Thirst For Firsts – A Girls’ Angle Raffle

For a PDF version, please click here.

(We will not use your contact information for any purpose other than to deliver your prize, should you win. After the winner has been selected, all emails received will be promptly deleted. At the winner’s discretion, we will let you know who won. Anyone who makes more than one submission will be disqualified! Sorry! Also, this offer is only valid in those states in the United States where such things are legal. There is no fee to enter this puzzle contest.)

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Girls’ Angle Bulletin, Volume 13, Number 5

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

At Girls’ Angle, we frequently have members who express interest in making their drawings look more realistic, and this often leads to a study of perspective drawing, which is a fabulous way to get into geometry. In this issue, we’re fortunate to feature an interview with Franklin & Marshall College Professor Annalisa Crannell, who recently wrote an entire book on perspective drawing together with Marc Frantz and Fumiko Futamura. And, special thanks to Princeton University Press, we include a two-page excerpt from their book, “Perspective and Projective Geometry.” Their book is highly engaging and offers a series of excellent perspective drawing exercises that are not only mathematically interesting, but also a lot of fun to do.

We have been incredibly fortunate to include a regular column, Needell in the Haystack, authored by Professor Deanna Needell on big data techniques. She is an amazing, benevolent force, and today, she’s also helping to fight the pandemic. However, in this issue, she writes about how bias creeps into mathematical analyses of data, which is an incredibly important and apt topic for our times. Normally, we remove her column from the electronic version, but we’re including the full article this time.

The cover might look like a perspective drawing, but it isn’t. It’s a design that Liliana Smolen and Isabel Wood created while the were playing around coming up with designs that illustrate mathematical identities. This issue’s Mathematical Buffet features four more of their designs. While Liliana and Isabel dreamed up these designs entirely from a blank white board, these particular designs have been seen before, and many more are collected in the books Proof Without Words: Exercises in Visual Thinking, by Roger Nelsen. The cover is a triangular version of the Fibonacci spiral. They even came up with a spiral that represents the mathematical constant e, but, unfortunately, there wasn’t enough space to include it. Perhaps you can come up with a way?

And we have our traditional Summer Fun Problem Sets. This summer, we present five: Cannonballs and Combinatorics by Girls’ Angle mentor Annie Yun, Tetrahedra with Congruent Faces by Ken Fan, Bernoulli Numbers by Matthew de Courcy-Ireland, Matrix Expedition by Girls’ Angle mentor Jasmine Zou, and Two Whole Squares by Ken Fan and Girls’ Angle Head Mentor Grace Work. As always, members and subscribers are encouraged to send us any questions and/or solutions.

We conclude with some Notes from the Club, which are authored by our Head Mentor Grace Work.

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!

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LCM Optimal Sequences

The best way to learn math is to do math, which is one of the reasons I’m thrilled about the work of Antonella Catanzaro, Jaemin Feldman, Mika Higgins, Bradford Kimball, Henry Kirk, Ana Chrysa Maravelias, and Darius Sinha, who were all eighth graders at the Buckingham, Browne, and Nichols Middle School when they embarked on their own personal mathematical adventure and discovered some curious results which I’d like to draw more attention to, especially because they left behind a number of conjectures which hopefully someone might be interested in pursuing.

Their adventure began when Ms. Higgins wondered aloud whether algorithms and number theory could be combined. After some brainstorming, the group came up with the following question:

What is the least expensive path through n cities, labeled 1 through n, starting at city 1 and allowing multiple visits to a city, if the cost to travel between city a and b is the least common multiple of a and b?

Here’s a table of some examples of optimal paths.

n Cost Sample Optimal Path
1 0 1
2 2 1, 2
3 7 1, 2, 1, 3
4 12 1, 3, 1, 2, 4
5 21 1, 3, 1, 2, 4, 1, 5
6 28 1, 3, 6, 2, 4, 1, 5
7 40 1, 3, 6, 2, 4, 1, 5, 1, 7
8 51 1, 5, 1, 7, 1, 3, 6, 2, 4, 8
9 65 1, 5, 1, 7, 1, 4, 8, 2, 6, 3, 9
10 79 1, 4, 8, 2, 6, 3, 9, 1, 7, 1, 5, 10

In each optimal path, note that in every pair of adjacent numbers, one is always a multiple of the other. This is something that the seven students proved, and they proved it by showing that for positive integers x and y,

LCM(x, y) < x + y if and only if x divides y or y divides x.

In other words, if x is not a factor or multiple of y, then it is cheaper to travel from x to y via 1 than it is to travel directly from x to y, because the cost of traveling from x to 1 to y is xy, whereas the cost of traveling from x straight to y is LCM(xy). A corollary of this fact is that prime numbers greater than n/2 will always be sandwiched by ones in an optimal path, unless the prime occurs at the very end of the path.

They showed that there always exists an optimal path that visits numbers greater than 1 exactly once, as all of the optimal paths in the table do. And they showed that when n is prime, all optimal paths end in n, but this is not necessarily true when n is composite as the example for n = 6 in the table above shows. Through their math department chair, the seven submitted the sequence of optimal costs to the Online Encyclopedia of Integer Sequences, and it was recently approved as sequence A333354. For details, see pages 13-19 of Volume 13, Number 4 of the Girls’ Angle Bulletin which can be accessed for free at the Girls’ Angle website.

Here are some conjectures and avenues for further investigation:

  • For a given n, do all optimal paths have the same length?
  • What are good upper and lower bounds on the minimal cost sequence?
  • What can be said if you replace the cost of travel between a and b with the highest power of 2 that divides one of the two numbers?
  • What is an efficient way to generate optimal paths for n > 20?

They determined the minimal cost for n up to 20, and for prime n below 20, the costs are:

n 2 3 5 7 11 13 17 19
Cost 2 7 21 40 100 138 238 295

In every prime case listed above, the minimal cost is equal to

\displaystyle n + 2(\sum_{k=(n+1)/2}^{n-1} k) + (\sum_{k=\lceil n/3 \rceil}^{(n-1)/2} k).

Is that true for all prime numbers n?

If you find anything, please do let us know at girlsangle “at” gmail.com!

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Girls’ Angle Bulletin, Volume 13, Number 4

blog_043020_01The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

We open with the concluding half of our interview with Draper Labs mathematician Erin Compaan. Dr. Compaan received her doctoral degree in mathematics from the University of Illinois Urbana-Champaign under the supervision of Nikolaos Tzirakis. She was a National Science Foundation Postdoctoral Fellow in Mathematics at the Massachusettes Institute of Technology prior to joining Draper Labs. In this second part, you can get a glimpse into one of Dr. Compaan’s hobbies: oil painting.

Deanna Needell has been organizing a major effort to apply mathematics to COVID-19 matters, and in this issue’s installment of Needell In The Haystack, she gives us a brief summary of these efforts. She outlines quite a number of ways data science can be applied to combating this dreadful pandemic. Perhaps a reader will be inspired to join that effort?

Next up is an intriguing article by Antonella Catanzaro, Jaemin Feldman, Mika Higgins, Bradford Kimball, Henry Kirk, Ana Chrysa Maravelias, and Darius Sinha, 7 eighth graders at the Buckingham, Browne, and Nichols Middle School. They merged algorithms and number theory by asking: What is the least expensive path through n cities, labeled 1 through n, starting at city 1 and allowing multiple visits to a city, if the cost to travel between city a and b is the least common multiple of a and b? In their quest for an answer, they unearthed a number of curious properties of “complete optimal paths” and taught themselves how to program in Python. There remain tantalizing mysteries to prove. Can you prove them?

One of their discoveries is that certain complete optimal paths have the structure of a multi-decker open-faced sandwich, hence the cover illustration, which was drawn by Mika Higgins.

This issue is the third in a row with student work, and we’d love to publish more! The Girls’ Angle Bulletin serves as a venue for students to showcase their mathematical work that goes beyond the classroom.

Emily and Jasmine are surprised by an email from Prof. Noam Elkies of Harvard University. In the previous issue, they proved the following theorem:

The ZigZags Theorem (Emily and Jasmine).  Let n and m be distinct positive integers.  Let a rectangle be crisscrossed by an n-zigzag and an m-zigzag, each bouncing back and forth between the top and bottom edges.  Then the region of the rectangle below both zigzags, the region above both zigzags, and the region between the two zigzags split the rectangle exactly in thirds.

However, their method of proof, which involved explicit computation of the areas of all parts of the pattern created by two zigzags, did not offer much by way of explanation. Prof. Elkies found an alternative proof that provides such an explanation.

The Emily and Jasmine series is fiction, however the theorem is real, and was discovered in the summer of 2017 by the author along lines that closely resemble the fictional development in the series. Surprised and amused by the result, the author challenged a number of his math friends to prove it. After telling Prof. Elkies, hours later, Prof. Elkies sent the author an email which forms the centerpiece of this last installment in the zigzags saga of Emily and Jasmine. The purpose of the Emily and Jasmine series is to illustrate how mathematics is created with the hope that a reader might be inspired to create some math herself. That is, instead of presenting mathematics as a body of known results, in the Emily and Jasmine series, math is presented in a way in which it could plausibly be discovered or created as Emily and Jasmine create mathematics out of nothing but their minds.

We conclude with some Notes from the Club, which are authored by our Head Mentor Grace Work.

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!

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Girls’ Angle Bulletin, Volume 13, Number 3

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

We open with the first half of an interview with Draper Labs mathematician Erin Compaan. Dr. Compaan received her doctoral degree in mathematics from the University of Illinois Urbana-Champaign under the supervision of Nikolaos Tzirakis. She was a National Science Foundation Postdoctoral Fellow in Mathematics at the Massachusettes Institute of Technology prior to joining Draper Labs. In this first part, Erin retraces her route into mathematics and her specialty, which is partial differential equations.

Next, Esmé Krom and Molly M. Roughan describe results they found last spring in Path Counting and Eulerian Numbers. The two enjoyed counting paths in various street networks so much that they decided to devise their own network of one-way streets and analyze it. They restricted to paths that never visit a node more than once and succeeded in finding formulas for the number of paths from their starting node to all the other nodes. In the process, they brushed up against a well-studied sequence of numbers known as the Eulerian numbers. Primary guidance and mentorship for this mathematical investigation was provided by MIT undergraduate Adeline Hillier.

Esmé and Molly’s network of one-way streets.

Deanna Needell continues making us wonder what computers, with the right algorithms are capable of today. This time, she asks, “Are Computers Artists?” She explains how people have gotten computers to replicate artworks in the style of other artists.

Emily and Jasmine’s investigation into the patterns created by two zigzags that bounce across the face of a rectangle reaches a climax as they discover and prove the following theorem:

The ZigZags Theorem (Emily and Jasmine).  Let n and m be distinct positive integers.  Let a rectangle be crisscrossed by an n-zigzag and an m-zigzag, each bouncing back and forth between the top and bottom edges.  Then the region of the rectangle below both zigzags, the region above both zigzags, and the region between the two zigzags split the rectangle exactly in thirds.

As far as we are aware, this result is new. In previous installments, Emily and Jasmine analyzed all the shapes formed by two zigzags and computed their areas. Applying these formulas and working through a lot of algebra, they were able to prove their result, though it seems to be some kind of miracle that all the rational expressions simplify to 1/3. Tune in for the next installment of Zigzags for a beautiful conceptual proof of their result by Harvard mathematician Noam Elkies. (For precise definitions and details, please see Zigzags, Parts 1 through 7 in the Girls’ Angle Bulletin.)

This also explains the cover, which is a recoloring of the drawing from the cover of Volume 12, Number 2. There, the coloring was produced using a random number generator and reflected Emily and Jasmine’s understanding of these patterns at that time. But with their theorem, order has been discovered and the new color scheme reflects this order.

If you’ve ever wondered why there are the 6 trigonometric functions sine, cosine, tangent, secant, cosecant, and cotangent, Lightning Factorial provides a Meditate to the Math on the topic.

Finally, we conclude with some Notes from the Club, which are authored by our Head Mentor Grace Work.

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!

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Happy New Year 2020!

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Girls’ Angle Bulletin, Volume 13, Number 2

Cover of the Girls' Angle Bulletin, Volume 13, Number 2The electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

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!

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Girls’ Angle Bulletin, Volume 13, Number 1

Cover of Volume 13, Number 1 of the Girls' Angle BulletinThe electronic version of the latest issue of the Girls’ Angle Bulletin is now available on our website.

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!

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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!

 

 

 

 

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Girls’ Angle Bulletin, Volume 12, Number 6

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

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!

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