## Girls’ Angle Bulletin, Volume 8, Number 6

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.

Does this make you go “Hmm…”?

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!

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!

## Origami-inspired Proof of the Pythagorean Theorem

While working with math teachers Richard Chang and Randi Currier of the Buckingham Browne & Nichols Middle School to come up with problems that use polynomials and rational expressions, we stumbled upon an origami-inspired proof of the Pythagorean theorem that I’d like to share. Continue reading

## Girls’ Angle Bulletin, Volume 8, Number 5

Volume 8, Number 5 of the Bulletin kicks off with an interview of Ivana Alexandrova. Ivana is an Assistant Professor of Mathematics at the State University of New York, Albany.  Among other things, she maintains a webpage of weekly problems for high school students. Check it out!

The topic of induction came up quite a few times this spring at the Girls’ Angle club, so next comes an article on this widely used proof technique.

This issue’s Learn by Doing features irrational numbers and culminates in a series of problems that let you reconstruct a proof of the irrationality of $\pi$ due to Charles Hermite.

Anna tackles one of Prof. Alexandrova’s weekly problems for high school students in Anna’s Math Journal, finding 3 different ways to solve the problem, which is to compute $\cos 72^\circ \cos 36^\circ$. Can you find your own solution?

Next comes our 4th installment on the derivative where we find the derivatives of the basic trigonometric functions. The way we deduce the derivative of sine is similar in spirit to the way we showed that the area under one hump of a sine curve is exactly 2.

Since this is our June issue, we include the 2015 Summer Fun problem sets. This batch contains problems pertaining to telescoping series (by Fan Wei), induction, the symmetric group (by Noah Fechtor-Pradines), and derivatives.

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!

## Conceptual Solution to 2008 AIME I, Problem 14

Here’s a solution to problem 14 on the 2008 AIME I contest that attempts to avoid computation as much as possible. Sometimes, it’s an amusing exercise to try to solve a contest problem entirely in your head. Doing so often forces one to see deeper into the math.

First, the problem:

Let $\overline{AB}$ be a diameter of circle $\omega$. Extend $\overline{AB}$ through $A$ to $C$. Point $T$ lies on $\omega$ so that line $CT$ is tangent to $\omega$. Point $P$ is the foot of the perpendicular from $A$ to line $CT$. Suppose $AB = 18$, and let $m$ denote the maximum possible length of the segment $BP$. Find $m^2$.

Here’s an illustration of the situation:

We must maximize the length of the red line segment $\overline{BP}$. Continue reading

## Girls’ Angle Bulletin, Volume 8, Number 4

Volume 8, Number 4 of the Bulletin kicks off with the concluding half of our 2-part interview with Hamilton College Assistant Professor of Mathematics Courtney Gibbons.  In this second half, one of the objects she describes are the Cayley graphs of groups.  This inspired the creation of several Cayley graphs by members of Girls’ Angle, which are featured in this issue’s Math Buffet.

The cover itself also shows a Cayley graph of $S_5$.

## Snowball Problems Melted

The snow from the record setting snowfalls in Boston are pretty much gone, so here are solutions to the two snowball problems from the February.

## Solution to Snowball Problem #1.

We claim that every point in space on the side of the plane with the rolling snowball and within a snowball diameter of the plane can be visited by the marked point.

Let C be the circumference of the snowball. Any positive distance less than 2C can serve as the base of an isosceles triangle with equal sides of length C. Draw this triangle in the plane and put the snowball at one endpoint of the base with its marked point touching the plane. Roll the snowball along the two sides of length C. It ends up at the other endpoint of the base with its marked point touching the plane. This shows that the marked point can visit any point within a circle of radius 2C from its starting point. By taking several such journeys, we see that every point in the plane can be visited. For any point P on the same side of the plane as the snowball and within a snowball’s diameter of the plane, position the snowball so that its marked point is at P. We can roll this snowball until the marked point is touching the plane and since all points in the plane are visitable, we conclude that P can also be visited.

## Solution to Snowball Problem #2 (Noah Fechtor-Pradines).

We claim that the locus of visitable points is a cardioid of revolution.

Let’s mark the point on the sphere where the marked point on the snowball touches it at the start with a golden dot. Look at the situation from the point-of-view of the common tangent plane. The two balls roll as mirror images of each other. The marked points are reflections of each other in the common tangent plane.

In other words, no matter how the snowball rolls about the sphere, the marked point will be located at the reflection of the golden dot in the common tangent.

We conclude that the locus of visitable points is a surface with rotational symmetry (about the axis one of whose poles is the golden dot). A vertical cross section by a plane containing the pole of the stationary sphere is the curve traced out by a dot on a circle as it rolls about a circle of the same diameter, and this is known to be a cardioid.