Volume 9, Number 5 kicks off with an interview with the ADVANCE Professor of Computing at Georgia Institute of Technology, Dana Randall. Prof. Randall is also the Director of the Algorithms and Randomness Center and an Adjunct Professor of Mathematics.

Next, comes the concluding half of Heidi Hurst’s article on how she used math to help find optimal locations for FEMA Disaster Recovery Centers.

Emily and Jasmine continue their exploration of nice triangles by commencing with an exploration of which triangles have 3 “nice” angles and 2 sides of integer length. This is the last major case in their investigation of “nice” triangles and involves the most advanced mathematics.

Anna manages to generalize the result she found in the last issue to the finite fields , where is prime. This result was known to Carl Friedrich Gauss.

We conclude with this summer’s batch of Summer Fun problem sets! This summer, we have contributions from Matthew de Courcy-Ireland, Aaron Levy, Lauren McGough (who is also a Girls’ Angle Advisory Board member and the Girls’ Angle Treasurer), Long Nguyen, and Zachary Sethna. Matthew, Aaron, Lauren, and Zachary are all graduate students at Princeton University. Long is an undergraduate at MIT.

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 subscr

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Volume 9, Number 4 begins with an interview with Oberlin’s Andrew and Pauline Delaney Professor of Mathematics Susan Jane Colley, although as has been our practice, we truncate the interview in the electronic version. For the full version, please subscribe!

After a meditation on Miquel’s theorem, find out how a frog fish, dragon, and mathematician end up in the same story with the concluding half of *The Mountain Clock*, a fictional math story about an ancient city that adopted a 32-hour day.

Anna manages to prove a theorem she has been leading up to ever since Prof. Judy Walker suggested some exercises involving polynomials over the finite field with 2 elements.

Next comes a neat application of mathematics to disaster recovery. Harvard University undergraduate Heidi Hurst spent a summer using math to figure out how to find optimal locations for Disaster Recovery Centers. She explains her work in a two-part *Math In Your World* series.

Emily and Jasmine continue their exploration of nice triangles and establish a direct link between integer-sided triangles with perimeter *p* and partitions of *p* – 3 into parts with sizes 2, 3, and 4.

We conclude by unraveling some secrets of a marvelous recurrence relation created by one of our 11-year-old members.

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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Volume 9, Number 3 begins with the concluding half of an interview with French mathematician Alice Guionnet. Prof. Guionnet is a professor of mathematics at MIT and an expert on random matrices.

The cover features an illustration by Julia Zimmerman for a fictional math story entitled *The Mountain Clock*. The Mountain Clock tells the story of an ancient King who decreed a 32-hour day.

Katherine Cliff, a graduate teaching fellow at the University of Colorado, Colorado Springs, contributes this issue’s *Math In Your World*. She brings the rational world of mathematics to the computation of probabilities for the Powerball lottery. If you work through this installment, you’ll be able to compute lottery probabilities and the value of a lottery ticket from scratch.

Anna continues her investigation of irreducible polynomials over the finite field with 2 elements. In this installment, she formulates 2 conjectures, works out implications of them, and proves one of them. Anna’s entire investigation traces back to an exercise suggested by Prof. Judy Walker in her interview from Volume 8, Number 6. Can you prove the conjecture she doesn’t prove?

Emily and Jasmine continue their quest for “nice” triangles. This time, they explore integer-sided triangles with no conditions at all on their angles. They determine the number of such triangles that have 2 given side lengths and begin to study the question of how many integer-sided triangles there are of a given perimeter.

In this issue’s *Learn By Doing*, you can try your hand at coming up with counterexamples. There are few things that so decisively prove a conjecture wrong than coming up with a counterexample!

There’s a lot more inside and 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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Thank you so much to Ken for allowing me to contribute a guest blog! I am a middle school math teacher at the Pollard Middle School in Needham. I have had the extreme pleasure of having Ken come to Pollard for the past four years to host a Math Collaboration. The events have been thoroughly amazing. From a teacher’s perspective, there is nothing I enjoy more than seeing students working tirelessly and collaboratively towards a common goal. The puzzles and problems that Ken puts together are the perfect balance of stretching students’ thinking while also making them attainable, and his gentle manner of prodding students without guiding them is something that I try to emulate on a daily basis in my own classroom. I cannot do justice to how exceptional these events are, so I asked two students who participated in the latest Collaboration to write their thoughts on the event. Here is what they wrote.

“Recently we participated in the math collaboration treasure hunt hosted by Girls’ Angle. This event was after school on an early release day, and beforehand it didn’t seem like the best way to spend our time out of school. We were surprised by the complexity of the puzzles and impressed at how fast we solved it. It was a change to not be given detailed instructions or told exactly what to do. Instead we had to figure out how to be organized and efficient, without relying on adults. At first everything was chaotic and nobody knew exactly what to do. But then we established a system with small groups working on each problem, and everybody helping those who needed it. This activity taught us to work together and help others to achieve a common goal. In the end, we were only able to solve the puzzle because every single person contributed in some way. We had to be independent, work together, use leadership skills, be organized and listen to everybody’s ideas in order to complete the treasure hunt and receive the prize. This collaboration was an amazing learning experience and built leadership and teamwork skills, but more importantly a fun way to spend an afternoon.”

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Volume 9, Number 2 begins with the first half of an interview with French mathematician Alice Guionnet. Prof. Guionnet is a professor of mathematics at MIT and an expert on random matrices.

In addition to the interview, there are 2 more articles in this issue that pertain to probability and statistics. One is the concluding half of Prof. Elizabeth Meckes articles on the laws of probability. This time, she pulls up the curtain on the central limit theorem. The other is this issue’s Math In Your World, which describes an activity led by Girls’ Angle Support Network visitor Jinger Zhao. Jinger is a financial modeler who works at TwoSigma, a hedge fund based in New York City. At the club, Jinger uses statistics to model the connection between wingspan and height.

Anna continues her investigation of irreducible polynomials over the finite field with 2 elements. In this installment, she works out the roots of all the irreducible polynomials of degrees 4 and 5. Anna’s entire investigation traces back to an exercise suggested by Prof. Judy Walker in her interview from Volume 8, Number 6. Do you think you can see where Anna might be headed? If you do, follow your thoughts and see where they lead. You’re invited to tell us about it; we’d love to hear from you. If you’re falling in love with polynomials over , check out this proposal for a new PolyMath project and the comments that follow.

Multiplication and exponentials are fundamental concepts in mathematics. For anyone working on learning these concepts, we hope Addie Summer’s *Thoughts on Multiplication* and our Learn by Doing on exponentials will be of use. The cover, which was created using MATLAB by MathWorks, honors multiplication.

Emily and Jasmine continue their quest for “nice” triangles. This time, they explore integer-sided triangles that have a 120 degree angle and establish a beautiful bridge between these and integer-sided triangles with a 60 degree angle.

We conclude with some notes from the club. If you’re a girl, aged roughly 10-18 in the Greater Boston Area, you’re welcome to join. Our next session begins January 28, 2016.

We hope you enjoy the Bulletin!

*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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Volume 9, Number 1 features material from two mathematicians: an extensive interview with Elizabeth Munch, Assistant Professor of Mathematics at the University of Albany and the first of a two-part article on the laws of probability by Elizabeth Meckes, Associate Professor of Mathematics at Case Western Reserve University.

Prof. Munch’s road to mathematics is interesting in that music plays a significant role in her life. In addition to her mathematical degrees, she also holds a degree in Harp performance from the Eastman School of Music. Find out how her musical studies affected her discovery of mathematics. In her interview, she mentions Takens embeddings, and the cover features a kind of Takens embedding. The embedding is actually a 4-dimensional Takens embedding consisting of two spatial dimensions and two that extend into the color space.

Statistics plays an important role in so many sciences. Research would grind to a halt without it. One of the most important results in statistics is the central limit theorem. Prof. Meckes has contributed an eloquent two-part article that explains the meaning of this theorem. In part 1, she explains the laws of probability.

In Math In Your World, Lightning Factorial uses statistics to improve at darts, following the lead of statisticians Ryan Tibshirani, Andrew Price, and Jonathan Taylor, who showed in their paper *A Statistician Plays Darts*, that depending on your dart throwing prowess, you might be better off not aiming for the bull’s-eye. Skeptical? Read Lightning’s article!

In Anna’s Math Journal, Anna continues her investigation of irreducible polynomials over the finite field with 2 elements. This investigation traces back to suggested exercises made by Prof. Judy Walker in the previous issue. If you’re unfamiliar with finite fields but want to follow along with Anna on her journey, this issue’s Learn by Doing is just for you. In it, finite fields are introduced assuming very little by way of prerequisites.

Meanwhile, Emily and Jasmine continue their quest for “nice” triangles. This time, they apply a technique that is often used to find Pythagorean triples to find formulas that yield the sides of all primitive triangles that contain a 60 degree angle, such as the 5-7-8 and 16-19-21 triangles.

We conclude with some notes from the club! We’ve got a wonderful group of members this semester and if you’re a girl in grades 5-12 who lives near Cambridge, MA, you’re welcome to attend!

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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(Added October 1, 2015: This raffle is now closed. Thank you to all who entered. Congratulations to Fran M. who won the general draw and Iris L. who won the member draw!)

Can you reconstruct Serena and Katherine’s path through the museum?

They start at the museum entrance. They choose a random direction, north, south, east, or west and walk together in that direction until they come upon the first exhibit that they had not yet visited and stop. When done looking at that exhibit, they again choose a random direction, north, south, east, or west and walk together in that direction until they come upon an exhibit that they had not yet visited and stop. They continue wandering in this manner. Miraculously, they manage to visit all 17 exhibits!

Figure out all 17 directions Serena and Katherine took and the order that they took them. Send your answer to girlsanglepuzzler “at” gmail.com by **midnight on September 30, 2015**. For example, you might send the string “NSEWNSEWNSEWNSEWN” if you think those are the directions they went in that order, left to right. On October 1, 2015, two random “winners” will be drawn from correct entries, one from among Girls’ Angle members and the other from among the general public. The two “winners” will receive a modest prize. (The member prize is bigger.)

Exhibits are marked by flag poles. Determine the path that goes from flag pole to flag pole. They start at the museum entrance and end at one of the exhibits.

Good luck!

(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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The object on the cover of Volume 8, Number 6 is called the Chebyshev Lollipop. It is based directly on an idea of Michael Trott. The mathematical content differs slightly from his creation which can be seen at MathWorld.

Chebyshev polynomials (of the first kind) appear in this summer’s batch of Summer Fun problem sets and in a new Emily and Jasmine series which commences in this issue.

In school, Jasmine happened by a geometry class where the teacher had the peculiar figure shown at right on the board. That figure turned out to be the launch point for an adventure in search of nice triangles.

This issue’s interview is with the Chair of the Department of Mathematics at the University of Nebraska-Lincoln, Professor Judy Walker. In her interview, Prof. Walker gives some pointers for how to learn mathematics well, saying “I absolutely must work through examples.”

In Anna’s Math Journal, Anna takes up Prof. Walker’s specific example suggestion and explores finite fields with 4 and 8 elements

In Part 5 of our series on the derivative, we explore the derivative of the exponential function.

We wrap up with detailed solutions to this summer’s batch of Summer Fun problem sets which include proofs of the arithmetic-geometric mean inequality, a derivation of the Taylor series of the arctangent function, and a proof of Lagrange’s theorem that, in finite groups, the order of any subgroup divides the order of the group.

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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I haven’t seen this proof before, but there are so many proofs of the Pythagorean theorem that it wouldn’t surprise me if this has already been discovered. If you’ve seen this proof published elsewhere, please send email to Girls’ Angle and let me know.

I am aware of other “origami” proofs of the Pythagorean theorem, although the ones I’ve seen don’t particularly use origami so much as that they take some diagram used to illustrate a proof and then render the lines in the diagram as creases in paper.

This particular proof is not elegant, but it does show how one might discover the Pythagorean theorem through a natural line of exploration starting with an origami square. Also, it really uses the notion of an origami fold as its key ingredient.

All you have to do is think about what happens when you fold a corner of the origami square to a point along one of the further edges, as illustrated below.

That single fold produces 3 triangles and a quadrilateral.

Notice that the 3 triangles are all similar right triangles. (Here, we must use the fact that a square has 4 right angles.)

Since the entire model is determined by the parameter , a natural question to ask is: What are the lengths of all the other line segments in terms of ?

To get the answer, we will repeatedly apply two facts. First, that the 3 triangles are similar to each other. And second, that the edge lengths along a side of the square add up to the side length of the square, which we’ll go ahead and take to be one unit for now.

We’ll start by letting be the length of the other leg of . We’ll then express the other lengths in terms of and and find an equation that relates the two. By solving for in terms of , we will effectively have succeeded in expressing the lengths in terms of .

Here goes!

We add the label to the diagram:

The hypotenuse of and the leg of length make up what was the right side of our square, so the hypotenuse of has length :

One of the legs of extends to the full side of the square:

Using the fact that and are similar, we can find the other side lengths of :

One leg of and the hypotenuse of form what was the upper side of the origami square:

Next, we use similarity of with to see that its other sides have length:

and .

So far, we’ve used the right, bottom, and top sides of the original square. So now we use the left side. Studying the figure, we see that the original left side of the square has become one leg of and the last two sides we computed of . Thus,

.

If you solve for in terms of using this equation, you will find that .

Thus, the legs of are and and its hypotenuse has length .

Notice that as varies from 0 to 1, the ratio of the leg lengths of covers the entire range from 0 to infinity (i.e. the range of is when is restricted to the interval ). This means that a representative of every similarity class of right triangle can be folded by making a suitable choice of .

Therefore, by scaling the origami square if necessary, we can make congruent to any given right triangle.

The form of the expressions for the lengths of the sides of in terms of suggests the possibility of finding an explicit relationship between these 3 side lengths, especially since one of the side lengths is just , and if we can find such a relationship that is also homogeneous in these lengths, then the relationship will hold for *all* right triangles since a homogeneous relationship is preserved by scaling. And, indeed, the desired algebraic identity is , hence, the Pythagorean theorem. (Making note of this algebraic identity without knowing the Pythagorean theorem might be a bit of a trick, but it doesn’t seem unreasonable to think it possible.)

(Note that as well, and this corresponds to the equation , where and are the legs of a right triangle with hypotenuse of length . However, this identity is *not* homogeneous in the leg lengths, so it is not true that for all right triangles. The equation holds only for right triangles where , i.e. those folded from a unit square.)

In an earlier post, we described David Gale’s method for constructing Pythagorean triples. There, our aim was to find Pythagorean triples, not prove the Pythagorean theorem, so when we spoke of finding the lengths of , we had in mind a method that made use of the Pythagorean theorem. One could say that the content of this blog post is that you don’t have to assume the Pythagorean theorem there because you can deduce it instead. In other words, one-fold origami can deliver the Pythagorean theorem from scratch as well as all Pythagorean triples.

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