover is a picture of the function , modulo the prime number 503. For more examples and an explanation of how the image was made, check out this issue’s *Mathematical Buffet*.

We feature a contribution from Villanova assistant professor of mathematics Beth Malmskog, *Quilt-Doku!* Prof. Malmskog shows how she went from a quilting friend’s request to unsolved problems in mathematics. She blogs for the American Mathematical Society – check it out!

We’d also like to take special note of a member contribution in *Member’s Thoughts*. Here, ** π** takes us along on her derivation of the volume of a regular

Emily and Jasmine continue their pursuit of triangles with 3 nice angles but only 2 sides of integer length. While they make some progress in their investigation of the minimal polynomials of cosines of nice angles, the journey begins to appear rather daunting!

Last issue, Anna B. found a recursive formula for the number of special tilings of a 1 by rectangle. In this issue, she manages to guess the closed-form formula, but is unable to prove it. Can you help her prove that the close-form formula she found is correct?

Addie Summer explains how she discovered the quadratic formula. In the process, she shows that the algebraic technique known as “completing the square” corresponds to a natural geometric idea: shifting the parabola sideways so that it is symmetric about the *y*-axis.

In Notes from the Club, we mention a few of the things that have been happening at the Girls’ Angle club and give a sampling of problems from our traditional end-of-session Math Collaboration.

We hope you enjoy it!

*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!

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