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 Girls’ Angle Bulletin, Volume 14, Number 2
 Girls’ Angle Bulletin, Volume 14, Number 1
 Girls’ Angle Bulletin, Volume 13, Number 6
 Thirst For Firsts – A Girls’ Angle Raffle
 Girls’ Angle Bulletin, Volume 13, Number 5
 LCM Optimal Sequences
 Girls’ Angle Bulletin, Volume 13, Number 4
 Girls’ Angle Bulletin, Volume 13, Number 3
 Happy New Year 2020!
 Girls’ Angle Bulletin, Volume 13, Number 2
 Girls’ Angle Bulletin, Volume 13, Number 1
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Girls’ Angle Bulletin, Volume 14, Number 2
We open Volume 14 with the concluding half of our twopart interview with Williams College Associate Professor of Mathematics Pamela E. Harris. Here, she discusses work/life balance and aspects of learning mathematics, as well as describing some of the math she discovered.
Further on in this issue, Prof. Harris coauthors a wonderfully written introduction to parking functions with her student Kimberly Hadaway in Honk! Honk!, Part 1. If you ever wanted to learn about parking functions, this article is a great place to start. Parking functions are a neat example of turning an everyday real life problem into interesting mathematics.
Also, included is the concluding half of Deanna Needell’s survey of the various ways in which she and her team has studied Lyme Disease using machine learning. Interspersed throughout are insights about how to handle the various tools.
In Toblerone Game, four students give a complete analysis of a combinatorial game involving the sharing of a Toblerone candy bar. The idea is you have any number of Toblerone bars before you, and they can even be of different lengths. You wish to share it with your friend, though you still want to eat as much of the candy as you can. But, being polite, you divvy up the bar using the following civilized rules: You take turns. On any given turn, If there happens to be an isolated triangular piece, then you are free to eat that piece. If not, then you must split a bar and let your friend go. What is the optimal strategy? If you have to split a bar, which bar should you split, and where? You can find all the answers in this paper.
Another allstudent paper, or, I should say, saga, is The Saga of Fran & Fred. Ostensibly about high politics in the Kingdom of ABBABA, it is actually a probability paper in disguise.
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!
Posted in math, Math Education
Tagged Antonella Catanzaro, Eleanor Bogosian, Elizabeth Cavatorta, Esmé Krom, game theory, gorillas, induction, Kateri Escober Doran, Kimberly Hadaway, Molly Roughan, Naomi Danison, Pamela E. Harris, parking functions, probability, Toblerone, Viola Shephard, Violet Freimark
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Girls’ Angle Bulletin, Volume 14, Number 1
We open Volume 14 with the first half of a twopart interview with Williams College Associate Professor of Mathematics Pamela E. Harris. Prof. Harris’s journey into mathematics is extraordinary and quite unique. She also coauthored a wonderful contribution to the Bulletin in Volume 11, Number 2, entitled “Partitions from Mars”. Prof. Harris is actively involved in promoting minorities in mathematics. Among her many talents, Prof. Harris is expert at involving undergraduates in mathematical research.
Next, Deanna Needell gives us a survey of the various ways in which she and her team has studied Lyme Disease using machine learning. With the vast amounts of data being produced every second today, there is no hope for humans to analyze it all without the aide of computers. But computers are only as good as the algorithms that run on them. Data Science is a burgeoning field full of opportunities and promise.
Have you ever noticed how difficult it can be to keep track of multiple characteristics, even if the characteristics only come in two flavors, especially if the various characteristics influence each other? In Switcheroo!, Lightning Factorial tests your ability to stay organized in the face of many influencing, twovalued, switches. All you have to do is determine whether a light bulb is on or off. Good luck!
We enjoy turning activities that lend themselves to collaboration into mathematical activities. For example, crossword puzzles and jigsaw puzzles work really well as group activities, and at Girls’ Angle, we’ve “mathefied” both. (The first instance of a “mathefied” jigsaw puzzle is due to Girls’ Angle mentor Elise McCormackKuhman.) Word searches also lend themselves to collaboration, so we “mathefied” that this fall to create an activity that works well in the virtual world. Addie Summer provides a few example for you to try your hand at, as well as posing a number of math questions pertaining to these “number searches”.
Emily and Jasmine spent an afternoon building an icosahedron out of toothpicks. If you’re interested, they provide detailed instructions. They also analyzed the shape of the column of water that falls from a tap in a new installment of Emily and Jasmine’s adventures in The Water Column.
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!
Posted in applied math, math, Math Education
Tagged combinatorics, crafts, Deanna Needell, icosahedron, Lathisms.org, logic, Lyme disease, modeling, number search, Pamela E. Harris, primes, squares, switch problems, water
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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.)
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 twopage 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 CourcyIreland, 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!
Posted in Uncategorized
Tagged Annalisa Crannell, Annie Yun, Bernoulli numbers, bias, Deanna Needell, Diophantine equations, disphenoids, Fibonacci, Grace Work, Isabel Wood, isosceles tetrahedra, Jasmine Zou, Liliana Smolen, matrices, Matthew de CourcyIreland, Pascal's triangle, Pell's equation, perfect squares, perspective drawing, Princeton University Press, Pythagorean triples, tetrahedra, vectors
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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 x + y, whereas the cost of traveling from x straight to y is LCM(x, y). 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 1319 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
.
Is that true for all prime numbers n?
If you find anything, please do let us know at girlsangle “at” gmail.com!
Girls’ Angle Bulletin, Volume 13, Number 4
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 UrbanaChampaign 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 COVID19 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 multidecker openfaced 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 nzigzag and an mzigzag, 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!
Posted in applied math, math, Math Education
Tagged Ana Chrysa Maravelias, Antonella Catanzaro, Bradford Kimball, COVID19, Darius Sinha, Deanna Needell, Erin Compaan, geometry, Henry Kirk, Jaemin Feldman, LCM, Mikayla Higgins, Multidecker openface meat sandwich, Noam Elkies, Optimal paths, Traveling salesman, zigzags, zigzags theorem
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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 UrbanaChampaign 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 oneway 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 wellstudied sequence of numbers known as the Eulerian numbers. Primary guidance and mentorship for this mathematical investigation was provided by MIT undergraduate Adeline Hillier.
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 nzigzag and an mzigzag, 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!