# Archive

### May 12, 2018

**Beginner Group:**

**Speaker: **Max Warshauer, Texas State University

**Title: **Modular arithmetic and divisibility

**Abstract:** In this Math Circle, participants will learn about modular arithmetic and use this to discover divisibility properties. This will be a hands-on exploration where participants investigate problems, make conjectures, and then give careful arguments to prove or disprove their conjectures.

**Intermediate Group:**

**Speaker: **Alex Sprintson, Texas A&M University

**Title: **Introduction to Finite State Machine Design

**Abstract:** We will discuss mathematical aspects of logic design. We will start with a review of the fundamentals of boolean algebra and design of K-maps. Next, we will discuss the fundamentals of design and implementation of the Finite State Machines (FSM) that solve engineering problems.

**Advanced Group:**

**Speaker: **Amudhan Krishnaswamy-Usha, Texas A&M University

**Title: **Rigid motions and symmetries in 2 and 3 dimensions

**Abstract:** Rigid motions are transformations which preserve angles and distances (for example, rotations). We will explore what these look like in two and three dimensions and attempt to classify them. If time permits, we will also look at rigid motions which preserve a given figure (the symmetries of the figure), and see what happens in the case of regular polygons.

### May 5, 2018

**Beginner Group:**

**Speaker: **Erica Metheney, Texas A&M University

**Title: **Intro to Data: What’s the Story?

**Abstract:** We will begin with discussing what statistics is and why it is useful. During the session, students will learn about different data types, summary statistics, and data visualization techniques then apply their new skills on data they will collect themselves. After the session, students should be able to identify data types, summarize different types of data, and create appropriate data visualization graphics.

**Intermediate Group:**

**Speaker: **Theodora Chaspari, Texas A&M University

**Title: **How is mathematics used in artificial intelligence and machine learning applications?

**Abstract: **Artificial intelligence (AI) is the field of study that focuses on machines or computer programs that are capable of thinking, acting, and learning like humans. Machine learning is the sub-field of AI that develops computer programs able to access and automatically learn from data without human assistance or intervention. AI and machine learning applications are found in every aspect of everyday life from health and well-being to entertainment and military applications. Examples of these include online games, self-driving cars, chatbots, fitness monitors, and others. Mathematics is one of the most important tools for developing AI and machine learning applications since they allow representation of real-world objects into numbers and derive meaningful interpretations from these numbers. For example, an image taken by our camera can be represented as a matrix of integers, whose structure can convey meaningful information regarding the content of the image (e.g., whether an image contains a cat or a dog). This lecture will cover basic AI and machine learning concepts and will discuss how mathematics is tightly connected to these, inspired by two applications from the computer vision and biomedical health informatics domains.

**Advanced Group:**

**Speaker: **John Weeks, Texas A&M University

**Title: **Groups, Cosets, and Lagrange’s Theorem

**Abstract:** We will explore the concepts of an algebraic group and subgroup and discuss some natural partitions of a group called cosets. If time allows, we will use modulo arithmetic to investigate how the number of elements in these groups compares with that of any subgroup contained within them. This will be an elementary talk with a goal of introducing students to the nature of upper-level mathematics.

### April 28, 2018

**Speaker: **David W. Gent, P.E. (Founder and Chairman of **S**of* T*est Designs)

**Title: **Amateur Radio Satellite Orbital Mechanics

**Abstract:** There are countless objects currently orbiting Earth. Some of these are Communication Satellites built by Amateur Radio operators like me.

To talk through these Satellites, we must chart their current locations in space and predict their future passes above our horizon. This tracking depends upon various concepts developed by several brilliant mathematicians over the years. This is the story of that development. Example calculations using the International Space Station (ISS) are given.

### April 21, 2018

**Beginner Group:**

**Speaker: **David Sykes, Texas A&M University

**Title: **The Icosian Game

**Abstract:** We will play variants of William Rowan Hamilton’s icosian game and explore its relation to other mathematical puzzles.

**Intermediate Group:**

**Speaker: ** Pedro Morales, The University of Texas at Austin

**Title: **Internet privacy: How secure communication workss

**Abstract: **We will explore one of the most used encryption methods, known as the RSA. This is one of the most important applications of number theory and prime numbers. By applying basic concepts of modular arithmetic, we can develop a public key system which enables the secure exchange of information between two strangers.

**Advanced Group:**

**Speaker: **Eviatar Procaccia, Texas A&M University

**Title: **Simulations of random walks

**Abstract:** We will learn basic MATLAB programming and then simulate a process called “simple random walk”, which models fundamental natural phenomena. We will use these simulations to compute some probabilities and verify the numerical results with theoretical calculations.

### April 14, 2018

**Beginner/Intermediate Group:**

**Speaker: **Josiah Coad, Texas A&M University

**Title: **Everyone Can Code!

**Abstract:** We are excited to announce a new opportunity for the students in TAMU Math Circle to learn how to code. We will be using a website code.org, which has developed many fun and educational programs for kids of all ages and skill levels to learn how to code, no previous experience necessary. In an hour, the students will learn through a drag/drop interface the basics of coding logic, all while programming their own game! With many opportunities to rank up fast, we hope to inspire students to a self-discovery of this exciting field.

**Advanced Group:**

**Speaker: **Roger Howe, Texas A&M University

**Title: **The medial triangle, the Euler line, and the nine-point circle.

**Abstract:** Studying the relationship between a triangle and the triangle formed by the midpoints of its sides (known as the medial triangle) gives deeper insights into the triangle geometry than were found by the Greeks. A key part of this development is the Euler line, which exhibits a beautiful relationship between circumcenters, orthocenters, and centroids. This in turn sheds light on the nine-point circle and beautiful related configurations.

### March 24, 2018

**Beginner Group:**

**Speaker: **Josiah Coad , Texas A&M University

**Title: **Everyone Can Code!

**Abstract:** We are excited to announce a new opportunity for the students in TAMU Math Circle to learn how to code. We will be using a website code.org, which has developed many fun and educational programs for kids of all ages and skill levels to learn how to code, no previous experience necessary. In an hour, the students will learn through a drag/drop interface the basics of coding logic, all while programming their own game! With many opportunities to rank up fast, we hope to inspire students to a self-discovery of this exciting field.

**Intermediate Group:**

**Speaker: ** Dimitar Grantcharov, The University of Texas at Arlington

**Title: **Invariants

**Abstract: **Invariants are special “mysterious” tools that play an important role in various mathematical fields. We will solve several problems from number theory, game theory, combinatorics, and geometry that use invariants.

**Advanced Group:**

**Speaker: **David Sykes, Texas A&M University

**Title: **Counting Polyominoes

**Abstract:** We will be counting the number of polyominoes sharing certain properties. The activity will present several fun combinatorial problems, some of which remain unsolved; for example, to date, there is no explicit formula for the number of polyominoes having a given size.

### March 3, 2018

**Beginner Group:**

**Speaker: **David Sykes , Texas A&M University

**Title: **Planar Graphs

**Abstract:** Given three cottages and three wells, can we find non-intersecting paths so that every cottage is connect to each well by a different path? We will consider this problem along with others that introduce the topic of planar graphs.

**Intermediate Group:**

**Speaker: **Nathan Green, Texas A&M University

**Title: **Polya Counting

**Abstract: **Polya counting theory allows us to count how many ways there are to arrange objects taking symmetry into account. For example, how many different bracelets can we make using only 3 colors of beads? How many ways can we color a cube using n colors? This counting technique has been used to count the number of different molecules which can be formed from certain sets of atoms and many other important applications.

**Advanced Group:**

**Speaker: **John Weeks, Texas A&M University

**Title: **Relations, Equivalence Classes, and Langrange’s Theorem

**Abstract:** We will give a few definitions related to the study of relations and introductory group theory, inquire into some examples of equivalence classes, and utilize this information to analyze the nature of subgroups in finite groups.

### February 17, 2018

**Beginner Group:**

**Speaker: **Amudhan Krishnaswamy-Usha , Texas A&M University

**Title: **Fractions and bases

**Abstract:** We will look at decimal expansions of fractions and try to determine when they terminate. We will then try to expand fractions in other bases (such as binary).

**Intermediate Group:**

**Speaker: **Frank Sottile, Texas A&M University

**Title: **Shape of Space

**Abstract: **In mathematics and science, we often need to think about high (3 or more) dimensional objects, called spaces, which are hard or impossible to visualize. Besides the question of what such objects are or could be, is the problem of how we can make sense of such spaces.

The goal of this discussion is to give you an idea of how mathematicians manage to make sense of higher-dimensional spaces. We will do this by exploring the simplest spaces, and through our explorations, we will begin to see how we may tell different spaces apart. Along the way, we will dissect donuts, and I ask that at least half of the participants bring a belt.

**Advanced Group:**

**Speaker: **Parth Sarin, Texas A&M University

**Title: **Pancakes, Ham Sandwiches, and Topology

**Abstract:** Topology is a field of mathematics that tries to understand the shape of things without regards to distance or angles. In this Circle, we’ll explore some famous and surprising concepts from Topology. For example, we’ll consider whether there are two opposite points of the Earth that have the exact same temperature and pressure. And, we’ll explore how these questions are related to things you wouldn’t expect, like whether or not you can cut two pancakes in half with a very large knife using a single cut.

### February 10, 2018

**Beginner Group:**

**Speaker: **David Sykes , Texas A&M University

**Title: **Patio Planning Problems

**Abstract:** How many ways can we configure non-overlapping square tiles to build a patio with a given shape and a given perimeter? We will explore variations of this problem by drawing patio designs with small perimeters and using what we will find to make informed guesses about the answer for larger perimeters and that we then prove or disprove.

**Intermediate Group:**

**Speaker: **Valentin Zakharevich, The University of Texas at Austin

**Title: **Symmetry and Affine Transformations

**Abstract:** One of the most important ideas in geometry is that of symmetry. Understanding the symmetry of a problem can often significantly simplify finding a solution. In this presentation, we will be considering the affine symmetries of the plane, i.e. the symmetries which preserve straight lines. We will apply these ideas to understand theorems of Ceva and Menelaus.

**Advanced Group:**

**Speaker: **Tom Gannon, The University of Texas at Austin

**Title: **How to Make Friends with Graph Theory

**Abstract:** We’re going to learn about a subject called graph theory, which will be sure to impress all your friends. Graph theory is a subject about dots and lines and the various ways you can draw them. We’ll talk about complete graphs and about how friendship can be modeled by graph theory. We’ll also discuss a problem that no one on earth knows the answer to!

### February 3, 2018

**Beginner Group:**

**Speaker: **Philip Yasskin , Texas A&M University

**Title: **Cell Phone Dropping

**Abstract:** You work for a cell phone company. For advertising purposes, you are assigned the task of testing a new model of phone protector by dropping a phone from various floors of a 100 story building to determine the highest floor from which it can be dropped and not break. What is the most efficient way to perform this task if you are given 1, 2, or 3 phones?

**Intermediate Group:**

**Speaker: **Frank Sottile, Texas A&M University

**Title: **Balls and Boxes: Common shapes in uncommon dimensions

**Abstract:** Today, we will explore how common shapes- balls, cubes, triangles, and others –behave strangely in high-dimensional space. This is also an explorations of regular solids in dimensions greater than four. This is independent of last week’s circle activity.

**Advanced Group:**

**Speaker: **Alex Sprintson, Texas A&M University

**Title: **Fun with Finite State Machines

**Abstract:** We will continue our discussion about design and analysis of Finite State Machines (FSM). We will talk about minimization and equivalence problems. Towards the end, we will attend to write a program that plays a short five note song. *Please bring your computer if at all possible.*

### January 27, 2018

**Beginner Group:**

**Speaker: **Alex Sprintson, Texas A&M University

**Title: **Pi Math Contest (PiMC)

**Abstract:** We will work on the Pi Math Contest (PiMC) written by an expert committee, many of whose members are from MIT/Harvard/Stanford (see pimathcontest.com). All students in 4th and 5th grade students will officially participate in the contest. However, all students in the beginner group will work on the problems. We will discuss the problem and their solutions later in the circle. Top scoring students in this round will be invited to a Final Round in Bay Area, California on April 28th 2018.

Rulers and compasses are allowed. Calculators are not allowed (no problem on the test will require the use of a calculator). The students are strongly encouraged to visit https://alphastar.academy/event/pimc/#PiMC_2017 to see prior year tests and solutions. Top students in the first round are invited to the final round.

**Intermediate Group:**

**Speaker: **Frank Sottile, Texas A&M University

**Title: **Regular solids in all dimensions

**Abstract:** While we are all familiar with regular polygons (equilateral triangles, squares, …), and many of us know about the Platonic, or regular solids (tetrahedron, cube, octahedron,….), few are familiar with their analogs in dimensions four and higher. Of course, this is because we are not equipped to perceive four-dimensional space directly.

Nevertheless, the description of the regular solids in all dimensions has been known for a long time. The purpose of my talk will be to introduce you to these objects, with an emphasis on how to think about them. This presentation will be spiced up with some models of four-dimensional regular solids, some of which you can build yourself. This is independent of last week’s circle activity.

There is a link to an animation:

http://www.math.tamu.edu/~sottile/talks/17/4D/index.html

**Advanced Group:**

**Speaker:** Philip Yasskin, Texas A&M University

**Title: **Point Set Topology

**Abstract:** We will discuss the basic definitions of point set topology.

### January 20, 2018

**Beginner Group:**

**Speaker: **Amudhan Krishnaswamy-Usha, Texas A&M University

**Title: **Counting with Aliens

**Abstract:** We will discuss different number system and discover how to convert numbers from one base to another.

**Intermediate Group:**

**Speaker: **Frank Sottile, Texas A&M University

**Title: **Archimedean Solids

**Abstract:** Most of us know the five Platonic (or regular) solids; next to the sphere, they are the most regular and beautiful objects in our three-dimensional world. Less well-known are the Archimedean or semi-regular solids. In this math circle activity, we will recall the Platonic solids, and then explore the Archimedean solids and some relations between them. We will be building them and then studying our constructions. If time, I will explain their relation to fair dice.

**Advanced Group:**

**Speaker:** Alex Sprintson, Texas A&M University

**Title:** Introduction to Finite State Machine Design

**Abstract:** We will discuss mathematical aspects of logic design. We will start with a review the fundamentals of boolean algebra and design of K-maps. Next, we will discuss the fundamentals of design and implementation of Finite State Machines (FSM) that solve engineering problems. If time permits, we will discuss the capabilities and limitations of FSM.

### December 9, 2017

**Elementary Group – Dean Baskin on Towers of Hanoi**

**Presenter:** Dean Baskin

Department of Mathematics, Texas A&M University

**Abstract: **This activity is aimed at students in second grade. We will introduce the Towers of Hanoi game, work out how to solve it in small cases and try to find a pattern in the number of moves required. If there is time left at the end, we will play some mathematically oriented games.

**Beginner/Intermediate Group- Jane Long on Factors and Primes**

**Presenter: **Jane Long

Department of Mathematics, Stephen F. Austin University

**Abstract: **Prime numbers, those counting numbers with exactly two distinct factors (themselves and one), are very special in mathematics. We’ll discuss ways to find prime numbers and other factors of counting numbers, and investigate perfect numbers, amicable numbers, and some really big numbers.

**Intermediate/Advanced Group – Nicholas Long on Counting Rectangles with Integer Sequences**

**Presenter: **Nicholas Long

Department of Mathematics, Stephen F. Austin University

**Abstract: **When you look at a chess board, you can see lots of squares. How many do you see? How many rectangles can you make with the blocks on a chess board? What if the chess board was a different size? Slightly different figures can lead to wildly different patterns when you think flexibly about where squares and rectangles come from. We will also look at how the sequence of integers generated by these other questions can lead to a lot of related problems on the OEIS (Online Encyclopedia of Integer Sequences).

### November 25, 2017

**Beginner Group – Amudhan Krishnaswamy-Usha on Fibonacci and Other Numbers**

**Presenter:** Amudhan Krishnaswamy-Usha

Department of Mathematics, Texas A&M University

**Abstract: **Continuing the discussion led by David Sykes’ last math circle, we will look at Fibonacci numbers and a few of their properties. If time permits, we will look at a few other sequences coming from ‘recurrence relations’.

**Intermediate Group- Phil Yasskin on When do 4 or more Points Lie on a Circle?**

**Presenter: **Phil Yasskin

Department of Mathematics, Texas A&M University

**Abstract: **We will first answer the question: “When do 4 points lie on a circle?”. Then we will prove the Nine Point Theorem and maybe some others.

**Advanced Group – Frank Sottile on The Five Color Theorem**

**Presenter: **Frank Sottile

Department of Mathematics, Texas A&M University

**Abstract: **Doodling on a map of England in 1852, Francis Guthrie noticed that only four colors were needed to color the counties. He conjectured that any map could be colored with only four colors. Several mathematicians tried and failed to prove this; notably in 1879 Kempe published a proof and only in 1890 was the flaw found by Heawood. This four color conjecture evaded a proof until 1972, when Appel and Haken gave a proof that required a computer. While there is as yet no Human readable proof, Kempe’s argument suffices to prove that five color suffice, and this gives a flavor of known proofs of the four color theorem. I will sketch this history and prove the five color theorem, and then discuss the coloring theorem for other surfaces (torus, projective plane, Klein bottle…).

### November 18, 2017

**Beginner Group – David Sykes on The Fibonacci Sequence**

**Presenter:** David Sykes

Department of Mathematics, Texas A&M University

**Abstract: **We will learn about the golden ratio while solving problems associated with Fibonacci Numbers.

**Intermediate Group- Jennifer Whitfield on Using Euler and Hamiltonian Paths to Get Around**

**Presenter: **Jennifer Whitfield

Department of Mathematics, Texas A&M University

**Abstract: **In this session, we will investigate the different paths that exist on a given graph. We will also discover some properties of Euler and Hamiltonian paths and then apply the properties to solve problems.

**Advanced Group – Doug Hensley on Putnam Problems**

**Presenter: **Doug Hensley

Department of Mathematics, Texas A&M University

**Abstract: **The “Putnam” is the William Lowell Putnam mathematical competition. It’s famous for being both challenging and fun. The hard part is that the problems always have a twist to where you never “know how to work that type”going in. The fun part is that there is always a sweet solution.

### November 11, 2017

**Beginner Group – Amudhan Krishnaswamy-Usha on The Euclidean Algorithm**

**Presenter:** Amudhan Krishnaswamy-Usha

Department of Mathematics, Texas A&M University

**Abstract: **We will explore the GCD, LCM, and the Euclidean algorithm.

**Intermediate Group- Abraham Martin del Campo on Probability and Algebra**

**Presenter: **Abraham Martin del Campo

Department of Mathematics, CIMAT

**Abstract: **We will explore some basic probability concepts through a coin tossing game and use a little bit of algebra to find if we can play a fair game.

**Advanced Group – Nathan Green on Polya Counting**

**Presenter: **Nathan Green

Department of Mathematics, Texas A&M University

**Abstract: **Polya counting theory allows us to count how many ways there are to arrange objects taking symmetry into account. For example, how many different bracelets can we make using only 3 colors of beads? How many ways can we color a cube using n colors? This counting technique has been used to count the number of different molecules which can be formed from certain sets of atoms and many other important applications.

### October 28, 2017

**Beginner Group – Philip Yasskin on Playing with Toilet Paper**

**Presenter:** Philip Yasskin

Department of Mathematics, Texas A&M University

**Abstract: **We will solve a series of problems associated with folding toilet paper.

**Intermediate Group- Maurice Rojas on Guessing, Sorting, and Optimizing**

**Presenter: **Maurice Rojas

Department of Mathematics, Texas A&M University

**Abstract: **The log function is something we should all know. In this activity, we’ll see how log pops up in the game of “high-low”, and in algorithms for sorting. We’ll then see log again appears in an interesting geometric problem: How do you find the rectangle with axis-parallel sides of largest area inside a polygon? We’ll see how this geometric problem is practically important in architectural design.

**Advanced Group – Edriss S. Titi on What is mathematics? A journey through examples.**

**Presenter: **Edriss S. Titi

Department of Mathematics, Texas A&M University

**Abstract: **Why the honeycomb has hexagonal cell shapes? Is it because bees are lazy, unlike what is commonly believed!! Remarkably, a new mathematical framework has to be invented, every now and then, in order to answer intriguing, yet simple, questions of the kind mentioned above. In this lecture, I will provide few other simple examples, that have played fundamental role in advancing mathematics, as an additional support of this observation.

### October 14, 2017

**Beginner Group – Amudhan Krishnaswamy-Usha on Primes**

**Presenter:** Amudhan Krishnaswamy-Usha

Department of Mathematics, Texas A&M University

**Intermediate Group- Philip Yasskin on Axiomatic Finite Geometries**

**Presenter: **Philip Yasskin

Department of Mathematics, Texas A&M University

**Abstract: **We will study geometries with a finite number of points and lines satisfying a set of axioms.

**Advanced Group – Igor Zelenko on Sums of kth Powers**

**Presenter: **Igor Zelenko

Department of Mathematics, Texas A&M University

**Title: **Sums of kth powers: from telescopic sums and Lagrange interpolations to Bernoulli numbers and Euler-Maclaurin formula

**Abstract: **The formula for the sum of first n positive integers is taught in school. What is the sum of their squares, cubes etc? During this class, we will learn various methods to derive the formulas for these sums from more elementary to more advance.

### September 30, 2017

**Beginner Group – Kagan Samurkas on Mathematical Games of Strategy**

**Presenter: **Kagan Samurkas

Department of Mathematics, Texas A&M University

**Abstract: **Some mathematical games that one part has a winning strategy.

**Intermediate & Advanced Groups – Sherry Gong on Algebra Tricks for Math Contests**

**Presenter: **Sherry Gong

Massachusetts Institute of Technology

### September 23, 2017

**Beginner Group – David Sykes on Euclid’s Algorithm**

**Presenter: **David Sykes

Department of Mathematics, Texas A&M University

**Abstract: **We will explore properties of common divisors. In particular we will discuss how to find greatest common divisors using the Euclidean algorithm, and we will investigate why the Euclidean algorithm works.

**Intermediate & Advanced Groups – Dr. Zuming Feng on An Example on Math Learning via Classroom, Extra Extracurricular, and Contest Activities**

**Presenter: **Dr. Zuming Feng

Phillips Exeter Academy

Proof School Board member

Cogito, part of Johns Hopkins University’s CTY SET program Board member

Former coach of the USA International Mathematics Olympiad (IMO) team

### September 16, 2017

**Beginner Group – Amudhan Krishnaswamy-Usha on A Few Easy Tests for Divisibility**

**Presenter:** Amudhan Krishnaswamy-Usha

Department of Mathematics, Texas A&M University

**Abstract:** I will present some tests for divisibility by small numbers, and introduce modular arithmetic and congruence to explain why they work.

**Intermediate Group – Frank Sottile on The Five Color Theorem**

**Presenter:** Frank Sottile

Department of Mathematics, Texas A&M University

**Abstract:** Doodling on a map of England in 1852, Francis Guthrie noticed that only four colors were needed to color the counties. He conjectured that any map could be colored with only four colors. Several mathematicians tried and failed to prove this; notably in 1879 Kempe published a proof and only in 1890 was the flaw found by Heawood. This four color conjecture evaded a proof until 1972, when Appel and Haken gave a proof that required a computer. While there is as yet no Human readable proof, Kempe’s argument suffice to prove that five color suffice, and this gives a flavor of known proofs of the four color theorem. I will sketch this history and prove the five color theorem.

**Advanced Group – Alex Sprinston on Cracking the Code**

**Presenter:** Alex Sprintson

Department of Engineering, Texas A&M University

**Abstract:** We will provide a brief overview of the fundamentals and applications of the coding theory. First, we will focus on efficient error and erasure correcting codes. Then, we will discuss network codes and codes for distributed storage.

### May 6, 2017

**Beginner group – Ola Sobieska**

**Presenter: **Ola Sobieska

Department of Mathematics, Texas A&M University

**Intermediate & Advanced Groups – Preston Wood on Variant – Limits Game**

**Presenter: **Preston Wood

Triseum – Game Designer

**Abstract:** Limits is an Educational Game developed by Triseum to help students learn about the Calculus topic of Limits. For more information see https://triseum.com/calculus/variant/

### April 29, 2017

**Beginner group – Alex Sprintson & Michael Sprintson on the Mathematics of Sorting Algorithms**

**Presenter:** Alex Sprinston & Michael Sprinston

Department of Electrical and Computer Engineering, Texas A&M University

AMCMS

**Abstract: **Sorting is a fundamental operation in the theory of algorithms and a building block for many computer programs. The activity will lead the students to think about efficient algorithms for sorting information. We will start with a simple exercise in strategic thinking that focuses on determining the ranking of football teams based an a partial information. Next, we will discuss systematic ways to design efficient sorting algorithms. Finally, we present tools for analyzing the complexity of sorting algorithms.

**Intermediate & Advanced Groups – Isaac Harris on Introduction to Fractals-The Concept of Measure and Dimension**

**Presenter:** Isaac Harris

Department of Mathematics, Texas A&M University

**Abstract: **We will look at how geometric quantities are measured. We normally think of length as 1 dimension, area as 2 dimensions and volume as 3 dimensions. Using a simple limiting process we will see that there are dimensions that are not whole numbers! This will lead us to consider the mathematical concept of Fractals and how one get these other dimensions for measurements.

### April 22, 2017

** Beginner Group: Maya Johnson on Humans vs. Aliens**

**Presenter:** Maya Johnson

Department of Mathematics, Texas A&M University

**Abstract: **A group of 6 humans are abducted by aliens in the night. Each of these 6 humans represent one sixth of the human population on the planet. The aliens tell the humans that in the morning they will order them in a single file line and place either a green or a purple hat on top of each person’s head. Each of the humans will then be able to see all of the hats a top the heads of all the persons in front of them, but will not be able to see their own hat or the hats of the people behind them. For example, the very last person in line will be able to see the hats of all five people in front of them, the second to last person can see the hats of all four people in front of them and so on. The aliens say they will then start at the back of the line and ask each person for the color of the hat on their own head. The person is only allowed to answer either green or purple, they are not allowed to say any other words. If the person answers correctly, then that person, along with the one sixth of the population that they represent, will live. However, if they answer incorrectly, the opposite will happen. The aliens are not entirely evil, however, and so they give the humans the night to come up with a strategy.

The problem facing the humans is this: what is the optimal strategy? That is, how can they save as many of themselves as possible, thereby saving as much of the human race as possible? There is a strategy that will guarantee the lives of all but one of them, but it requires a brave sacrifice from one of the 6 humans. Of course this human would jump at the chance to save five sixth of the human population, but what is the strategy? Also, what would be the minimum number of humans that would need to make the ultimate sacrifice if there were more than two color options for the hats? Help the humans out smart the aliens and save the human race with Math and logic!

**Intermediate & Advanced Groups: Roger Howe on Rules of Arithmetic**

**Presenter:** Roger Howe

Department of Mathematics, Texas A&M University

**Abstract: **We will take a more in-depth look at the Rules of Arithmetic than is usual in school. They have some very important implications for arithmetic, and they can lead to some fun mathematics.

### April 8, 2017

**Beginner Group: Philip Yasskin on Eleusis**

**Presenter:** Dr. Philip Yasskin

Department of Mathematics, Texas A&M University

**Abstract: **We will play a game that models scientific research.

**Intermediate & Advanced Groups: Doug Hensley on Why is Long Division Serious Mathematics?**

**Presenter:** Dr. Doug Hensley

Department of Mathematics, Texas A&M University

**Abstract: **The short answer is that it’s at the heart of the Euclidean algorithm, and that this algorithm is, in turn, the key to such computational mathematical challenges as, given integers a, b, and p (p prime or failing that, a and b relatively prime to p), finding c so that bc is congruent to a mod p. From a certain point of view, this is again division, as we can say c=a/b mod p.

### March 4, 2017

**Beginner & Intermediate Group: Philip Yasskin on Unexpected Probabilities**

**Presenter:** Philip Yasskin

Texas A&M University, Department of Mathematics

**Abstract: **We will look at 2 probability problems. First we will guess the answer. Second we will find the probability experimentally. And third we will compute the probability theoretically.

**Advanced Group: Nathan Green on Primes**

**Presenter:** Nathan Green

Texas A&M University, Department of Mathematics

**Abstract: **Prime numbers have been studied since ancient history, and in modern times they are doubly important, having crucial applications to cryptography and computer security. We will discuss some of the basic theory of prime numbers, with particular emphasis on large prime numbers which come up in computer applications.

### Feb. 25, 2017

**Beginner Group: Janice Epstein on Magic Squares**

**Presenter:** Janice Epstein

Texas A&M University, Department of Mathematics

**Intermediate Group: Yeong Chung on The Math of Origami**

**Presenter:** Yeong Chung

Texas A&M University, Department of Mathematics

**Abstract: **It is easy to divide a square sheet of paper into two equal parts, but how can we divide a square sheet of paper into three (or five or six) equal parts without using any tools? By investigating some ways of folding the paper, we will come up with a way to divide the paper into various numbers of equal parts. We may then also try to divide a rectangular sheet of paper into equal parts both horizontally and vertically.

**Advanced Group: Maurice Rojas on Counting Lattice Points in Polygons**

**Presenter:** Maurice Rojas

Texas A&M University, Department of Mathematics

**Abstract: **If you draw a polygon on a grid, you can try counting

the grid points (also called lattice points) insie the polygon.

This simple problem is at the heart of many deep ideas in combinatorics

and optimization. We’ll work out some basic examples, and see surprising

connections to geometric series, the computation of area, clever ways

to chop up regions into weighted regions. Be prepared to count!

### Feb. 18, 2017

**Beginner Group: Jane Long on A Math Without Words Puzzle**

**Presenter:** Jane Long

Stephen F. Austin State University, Department of Mathematics

**Abstract: **Many people who enjoy mathematics also enjoy games and puzzles. Generally, when people meet a new puzzle or game, they begin by reading or talking about rules or instructions. In this session, we will take a different approach: we will examine an intriguing puzzle in the form of a picture with no description or instructions. It will be up to us to discover the rules and solve the puzzle!

**Advanced and Intermediate Group: Nick Long on Which One Doesn’t Belong**

**Presenter:** Nick Long

Stephen F. Austin State University, Department of Mathematics

**Abstract: **When you look at the set of letters {A, B,C, D}, which one doesn’t belong? Your answer might be that A is a vowel or that C does not contain a closed loop. Can you come up with a way that B doesn’t belong? What about D? We will look more at how to distinguish the elements of a set by which one does not belong and how to build interesting sets for this kind of discussion.

### Feb. 11, 2017

**Physics Show**

**Presenter:** Tatiana Erukhimova

Texas A&M University, Department of Physics & Astronomy

**The Math Circle will be visiting the Physics Department this week for their famous Physics Show.**

### Feb. 4, 2017

**Beginner Group: Kun Wang on Penny Problems**

**Presenter:** Kun Wang

Texas A&M University, Department of Mathematics

**Abstract:** We will play some games with pennies. Those games are about geometry, combinatorics, probability, etc.

**Intermediate Group: Alan Demlow on An Introduction to Floating Point Arithmetic**

**Presenter:** Alan Demlow

Texas A&M University, Department of Mathematics

**Abstract:** Computers are used in almost every facet of life. They enable us to predict the weather, how planes will behave in flight, and whether a bridge design will be sturdy. They also are used to control many systems, such as cars and guided missiles. Modern computers use a number system called the floating point system in order to do these calculations. We will describe floating point numbers. Students will investigate some examples where floating point arithmetic has different properties than the arithmetic we are used to. We’ll also give some examples of computer simulations that failed, leading to disastrous results!

**Advanced Group: Philip Yasskin on Domino Circle & Diagonals**

**Presenter:** Philip Yasskin

Texas A&M University, Department of Mathematics

**Abstract:** Problem (1) Each Domino has two halves and each half has a number usually from 0 to 6. A full set has one of each pair of numbers from double 0 to double 6. Can a full set of 0-6 dominoes be placed end to end in a circle so that every two adjacent dominoes have the same number on the adjacent halves?

Problem (2) We will count the number of diagonals in a rectangular grid with certain restrictions on which diagonals to count.

** Jan. 28, 2017**

**Beginner Group: David Manuel on Tangram Origami**

**Presenter:** David Manuel

Texas A&M University, Department of Mathematics

**Abstract:** Given seven identical square sheets of paper, is it possible using simple origami folding techniques to create each of the seven tangram pieces used to build the square?

**Intermediate Group: Dean Baskin on Euler Numbers**

**Presenter:** Dean Baskin

Texas A&M University, Department of Mathematics

**Abstract:** The Euler number of a shape is the sum V + F – E, where V is the number of vertices in the shape, E is the number of edges, and F is the number of faces. How does this number depend on the shape we draw (or build)?

**Advanced Group: Volodymyr Nekrasheyvich on A Diophantine Equation and Uniform Tilings**

**Presenter:** Volodymyr Nekrasheyvich

Texas A&M University, Department of Mathematics

**Abstract:** I will to talk about the equations in natural numbers of the form

1/a+1/b+1/c+1/d=1 and its relation to geometry.

** Jan. 21, 2017**

**Beginner Group: Kun Wang on Card Games and Combinatorial Problems**

**Presenter:** Kun Wang

Texas A&M University, Department of Mathematics

**Abstract:** We will find a way to order poker cards so that the numbers

appear in a magical way. After that we will solve some combinatorial

problems.

**Intermediate Group: Philip Yasskin on Domino Circles**

**Presenter:** Philip Yasskin

Texas A&M University, Department of Mathematics

**Abstract:** Each Domino has two halves and each half has a number usually from 0 to 6. A full set has one of each pair of numbers from double 0 to double 6. Can a full set of 0-6 dominoes be placed end to end in a circle so that every two adjacent dominoes have the same number on the adjacent halves?

**Advanced Group: Alexander Engel on Zero-Knowledge Proofs**

**Presenter:** Alexander Engel

Texas A&M University, Department of Mathematics

**Abstract:** In a zero-knowledge proof one proves to someone else that one has a certain secret information or that a certain statement is true without conveying any other information, i.e., the other party does not get any knowledge about the secret information or the statement. We will discuss examples of such zero-knowledge proofs in a variety of contexts.

### Dec. 3rd, 2016

**Beginner Group: Philip Yasskin on Domino Circles**

**Presenter: ** Philip Yasskin

Department of Mathematics , Texas A&M University

**Abstract: **Each Domino has two halves and each half has a number usually from 0 to 6. A full set has one of each pair of numbers from double 0 to double 6. Can a full set of 0-6 dominoes be placed end to end in a circle so that every two adjacent dominoes have the same number on the adjacent halves?

**Intermediate Group: Tamara Carter on CLUE in the Math Department**

**Presenter: **Tamara Carter

Department of Mathematics, Texas A&M University

**Abstract: ** Students will explore ciphers, decipher clues, and use those clues to find the prize.

**Advanced Group: Konrad Wrobel on Distinct Distances in the Plane**

**Presenter: **Konrad Wrobel

Department of Mathematics, Texas A&M University

**Abstract: **We will look at collections of points with exactly 2 distinct distances between them and try to investigate all such collections. We’ll also work on some other problems in Euclidean geometry.

### Nov. 19th, 2016

**Beginner Group: Alex Sprinston on Design of Combinational Circuits Using Boolean Algebra**

**Presenter: ** Alex Sprinston

Department of Electrical and Computer Engineering , Texas A&M University

**Abstract: **We will start with a quick introduction to Boolean Algebra. Then, we will show how to use the rules of Boolean Algebra to construct simple logic circuits. Finally, we will introduce Karnaugh maps and show how to use them to design more efficient circuits.

**Intermediate Group: R. Saravanan on Hash functions, Cryptography**

**Presenter: **R. Saravanan

Department of Atmospheric Sciences, Texas A&M University

**Advanced Group: Peter Kuchment on Unreasonable Effectiveness of Mathematics**

**Presenter: **Peter Kuchment

Department of Mathematics, Texas A&M University

**Abstract: **Since antiquity, and especially nowadays mathematicians have been developing extremely abstract concepts, having no clear relation to reality, and “play” with them according to seemingly rather arbitrarily invented rules. In many (maybe most of) cases, the trigger for such developments is the aesthetic feeling of mathematical beauty. In this regard, mathematics is similar to other games, such as chess, go, and others. However, for some inexplicable reason, unlike other games, the mental math constructions eventually are applicable for producing practically useful results in natural sciences and engineering. The talk will be addressing this intriguing issue.

### Nov. 5th, 2016

**Beginner Group: Eviatar Procaccia on Folding the Platonic Solids**

**Presenter:** Eviatar Procaccia

Department of Mathematics , Texas A&M University

**Abstract:** The Greek philosopher Plato believed true beauty exists only in a few geometric shapes we now call the Platonic solids. We will learn why there are only five of them, and fold some of them in paper.

**Intermediate Group: Parth Sarin on How Fast Can You Gossip?**

**Presenter:** Parth Sarin

TAMU Math Circle Organizer

Undergraduate in Department of Mathematics, Texas A&M University

**Abstract:** From visiting a website to making a call, modern society depends on our ability to exchange information online. But, modern computers can’t multi-task well – they can only exchange one piece of information at a time. We’ll explore how even with this limitation, networks of computers exchange information quickly and intelligently in order to keep our lives up to date.

### Oct. 15, 2016

**Beginner Group: Tamara Carter on CLUE in the Math Department**

**Presenter: **Tamara Carter

Department of Mathematics, Texas A&M University

**Abstract: **Students will explore ciphers, decipher clues, and use those clues to find the prize.

**Intermediate Group: Jens Forsgård on The a+b+ab Problem**

**Presenter: **Jens Forsgaard

Department of Mathematics, Texas A&M University

**Abstract: **Write down the numbers from 1 to 100. Randomly select 2 numbers from the list, say a and b, and cross them off, but add to the list the number a+b+ab. You now have 99 numbers. Repeat this process until you have only 1 number left. What are all possible final numbers?

**Advanced Group: Kim Currens & Dr. Sandra Nite on Modeling Sound Waves with Periodic Functions**

**Presenters: **Kim Currens & Dr. Sandra Nite

Department of Mathematics, Texas A&M University

**Abstract: **We will use graphing calculators, calculator based laboratory (CBL), and probes to collect sound wave data. Then we will use at least two methods to model the data with a periodic function.

**Oct. 8, 2016**

**Beginner Group: Dr. Robert Capraro on Counting Cows & 3 Bean Salad**

Presenter: Dr. Robert Capraro

Department of TLAC, Texas A&M University

Abstract: For “Counting Cows” students will use cubes to organize thinking and solve algebraic problems in the context of cows in different pastures. In “3 Bean Salad” students use several types of beans to represent salad mixtures and solve equations to determine the total number of beans in the salad.

**Intermediate Group: Dr. Mary Margaret Capraro on Locker Problem, Arithmagons, Magic Squares**

Presenter: Dr. Mary Margaret Capraro

Department of TLAC, Texas A&M University

Abstract: These 3 problems use algebraic thinking by building habits of mind. The locker problem will focus on building rules to represent functions and doing-undoing. Arithmagons use a simple system of equations, and students will utilize intuitive and informal operation sense. The magic square problems will help develop symbol sense by requiring decisions as to when it is appropriate to invoke the use of symbols and also understand the meaning of symbolic solutions.

**Advanced Group: Dr. Luciana Barroso & Dr. Sandra Nite on Exploring Lung Capacity**

Presenter: Dr Luciana Barroso & Dr. Sandra Nite

Department of TLAC and Mathematics, Texas A&M University

Abstract: Students will use graphing calculators and calculator based laboratory (CBL) to gather and examine data for lung capacity.

### Oct. 1, 2016

**Beginner Group: David Kerr on Random Walks and Search Engines**

**Speaker:** David Kerr

Department of Mathematics, Texas A&M University

**Topic:** *Random Walks and Search Engines*

**Abstract:** Abstract:

We will investigate the notion of chance by performing experiments with random walks, and see how this can be applied to the problem of internet search.

**Intermediate Group: Riad Masri on Explorations with Prime Numbers**

**Speaker: **Riad Masri

Department of Mathematics, Texas A&M University

**Title: ***Explorations with Prime Numbers*

**Abstract: **In this activity we will explore some of the many interesting properties of prime numbers. First, we will learn how to find prime numbers using a “sieve”. We will then study questions related to differences between consecutive primes, and the distribution of primes in residue classes.

**Advanced Group: Zoran Sunic on “Wait, was I supposed to turn left or right?”**

**Speaker:** Zoran Sunic

Department of Mathematics, Texas A&M University

**Topic: ***Wait, was I supposed to turn left or right?*

**Abstract:** We will consider journeys through a kingdom in which there are three roads out of every town, and the roads only intersect at the towns. Our knight will travel around, do a good deed here and there, and will have strange ideas how to get home. We will try to find out if he ever does get home, how many times he visits the same town along the way, and how long his journeys could be.

### Sept. 24, 2016

**Beginner & Intermediate Group: Philip Yasskin on Trapezoid Numbers**

Presenter: Philip Yasskin

Department of Mathematics, Texas A&M University

Abstract: Modern cryptography gives us intricate ways to safely share secrets and protect private information. But some of the underlying ideas are very simple. We’ll see how these ideas come together in a method to share a private key when communicating over a public channel.

**Advanced Group: Maurice Rojas on Gift Boxes, Mongoose in the Middle, and Secret Codes**

Presenter: Maurice Rojas

Department of Mathematics, Texas A&M University

Abstract: Modern cryptography gives us intricate ways to safely share secrets and protect private information. But some of the underlying ideas are very simple. We’ll see how these ideas come together in a method to share a private key when communicating over a public channel.

### Sept. 17, 2016

**Beginner Group: Ola Sobieska on Even and Odd Numbers**

Presenter: Ola Sobieska

Department of Mathematics, Texas A&M University

Abstract: In this activity, we will explore the topic of odds and evens, including various ways to define these numbers, learn several useful properties, and investigate how to apply them to problem solving.

**Intermediate Group: Dr. Ali Bicer & Dr. Sandra Nite on Dilutions**

Presenters: Dr. Ali Bicer & Dr. Sandra Nite

Department of Mathematics and Department of Teaching, Learning and Culture, Texas A&M University

Abstract: This activity will use food coloring and water to perform dilutions at several levels and then decide what level water with poisons will be safe to drink.

**Advanced Group: Philip Yasskin on Axiomatic Finite Geometries**

Presenter: Philip Yasskin

Department of Mathematics, Texas A&M University

Abstract: We will study geometries with a finite number of points and lines satisfying a set of axioms.

### May 14, 2016

**Beginner and Intermediate Groups: Kaitlyn Phillipson on Math Games**

Presenter: Kaitlyn Phillipson

Department of Mathematics, Texas A&M University

Abstract: We’ll discuss some games and try to come up with winning strategies.

**Advanced Group: Riad Masri on the Arithmetic of Integer Partitions**

Presenter: Riad Masri

Department of Mathematics, Texas A&M University

Abstract: The goal of this project is to explore some arithmetic aspects of integer partitions. In particular, we will focus on Ramanujan’s famous congruences for the partition function, and study how the Dyson rank and the Andrews/Garvan crank can be used to give a combinatorial explanation for these congruences.

### May 7, 2016

**Beginner Group: Philip Yasskin on Balance Beams**

Presenter: Philip Yasskin

Department of Mathematics, Texas A&M University

Abstract: We use a meter stick as a balance beam with a pencil at the 50 cm mark. In each problem, we put weights at the locations indicated and experiment to figure out where to put the extra weights. We will progress to using equations to figure out where to put the weights.

**Intermediate Group: Gregory Berkolaiko on Icosahedron Made from Scratch**

Presenter: Gregory Berkolaiko

Department of Mathematics, Texas A&M University

Abstract: The task is to make an icosahedron from scratch using only paper and glue (plus compass, ruler, scissors and pencil). Along the way we will need to solve the problem of dividing the circle into the equal parts lengthwise.

**Advanced Group: Timo de Wolff on Public Key Cryptography**

Presenter: Timo de Wolff

Department of Mathematics, Texas A&M University

Abstract: Cryptography handles with the secure transmission of secret

messages. More precisely, a third party is supposed to be unable to

understand the content of an intercepted message if it is encrypted.

Classically, both parties exchange secret keys for a secure en- and

decryption. Nowadays, however, a lot of communication happens via

insecure channels like the internet. Thus, secret keys often cannot be

exchanged securely. Thus, one needs a new type of crypto system, in

which parts of the keys do not need to be hidden anymore. This is called

public key cryptography.

In this talk we will first review a couple of classical symmetric crypto

systems like the Ceasar cipher. In the second part I will explain and

show the RSA crypto system, which is the current industry standard for

public key cryptography.

### April 30, 2016

**Beginner Group: Frank Sottile on Meet the Cube**

Presenter: Frank Sottile

Department of Mathematics, Texas A&M University

Abstract: We will investigate the familiar cube, using it

to study three-dimensional geometry.

**Intermediate Group: Christopher O’Neill on When Can You Draw a Picture Without Picking Up Your Pencil?**

Presenter: Christopher O’Neill

Department of Mathematics, Texas A&M University

Abstract: Suppose someone hands you a picture and asks you to trace it in one continuous motion, that is, without picking up your pencil or backtracking. When is it possible to succeed? How should you decide where to start tracing?

**Advanced Group: Eric Rowell on Mathematical Knots and Links**

Presenter: Eric Rowell

Department of Mathematics, Texas A&M University

Abstract: Knots and links have been used as decorations for centuries, but their mathematical study only began in the 19th century. For a brief period it was believed that atoms were just knotted bits of swirling ether, and physicists set to work to tabulate them. It turned out they were completely wrong, but this led to the development of topology. More than 100 years later, knots may again be useful in physics though Topological Quantum Computation. We will explore important questions surrounding knots and links, such as: how do we know when two knots are actually the same? How can we tell that they are genuinely different?

### April 23, 2016

**Beginner Group: Roger Howe on Around the Pythagorean Theorem**

Presenter: Roger Howe

Department of Mathematics, Yale University

Abstract: This session will discuss a few of the many applications of the Pythagorean Theorem in the real world. Among the questions to be considered will be, why do ladders work, taking shortcuts, and how far can we see?

**Advanced Group: David Dynerman on Normal Mapping and Video Games**

Presenter: David Dynerman

Department of Mathematics, University of California, Berkley

Abstract: Normal mapping is a way of increasing surface detail when rendering 3D graphics and has become a standard technique in the video game industry. Normal mapping sneaks in higher quality lighting detail over a lower-quality polygonal model. This talk will give an overview on how this interesting application of math, computer science and physics creates better looking video games.

### March 26, 2016

**Beginner Group: Ola Sobieska on Weighings and Counterfeit Coins**

Presenter: Ola Sobieska

Department of Mathematics, Texas A&M University

Abstract: This session will focus on problems about balance scales and weights. The students will learn to identify counterfeit coins, discover tricky ways to weigh objects, and solve other puzzles.

**Intermediate Group: David Sykes on Dinner Party Problems and Graph Coloring**

Presenter: David Sykes

Department of Mathematics, Texas A&M University

Abstract: We will be discussing the Dinner Party Problem along with some of its generalizations while exploring the concept of graph coloring. The problems are special cases of a theorem established by Frank Ramsey around 1930. The discussion will build towards the solution to a challenging Ramsey Theory problem along with the statement of problems that remain unsolved today.

**Advanced Group: Matt Young on Which Numbers are Sums of Two Squares?**

Presenter: Matt Young

Department of Mathematics, Texas A&M University

Abstract: Some numbers are the sum of two squares, and some numbers aren’t. For example, 5 is (since 5 = 1 +4) but 7 isn’t. Numbers that can be expressed as the sum of two squares have many amazing properties, and we will discover many of these patterns in this math circle.

### March 5, 2016

**Beginner Group: Phil Yasskin on Splitting Piles and Handshakes**

Presenter: Phil Yasskin

Department of Mathematics, Texas A&M University

Abstract: We will consider 2 problems and ultimately see how they are related.

1) Take a pile of coins, say 10 coins. Split it into two piles, with say 4 and 6 coins. Write down the product 4*6=24. Split each of those piles into two piles, with say 1 and 3, and say 2 and 4. Write down those products 1*3=3 and 2*4=8. Continue in this way until you have ten plies each with 1 coin. Then add all the products, say 24+3+8+… What are all possible sums?

2) If 10 people are in a room, how many ways can they shake hands?

**Intermediate Group: Eviatar Procaccia on The Gambler’s Ruin and a Disoriented Bird**

Presenter: Eviatar Procaccia

Department of Mathematics, Texas A&M University

Abstract: Probability theory is the mathematical framework to study

randomness in the universe. We will learn how to use one source of

randomness to create another and why a disoriented bird will never find its

nest.

**Advanced Group: Volodymyr Nekrashevych on Binomial Coefficients and Their Properties**

Presenter: Volodymyr Nekrashevych

Department of Mathematics, Texas A&M University

Abstract: We will discuss Pascal’s triangle, binomial coefficients,

combinations, triangular numbers, and different interesting facts

about them.

### February 27, 2016

**Beginner and Intermediate Group: Michelle Pruett on Code-Breaking Through the Years**

Presenter: Michelle Pruett

Texas State University at San Marcos

Abstract: A variety of codes have been used throughout history. We will discover how

to code and decode messages using several techniques.

**Advanced Group: Konrad Wrobel on Dealing with Infinite Sets**

Presenter: Konrad Wrobel

Department of Mathematics, Texas A&M University

Abstract: The roots of modern day set theory stem from Georg Cantor’s work in 1874, when he introduced several concepts that many mathematicians of the time found disconcerting. We’ll delve into his notion of size, or cardinality, and what it means when applied to infinite sets.

### February 20, 2016

**Beginner Group: Anneliese Slaton on The Bridge Problem: A Puzzle That Changed the Mathematical World**

Presenter: Anneliese Slaton

Undergraduate Student at George Mason University

Abstract: We will be discussing The Koningsberg Bridge Problem, a seemingly simple problem that was solved by Euler and opened the door to the development of graph theory as we know it. We will not only look at Euler’s original proof, but will explore variations of the problems in physically get up and try to walk to the path!

**Intermediate Group: Igor Zelenko on Domino Puzzles, Invariants, and Walking Along the Grids and Bridges**

Presenter: Igor Zelenko

Department of Mathematics, Texas A&M University

Abstract: During the activity we will try to solve various problems regarding covering grids by dominos, trominos, transforming the tables of numbers according to certain rules, moving along grids without raising a pencil, or walking along the bridges of cities with many bridges. In all these problems we will ask whether we can complete certain tasks, the answer will often follow from certain nontrivial observations or properties of certain quantities, called invariants, that are preserved by natural transformations allowed in the problem.

**Advanced Group: Ramalingam Saravanan on Predictability of Weather and Climate**

Presenter: Ramalingam Saravanan

Department of Mathematics, Texas A&M University

Abstract: The discovery of the limits to weather predictability by Edward Lorenz was a seminal event both in theoretical meteorology and in nonlinear dynamics. The mathematical and physical basis for the predictability of weather and climate will be discussed in the context of this discovery. Topics to be covered will include trigonometric functions, limit cycles, and chaotic attractors.

### February 13, 2016

**Beginner Group: Frank Sottile on Word Problems and Common Sense**

Presenter: Frank Sottile

Department of Mathematics, Texas A&M University

Abstract: While we are taught to use algebra to solve word problems, many can be solved just using common sense. In this circle, we will use our common sense to solve word problems.

**Intermediate Group: Maurice Rojas on Hats, Codes, and Lattice Points**

Presenter: Maurice Rojas

Department of Mathematics, Texas A&M University

Abstract: We will see how a puzzle involving hats relates to codes that help protect data from noise. We’ll then see how lattice points come up in many different mathematical puzzles, as well as the modern study of secret codes.

**Advanced Group: William Rundell on A 5,000 Year History into Mathematical Innovation**

Presenter: William Rundell

Department of Mathematics, Texas A&M University

Abstract: Here are a series of questions. How does your calculator come up with it’s approximation to the square root of, say, 2? How were the square roots calculated in antiquity? Is there anything new to say about the problem? This talk will explore some of the answers.

### February 6, 2016

**Beginner Group: Frank Sottile on Mathematical Auction**

Presenter: Professor Frank Sottile

Department of Mathematics, Texas A&M University

Abstract: We will be doing a team contest called ‘mathematical auction.’

**Intermediate Group: Phil Yasskin on The a+b+ab Problem**

Presenter: Professor Phil Yasskin

Department of Mathematics, Texas A&M University

Abstract: Write down the numbers from 1 to 100. Randomly select 2 numbers from the list, say a and b, and cross them off, but add to the list the number a+b+ab. You now have 99 numbers. Repeat this process until you have only 1 number left. What are all possible final numbers?

**Advanced Group: Maurice Rojas on Polygons, Lattice Points, and Equations**

Presenter: Professor Maurice Rojas

Department of Mathematics, Texas A&M University

Abstract: We’ll see how geometric series and the Triangle Inequality allow us to understand hard equations with simple pictures. We’ll then see how counting lattice points in polygons leads us to some beautiful and unexpected applications of mathematics.

### September 19, 2015

**Beginner Group: Frank Sotille on How to Solve a Problem**

Presenter: Frank Sotille

Department of Mathematics, Texas A&M University

Abstract: We will work together to solve and discuss some

interesting puzzles and problems.

**Intermediate Group: Phillip B. Yasskin on The Candy Conundrum**

Presenter: Philip B. Yasskin

Department of Mathematics, Texas A&M University

Abstract: A candy company wants to a advertise the large number of flavors that can be made by mixing candies in your mouth. Let’s figure it out.

**Advanced Group: Kaitlyn Phillipson on Catalan Structures**

Presenter: Kaitlyn Phillipson

Department of Mathematics, Texas A&M University

Abstract: The Catalan numbers are one of the most common sequences in mathematics. There are many structures counted by the Catalan numbers, and in this activity we take a look at several of them.

### April 25, 2015

**Beginner – Matthew Barry – Turk’s Head Knots**

Title:

Turk’s Head Knots

Speaker:

Matthew Barry (with help from Philip Yasskin and Michael Sprintson)

Texas Engineering Extension Station

TAMU Class of 2014

Abstract:

The Turk’s head knot, flat mat, and pineapple knot all belong to a family of interwoven decorative knots favored by many people for many centuries, notably the Celtics. In its final form, the turks head knot is a symmetric prime knot that can be classified by the number of intersections the rope makes with itself. In the knot-tying community, Turk’s head knots are classified by counting leads and bights: the lead count is the number of times the rope goes around the knot, and the bight count is the number of loops at each end. For example a 3×5 Turk’s head knot has three leads and five bights. Here we explore the math theory behind these knots and use it to plan and tie Turk’s head knots of any size.

**Intermediate**

Title:

Turk’s Head Knots

Speaker:

Matthew Barry (with help from Philip Yasskin and Michael Sprintson)

Texas Engineering Extension Station

TAMU Class of 2014

Abstract:

The Turk’s head knot, flat mat, and pineapple knot all belong to a family of interwoven decorative knots favored by many people for many centuries, notably the Celtics. In its final form, the turks head knot is a symmetric prime knot that can be classified by the number of intersections the rope makes with itself. In the knot-tying community, Turk’s head knots are classified by counting leads and bights: the lead count is the number of times the rope goes around the knot, and the bight count is the number of loops at each end. For example a 3×5 Turk’s head knot has three leads and five bights. Here we explore the math theory behind these knots and use it to plan and tie Turk’s head knots of any size.

**Advanced**

Title:

Turk’s Head Knots

Speaker:

Matthew Barry (with help from Philip Yasskin and Michael Sprintson)

Texas Engineering Extension Station

TAMU Class of 2014

Abstract:

The Turk’s head knot, flat mat, and pineapple knot all belong to a family of interwoven decorative knots favored by many people for many centuries, notably the Celtics. In its final form, the turks head knot is a symmetric prime knot that can be classified by the number of intersections the rope makes with itself. In the knot-tying community, Turk’s head knots are classified by counting leads and bights: the lead count is the number of times the rope goes around the knot, and the bight count is the number of loops at each end. For example a 3×5 Turk’s head knot has three leads and five bights. Here we explore the math theory behind these knots and use it to plan and tie Turk’s head knots of any size.

### March 7, 2015

**Beginner**

Speaker Ms. Kaitlyn Phillipson

Department of Mathematics

Texas A&M University,

Title: Guarding an Art Gallery

Abstract: We will discuss the “Art Gallery Problem,” a well-studied problem in mathematics.

**Intermediate**

Speaker Ms. Kaitlyn Phillipson

Department of Mathematics

Texas A&M University,

Title: Guarding an Art Gallery

Abstract: We will discuss the “Art Gallery Problem,” a well-studied problem in mathematics.

**Advanced**

Speaker: Dr. Nicholas Long

Department of Mathematics

Stephen F. Austin State University

Title: “Pressing Buttons on a Calculator.”

Abstract: One of the first things kids do when they start playing with a calculator is explore what happens to the screen when you keep hitting the same button over and over. We can figure out pretty quickly what happens when we keep pressing the addition or multiplication buttons. What happens if we had some buttons on a calculator that used multiplication and addition together? What would the result be if we keep pressing a button like that?

### February 28, 2015

Speaker: Dr. Altha Rodin

Department of Mathematics

University of Texas

Title: The Next Move: Some Theory and Practice with Impartial Games

We will discuss combinatorial impartial games defined as follow.

Combinatorial games are two-player games with the following characteristics:

* Two players alternate moves.

* Play continues until there are no legal moves remaining.

* No element of chance is involved (i.e. dice, spinners, etc.).

* Each player has full knowledge of the game position at all times.

In normal play, the last player to make a legal move wins. In misère play, the last player to make a legal move loses. A combinatorial game is called impartial if both players have the same set of allowable moves at each position of the game. A game in which the allowable moves depends on the player is called a partisan game.

**Intermediate**

Speaker: Dr. Altha Rodin

Department of Mathematics

University of Texas

Title: The Next Move: Some Theory and Practice with Impartial Games

We will discuss combinatorial impartial games defined as follow.

Combinatorial games are two-player games with the following characteristics:

* Two players alternate moves.

* Play continues until there are no legal moves remaining.

* No element of chance is involved (i.e. dice, spinners, etc.).

* Each player has full knowledge of the game position at all times.

In normal play, the last player to make a legal move wins. In misère play, the last player to make a legal move loses. A combinatorial game is called impartial if both players have the same set of allowable moves at each position of the game. A game in which the allowable moves depends on the player is called a partisan game.

**Advanced**

Speaker: Dr. Lucas Macri

Department of Physics & Astronomy

Texas A&M University

Title: The Mathematics of Astronomy (part I)

In this class, we will talk about the math used by ancient astronomers to learn about the Universe even before the telescope was invented. How did they determine the size of Earth, the distance to the Moon and the Sun? We will also talk about how we can measure the distances to other stars and figure out how much light they produce.

### February 21, 2015

**Beginner**

Speaker: Dr. Lucas Macri

Department of Physics & Astronomy

Texas A&M University

Title: The Mathematics of Astronomy (part I)

In this class, we will talk about the math used by ancient astronomers to learn about the Universe even before the telescope was invented. How did they determine the size of Earth, the distance to the Moon and the Sun? We will also talk about how we can measure the distances to other stars and figure out how much light they produce.

**Intermediate**

Speaker Mr. Trevor Olsen

Department of Mathematics

Texas A&M University

Title: Kinetic Origami (Curlicue)

Abstract: Are you ready to make amazing shape changing origami? Well I sure am! We will be making Curlicues that go from being flat paper to different 3D shapes. We will understand how these structures work and learn what other types of Curlicues we can make.

**Advanced**

Speaker Dr. Igor Zelenko

Department of Mathematics

Texas A&M University

Title Sums of k’th powers and other interesting sums

Abstract: The formula for the sum of first n positive integers is taught in school. What is the sum of their squares, cubes etc? We will learn how to derive formulas for these sums and other interesting sums and give applications for calculating areas.

### February 14, 2015

**Beginner**

Speaker: Dr. Phil Yasskin

Department of Mathematics

Texas A&M University

Title: GCD, LCM, Prime Factorization, and the Division and Euclidean Algorithms

Abstract:

I will present a series of problems whose solutions involve the Greatest Common Divisor, the Least Common Multiple, the Unique Prime Factorization Theorem, the Division Algorithm and/or the Euclidean Algorithm. For example:

Problem 1: You have an unmarked 5 liter bucket and an unmarked 9 liter bucket and an unlimited amount of water. Can you measure out exactly 2 liters of water? How?

Problem 2: How many 12 cent and 27 cent postage stamps should you buy to put exactly 83 cents worth of postage on an envelope?

Problem 3: You have a 3 foot by 5 foot pool table. The cue ball is located at a point which is 1 foot from the 5 foot side and 2 feet from the 3 foot side. You hit the ball at 45 degrees. Every time the ball hits a side it bounces back at 45 degrees with no loss of velocity. Will the ball eventually hit the corner of the pool table?

**Intermediate**

Speaker: Dr. Phil Yasskin

Department of Mathematics

Texas A&M University

Title: GCD, LCM, Prime Factorization, and the Division and Euclidean Algorithms

Abstract:

I will present a series of problems whose solutions involve the Greatest Common Divisor, the Least Common Multiple, the Unique Prime Factorization Theorem, the Division Algorithm and/or the Euclidean Algorithm. For example:

Problem 1: You have an unmarked 5 liter bucket and an unmarked 9 liter bucket and an unlimited amount of water. Can you measure out exactly 2 liters of water? How?

Problem 2: How many 12 cent and 27 cent postage stamps should you buy to put exactly 83 cents worth of postage on an envelope?

Problem 3: You have a 3 foot by 5 foot pool table. The cue ball is located at a point which is 1 foot from the 5 foot side and 2 feet from the 3 foot side. You hit the ball at 45 degrees. Every time the ball hits a side it bounces back at 45 degrees with no loss of velocity. Will the ball eventually hit the corner of the pool table?

**Advanced**

Speaker Ms. Kaitlyn Phillipson

Department of Mathematics

Texas A&M University,

Title: Guarding an Art Gallery

Abstract: We will discuss the “Art Gallery Problem,” a well-studied problem in mathematics.

### January 24, 2015

**Beginner**

Speaker: Dr. Jane Long

Department of Mathematics

Stephen F. Austin State University

Title: The Mathematics of Sona, Sand Drawings from Africa

Abstract: Many cultures around the world tell stories with the help of drawings made in sand. This activity will investigate interesting mathematics involved in some traditional sand drawings from Angola.

**Intermediate**

Speaker: Dr. Jane Long

Department of Mathematics

Stephen F. Austin State University

Title: The Mathematics of Sona, Sand Drawings from Africa

Abstract: Many cultures around the world tell stories with the help of drawings made in sand. This activity will investigate interesting mathematics involved in some traditional sand drawings from Angola.

**Advanced**

Speaker: Dr. David Manuel

Department of Mathematics

Texas A&M University,

Title: The Algebra of Rubik’s Cubes, part 3

Abstract: Many of us have learned how to solve the (3×3) Rubik’s Cube from solutions presented in a book or online. But how does one come up with their own solution? In this final session, we will apply what we have learned about groups, permutations, and partial commutativity to the movements of the Rubik’s Cube to develop our own strategies to solve the Cube. Bring your cubes, and, if possible, movements which exchange 2 cubes or rotate 1 cube in one row (regardless of what the other rows look like).