Girls’ Angle Bulletin, Volume 17, Number 4

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

Can you figure out why the cover image is entitled Complete Domination?

We open with the conclusion of our four-part interview with the Theresa Mall Mullarkey Associate Professor of Mathematics at Wellesley College, Karen Lange. In this installment, Prof. Lange goes into further depth about her field of research: complexity theory. She discusses the importance of persistence in mathematics. And, she talks about a good frame of mind to have when doing mathematics. Her concluding words: “Where do I get stuck or frustrated? It’s where I’m starting to think of where I should be with the math, which is exactly where anybody doing math gest frustrated. What’s actually going to help you get there is realizing we’re all on this journey, and it’s a lifelong journey. You’re never there. There’s always more to be learned and experienced.” Cheers!, Prof. Lange, and thank you for this wonderful interview!

Next, comes Part 2 of Optimal Resource Placement by Jillian Cervantes and Prof. Pamela E. Harris, both at the University of Wisconsin at Milwaukee. Their article inspired the cover image of this issue.

Robert Donley extends his series to a fourteenth installment that covers the basic of permutations and group theory.

Lightning Factorial continues tackling cubic equations in Cubics, Part 2 and succeeds in finding a way to find the solutions, but the method leaves more questions than answers…

In Member’s Thoughts we describe a very nice way of deriving the quadratic formula that one of our member’s found last fall.

We conclude with 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 17, Number 3

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

Can you figure out the connection between the cover image and cubic polynomials?

We open with the third of four installments of our interview with the Theresa Mall Mullarkey Associate Professor of Mathematics at Wellesley College, Karen Lange. In this installment, Prof. Lange describes more technical aspects of her field of expertise: the theory of computation. The Halting Problem and diagonalization arguments feature prominently!

Emily and Jasmine’s long romp through rational rompers, which began with an intriguing social media post, reaches a climax as they succeed in proving one of their main conjecture about them. Specifically, they showed the following: Suppose you have two sequences of whole numbers and , . And suppose that the sets and are equal and have elements, and, also, that and , then either sequence can be transformed into the other by a finite sequence of operations that they call “swaps.”

Next, undergraduate Jillian Cervantes and Prof. Pamela E. Harris, both at the University of Wisconsin at Milwaukee, introduce domination sets in graph theory. This is a highly accessible topic and can quickly get you doing mathematics.

Robert Donley introduces standard tableaux in his latest installment of his ongoing series on Shortcuts to Counting.

In a similar spirit to the previous two issue’s Follow Your Nose, we asked Lightning Factorial to try to find the cubic formula without looking it up. We’re not concerned about whether Lightning succeeds or not. What we expect is that by searching for the formula, Lightning will come up with more math along the way. We’ll see!

In Meditate to the Math, we present an opportunity for you to think yourself to understanding and/or proving Descartes’ Rule of Signs.

We conclude with 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 17, Number 2

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

The cover shows the Hasse diagram for the partially ordered set of partitions show Young diagrams fit inside a 4 row by 3 column rectangle. For more on these beautiful structures, see Robert Donley’s latest installment in his multipart series on partitions on Page 8.

For our interview, we continue with the second installment of our 4-part interview with the Theresa Mall Mullarkey Associate Professor of Mathematics at Wellesley College, Karen Lange. There’s lots of great advice in this installment!

We continue exploring the ramifications of an approach a group of 8th graders at the Buckingham, Browne, and Nichols Middle School came up to find Pythagorean triples in Follow Your Nose, Part 2.

Emily and Jasmine continue their investigation of rational rompers, this time focusing on finite rational rompers that represent the same rational numbers.

We have a Learn by Doing on random numbers. Remember: If you ask someone to pick a random number, you have to specify a mechanism or a probability distribution!

In Alphanumerics, we imagine what an alien mathematician might think upon encountering the alphabet for the first time.

Some members at the club worked on ways to fold regular polygons with the techniques of origami. In this issue, we provide folding instructions for a regular pentagon. Can you prove that it works?

We conclude with 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 17, Number 1

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

In this issue, we open with the first part of a four-part interview with the Theresa Mall Mullarkey Associate Professor of Mathematics at Wellesley College, Karen Lange. In this first part, Prof. Lange discusses her path into mathematics and her primary mathematical interest: computability theory.

The cover features a lamp designed and built by Sofia Egan, a high school student at the Buckingham, Browne, and Nichols Upper School. The lamp is a cuboctahedron which has 6 square and 8 equilateral triangular faces. How many vertices and edges does it have?

Next is an article by Emily Caputo, Sophie Harteveldt, and Alina Patwari on a tiling problem created by Sophie.

Robert Donley continues his tour of partitions with more on Young diagrams and related generating functions.

In Prof. Lange’s interview, she relates an anecdote from her youth where she solved a problem of her own making, but when she presented it to her teacher, her teacher assumed that she just read it in a book. In math class at school, nobody is expected to create math. It is easy to get the impression that math is just about what has been done in the past. Contrast this with a painting class, where is considered normal for students to make paintings. It is also easy to believe that one is incapable of creating mathematics. However, you definitely can create math. In Follow Your Nose, we follow a group of 8th graders as they set themselves the challenge of finding a way to generate Pythagorean triples without reading about it.

Next, Emily and Jasmine are back, furthering their understanding of rational rompers – sequences of nonnegative integers such that consecutive terms are relatively prime and the ratios of each term to the next yield every single nonnegative rational number exactly once.

We conclude with Notes from the Club which includes a brief summary of a very special Support Network visit from former Girls’ Angle mentor, now Assistant Professor of Mathematics at Brown University, Isabel Vogt. Isabel recently proved a beautiful theorem which she spent part of her visit explaining to us. Her theorem has its roots in the ancient observations that through any two points, there is a unique line, and through any three non-collinear points, there is a unique circle. Her work was written about in Old Problem About Mathematical Curves Falls to Young Couple by Jordana Cepelewicz for Quanta Magazine.

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 6

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

In this issue, we present the concluding part of our 2-part interview with Sarah Spence Adams, Professor of Mathematics and Electrical & Computer Engineering at Olin College. In this second part, Sarah tells us more about her own achievements and background history. She also gives excellent advise for students, both those interested in math and those interested in other things.

Our first article is by Tara Gall, a recent graduate of Clearview Regional High School in New Jersey. During her senior year, Tara decided to challenge herself by taking a unique math class created and taught by Anne Paoletti. Completely from scratch, Tara created a math problem for herself: What is the probability of winning a tennis game from a given score? She solved the problem and generalized it by giving a recurrence relation that enables one to compute the probability of winning an n-point rally scoring game (where the winner must win by a margin of 2 points), given that the score is A–B. At this year’s French Open women’s final between Iga Świątek (a math lover!) and Karolína Muchová, Tara’s formula predicts that Iga would win 69% of her service games. In fact, she won 10 out of 15, or about 67%, which is as close as the experimental result could be to Tara’s theoretical prediction! Tara’s formulas also quantify the importance of winning the next point. For example, you might be up 40-15 in a game and wonder if you might try a riskier strategy to win the point. Tara’s work gives a quantitative measure of how much your probability of winning drops should you lose that point and the score becomes 40-30.

Next, we have a fun self-referential multiple choice test created by Sascha McHugh. If you enjoy logic, you’ll love self-referential tests. You don’t have to have any general knowledge to solve these tests…just a sure-footed sense of logic.

Robert Donley finds analogues of the hockey stick rule in Pascal’s triangle to partition numbers.

And then we describe a beautiful mathematical theorem of Katherine Knox, just published (August 2, 2023) in the American Mathematical Monthly. Katherine began thinking about what she called “light paths” in polygons when she was in 6th grade. Our cover illustrates two light paths bouncing around the interior of a quadrilateral. If you like billiard trajectories or cyclic quadrilaterals, you’ll love this elegant gem.

Finally, we present this summer’s batch of Summer Fun solutions.

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 5

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

In this issue, we present the first part of our 2-part interview with Sarah Spence Adams, Professor of Mathematics and Electrical & Computer Engineering at Olin College. In this first part, Prof. Adams discusses how finite rings are used to create codes. Prof. Adams will be new to our members today, though she visited Girls’ Angle back in 2012 as a member of our Support Network and ran an activity involving secret codes.

Anna Ma continues her series on large language models moving us closer to understanding how programs like chatGPT work.

Robert Donley makes the counting of partitions fun and simple with aerated summation.

Next, we present this summer’s batch of Summer Fun problem sets, this year all with authors that are current mentors at Girls’ Angle. The first, by Clarise Han, will make you want to eat ice cream (and also inspired our cover). The second, by Matthew Bates and Jane Wang, is about scissors congruence. And the third, by Hanna Mularczyk, is about simple graphs, planar graphs, and graphs on surfaces other than the plane.

We conclude with Notes from the Club, presenting a few problems from an nonogram-themed Math Collaboration designed by Elisabeth Bullock and AnaMaria Perez…also both current mentors 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 16, Number 4

This issue’s interviewee is Katharine Ott, who is an Associate Professor of Mathematics at Bates College. In this interview, which was conducted by Ken Fan and Raegan Phillips, Katharine discusses a wide range of topics, including her role as the Director of GirlsGetMath@ICERM as well as her writing/journalism career. The interview contains lots of wonderful advice for students interested in mathematics.

Our cover shows all 196 domino tilings of a 5-by-5 square with its central square removed. This spring, two of our members found a formula for the number of tilings of a family of annuli. Unfortunately, they didn’t know until after they got their results that these results were published by Tauraso in 2004. This material is related to a lesser known algebraic identity called “Candido’s identity”:

For details, see Member’s Thoughts.

Robert Donley brings together several ideas from recent installments of his series on counting and partially ordered sets, this time delivering a beautiful bijection between compositions of positive integers into parts of size 1 or 2, into parts of odd size, and into parts of size greater than 1. There’s a lot of math to savor in this topic!

Anna Ma continues her discussion of natural language processing, illustrating language models with a toy example. If you know how to program, it would be easy to scale up the model described in her article.

After resolving the mystery of the rational number romping sequence that they learned through social media, Emily and Jasmine returned to an idea they had in Part 2 of Romping Through the Rationals, finding integer sequences such that is relatively prime to and every nonnegative rational number is equal to for a unique index .

We close with some Notes from the Club where you can read about former Girls’ Angle mentor, now graduate student at the University of Chicago, Karia Dibert’s Support Network presentation after returning from her first trip to the South Pole to prepare for the installation of a cosmic background radiation detector of her own design!

Special thanks to Margaret Lewis for permission to share her wonderful picture of MK the eagle (see page 29).

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 3

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

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 ?” Often, people will say that it is 1 and explain how this is because . However, it’s important to understand that this is not a proof that . (Likewise, observing that does not imply that !) 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

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

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 appears 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 such that for every rational number , there exists a unique positive integer such that . 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

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

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 , where is the greatest integer less than or equal to and is the fractional part of , namely . According to what they read, the sequence , , , , … 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!