Archived Math Circle Material
Contents
 1 Meetings for Fall 2014 and Spring 2015
 1.1 Abstracts
 1.2 Previous Math Circle Meetings for Spring 2014
 1.3 Previous Math Circle Meetings Fall 2013
 1.4 Previous Math Circle Meetings Spring 2013
 1.4.1 Infinitely Often
 1.4.2 Making taffy with the Golden mean
 1.4.3 Guarding an Art Gallery
 1.4.4 The Coming of Enigma
 1.4.5 Origami
 1.4.6 The Game of Nim
 1.4.7 Pythagorean Triples: A Personal Interview
 1.4.8 Doodling and Graph Theory
 1.4.9 A Mathematician's April Fools
 1.4.10 String puzzles
 1.4.11 How to Win (or not) at TicTacToe
 1.4.12 Playing with Zomes
 1.5 Previous Math Circle Meetings Fall 2012
 1.6 Previous Math Circle Meetings Spring 2012
Meetings for Fall 2014 and Spring 2015
All talks are at 6pm in Ingraham Hall room 120, unless otherwise noted.
Fall 2014  

Date and RSVP links  Speaker  Topic  Link for more info 
September 8, 2014  Philip Matchett Wood  Pictures and Puzzles  
September 15, 2014  Jen Beichman  Playing with geometric sums  
September 22, 2014  DJ Bruce  Is any knot the unknot?  
September 29, 2014  Uri Andrews  The games of Criss Cross and Brussels Sprouts  
October 6, 2014  David Sondak  Fluids, Math, and Oobleck!  
October 13, 2014  George Craciun  Proofs without words (but with plenty of pictures)  
October 20, 2014  Scott Hottovy  Coming soon!  
October 27, 2014  Daniel Hast  Clock arithmetic and perfect squares: a "Golden Theorem" of reciprocity  
November 3, 2014  Alisha Zachariah  Infinity  
November 10, 2014  Marko Budisic  Mathematics of epidemics  
November 17, 2014  Nigel Boston  Same bad channel  




December 1, 2014  Daniel Erman  How to catch a (data) thief  
Spring 2015  

TBA  Coming soon!  
February 2, 2015  Soledad Benguria  Exploring Palindromes  
February 9, 2015  Jeff Linderoth  Coming soon!  
February 16, 2015  Simon Marshall  The Ant Walk  
February 23, 2015  Uri Andrews  Coming soon!  
March 2, 2015  Jordan Ellenberg  Coming soon!  
March 9, 2015  Ali Lynch  Mathematical Games and Winning Strategies  
March 16, 2015  Daniel Schultheis  Picture Hanging and Secret Algebra  
March 23, 2015  Betsy Stovall  Divisibility Cheats  
March 30, 2015  No meeting  UW Spring Break  
April 6, 2015  Julie Mitchell  Protein Folding and Robot Dances: Understanding the Basics of Kinematic Motion  
April 13, 2015  Jessica Lin  Coming soon!  
April 20, 2015  DJ Bruce  Coming soon!  
April 27, 2015  David Anderson  Let’s make a deal!  
May 4, 2015  Daniel Ross  Coming soon!  
May 11, 2015  Grace Deane  Last meeting of semester! 
Abstracts
Philip Matchett Wood
Pictures and Puzzles
When does a simple picture solve a tricky puzzle? Come and learn about how lineanddot drawing can solve complex puzzles, and create some new puzzles besides!
DJ Bruce
Is any knot the unknot?
Abstract: You're walking home from school, and you pull out your head phones to listen to some tunes. However, inevitably they are a horribly tangled mess, but are they really a knot? We'll talk about what exactly is a knot, and how we can tell when something is not the unknot.
David Sondak
Fluids, Math and Oobleck!
We will explore the magical world of fluids and their relationship to mathematics. As an example of fluids and math in the real world, we will make the living fluid oobleck and discuss some of its mathematical properties.
George Craciun
Proofs without words (but with plenty of pictures)
We will discuss mathematical proofs that can be done using only pictures or figures. If you want to see many such examples you can check out the book "Proofs without Words: Exercises in Visual Thinking" by Roger B. Nelsen. For more information also look at the wikipedia page http://en.wikipedia.org/wiki/Proof_without_words , where you can find links to Java Applets that show animations of proofs without words, such as http://usamts.org/Gallery/G_Gallery.php .
Daniel Hast
Clock arithmetic and perfect squares: a "Golden Theorem" of reciprocity
We'll explore systems of arithmetic where numbers loop back around to zero (like the hours on a clock!), called "modular arithmetic". Which numbers are perfect squares in such systems? Gauss, one of the greatest mathematicians in history, called the remarkable answer the "golden theorem".
Alisha Zachariah
What is infinity anyway
Infinity has a long history of having confounded and fascinated thinkers. We will take a look at some fundamental problems that early mathematicians grappled with and see some ways to understand infinity that have contributed to how we do math today.
Marko Budisic
Mathematics of epidemics
Infectious diseases in our communities often make it into daily conversation: "There's a nasty cold going around.", "It's the flu season, get your shots.", and even, "There are news of a zombie outbreak!" Come hear how math gets applied to something as messy as spread of disease. We will use our wits, pencils, and computers to understand the progress of headaches, common cold, zombie outbreaks, and even ebola, a disease that is currently making the news.
Nigel Boston
Same bad channel
How do we get such clear photos of the comet in the news? A 20 watt transmitter sends signals 500 million km through space to us and yet amazingly they survive this ordeal errorfree. What's behind this is errorcorrecting codes. I'll give some of the basics, some related puzzles, and some challenges.
Soledad Benguria
Exploring Palindromes
A Palindrome is a word or a number that reads the same forward and backwards. For example, Hannah, radar and civic are palindromic words, and 34743, 6446 are palindromic numbers. We will explore some curious properties of palindromes, and talk about what makes the number 196 special.
Simon Marshall
The Ant Walk
An ant is walking on a grid in the plane, but it can only move north or east. How many ways are there for it to get from one square to another? The numbers that appear when we answer this question have a wealth of interesting properties.
Betsy Stovall
Divisibility Cheats
We will discuss simple ways to determine whether one number is evenly divisible by a smaller one and also how to prove these facts. If time permits, we will also look at divisibility rules in bases other than 10.
Julie Mitchell
Protein Folding and Robot Dances: Understanding the Basics of Kinematic Motion
We will learn about motion subject to constraints. Mathematics based on these principles helps us build robots, explains human motion, and helps us model the shape of proteins like enzymes and antibodies.
David Anderson
Let’s make a deal!
We will explore a famous problem, called the Monty Hall Problem, that was inspired by a game show and became well known after a number of mathematicians incorrectly solved the problem in a very public manner. We will discuss and solve the problem, learn some probability, play the game, and invent some variants of the game that may, or may not, have similar counterintuitive behaviors.
Previous Math Circle Meetings for Spring 2014
All talks are at 6pm in Ingraham Hall room 120, unless otherwise noted.
Date and RSVP links  Speaker  Topic  Link for more info 

January 27, 2014  
Cancelled for weather  
February 3, 2014  Daniel Ross  Encryption  
February 10, 2014  Betsy Stovall  Geometric addition  
February 17, 2014  Mimansa Vahia  Origami and Mathematics  Origami video 
February 24, 2014  Jon Kane  Rows of Roses  
March 3, 2014  Matthew Johnston  Surprising results in games of chance  
March 10, 2014  Jordan Ellenberg  Why the card game Set should actually be called Line, and other comments on finite geometry  Set 
March 17, 2014  NO MEETING  UW Spring Break  
March 24, 2014  Reese Johnston  The Mathematics of Lying  
March 31, 2014  Reese Johnston  The Mathematics of Lying, part 2  
April 7, 2014  Daniel Erman  Josephus Problem  Platonic solids 
April 14, 2014  NO MEETING  MMSD Spring Break  
April 21, 2014  Chris Janjigian  Pirates and prisoners: an introduction to game theory (with candy!) 
Abstracts
Betsy Stovall
Geometric Addition
Abstract: We will learn some neat geometric tricks for quickly and painlessly computing some surprisingly large sums.
Jon Kane
Rows of Roses
Abstract: Let’s talk about the sine and cosine functions. One does not need to use very much information about these commonly seen functions in order to understand a large number of curves which can be drawn by graphing sine and cosine in Cartesian and polar coordinates. We will see sine curves, sums of sine curves, Lissajous figures, cycloids, hypocycloids, epicyclodes, and, of course, many rows of roses.
Previous Math Circle Meetings Fall 2013
All talks are at 6pm in Ingraham Hall room 120, unless otherwise noted.
Date and RSVP links  Speaker  Topic (click for more info) 

October 7, 2013  Gheorghe Craciun  Games Puzzles and Theorems in Geometry 
October 14, 2013  Gheorghe Craciun  Games Puzzles and Theorems in Geometry 
October 21, 2013  Uri Andrews  King Chicken Theorems 
October 28 2013  Uri Andrews  King Chicken Theorems 
November 4 2013  JeanLuc Thiffeault  The Mathematics of Juggling 
November 11 2013  Theodora Hinkle  TBA 
November 18 2013  Theodora Hinkle  TBA 
November 25 2013  TBA  TBA 
Previous Math Circle Meetings Spring 2013
More details about each talk to follow soon. All talks are at 6pm in Van Vleck Hall, room B231, unless otherwise noted.
Date and RSVP links  Speaker  Topic (click for more info) 

February 4, 2013 Register!  Jonathan Kane  Infinitely Often 
February 11, 2013 Register!  JeanLuc Thiffeault  Making taffy with the Golden mean 
February 18, 2013 Register!  Alison Gordon Lynch  Guarding an Art Gallery 
February 25, 2013 Register!  Mimansa Vahia  Origami 
Wed., Feb. 27, 2013 (Public Lecture, 5pm, B239)  David Perry  The Coming of Enigma 
March 4, 2013 Register!  Betsy Stovall  The Game of Nim 
March 11, 2013 Register!  Greg Shinault  Pythagorean Triples: A Personal Interview 
March 18, 2013 Register!  Elaine Brow  Doodling and Graph Theory 
March 25, 2013 Register!  Spring Break  No Meeting 
April 1, 2013 Register!  Uri Andrews  A Mathematician's April Fools 
April 8, 2013 Register!  Daniel Ross  String puzzles 
April 15, 2013 Register!  Silas Johnson  How to Win (or not) at TicTacToe 
April 22, 2013 Register!  Lalit Jain  Playing with Zomes 
Infinitely Often
February 4th, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
Infinitely Often
So you think you can add two numbers, three number, even a lot of numbers together? Well, can you add an infinite number of numbers together? See how thinking about infinite processes can be used to add infinite sums, evaluate repeating decimals, understand infinite continued fractions, and calculate areas and volumes. Also see what strange things can go wrong when dealing with infinity.
Making taffy with the Golden mean
February 11th, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
Making taffy with the Golden mean
Taffy pullers are devices used to make candy or bread. They are very interesting mathematically: we can relate the number of folds of dough to some famous mathematical sequences. Some surprising numbers pop up, like the Golden mean but also its lesserknown cousins. We can use this knowledge to improve existing devices. (Warning: no actual taffy will be made. Sorry.)
Guarding an Art Gallery
February 18th, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
Guarding an Art Gallery
How many guards does it take to guard an art gallery so that every spot in the gallery can be seen by at least one guard? We will explore this question and find an upper bound on the number of necessary guards based only on the number of walls in the gallery.
The Coming of Enigma
Special Public Lecture: Wednesday, February 27th, 2013, 5pm, Van Vleck Hall room B239, UWMadison campus
The Enigma machine was a cryptodevice used by the Germans before and during World War II and was considered to provide unbreakable security. This belief was founded on very solid principles which will be outlined in this talk. Taking a twomillennia tour through the history of cryptology, we will come to understand the design principles that went into the Enigma and understand how it worked and how it was used. We will also touch on how espionage, treason, and sibling rivalry provided Polish mathematicians the necessary ingredients to break the unbreakable. This talk is geared towards the general public, with no specific expertise in mathematics assumed.
Origami
February 25th, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
Origami is the art of folding paper, and it involves some cool math, too. Come to find out more!
The Game of Nim
March 4th, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
Nim is a twoplayer game wherein the players alternate taking one or more stones from a pile (there are two or more piles at the beginning). The player who takes the last stone wins. We will spend most of the time playing and trying to come up with winning strategies. At the end, we will talk a little about the history a general strategy to win the game.
Pythagorean Triples: A Personal Interview
March 11th, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
We all know the Pythagorean theorem from geometry, which tells us the relationship between the side lengths of any right triangle: a^2 + b^2 = c^2, where c is the length of the hypotenuse. Sometimes we are very lucky, when a, b, and c are natural numbers such as 3, 4, and 5. That is called a Pythagorean triple. We're going take a close look at these characters and figure out a few of their lessthanobvious traits.
Doodling and Graph Theory
March 18th, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
Some of the pencilandpaper games we play in notebook margins use more math than meets the eye. We'll try out a few fun and simple doodling puzzles, and see how they translate to some basic questions in graph theory. Then we'll harness our new theory to find quick solutions to whole groups of puzzles.
A Mathematician's April Fools
April 1st, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
A paradox is a seemingly logically valid argument that leads to absurd conclusions. Mathematicians are always very careful to avoid accidentally using one, but they can be useful and fun to play with.
String puzzles
April 8th, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
They may be familiar from novelty shops or even your mantlepiecea bit of string wound around and through some configuration of objects, asking you to accomplish some apparently impossible rearrangement or removal. Part of their difficulty comes from hopelessly infinite array of available movesdo you perhaps tie a clever knot here? Maybe pass a bight through there? We'll look at a few examples and see how to distill them to something more manageable, and even turn some into puzzles that can be solved instead only on paper (no drawing skills required).
How to Win (or not) at TicTacToe
April 15th, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
Playing with Zomes
April 22nd, 2013, 6pm, Van Vleck Hall room B231, UWMadison campus
This week we will and study the symmetric and magic of 3 dimensional shapes. Be ready to get your hands dirty and make your own stellated icosahedron!
Previous Math Circle Meetings Fall 2012
Date and RSVP links  Speaker  Topic (click for more info)  Event and poster links 

October 1, 2012: Register  Richard Askey  Counting: to and then beyond the binomial theorem  Combined High School Math Night & Math Circle (Poster) 
October 8, 2012: Register  Philip Matchett Wood  Proofs with Parity  Math Circle 
October 15, 2012: Register  Philip Matchett Wood  Fun Flipping Coins  Math Circle (Poster) 
October 22, 2012: Register  Saverio Spagnolie  Random walks: how gamblers lose and microbes diffuse  Combined High School Math Night & Math Circle (Poster) 
October 29, 2012: Register  Beth Skubak  nonEuclidean geometry  Math Circle (Poster) 
November 5, 2012: Register  Mihai Stoiciu  Rubik's Cubes  Combined High School Math Night & Math Circle (Poster) 
November 12, 2012: Register  Alison Gordon  Curious Catalan Numbers  Math Circle (Poster) 
November 19, 2012: Register  Gregory Shinault  Tiling Problems  Math Circle 
November 26, 2012: Register  Claire Blackman  Binary Numbers  Math Circle 
Counting: to and then beyond the binomial theorem
October 8th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Richard Askey. How many ways can zeros and ones be put into n places? It is easy to see this is 2^n. It is also easy to show that there are n! ways of ordering n different objects. There are problems which go beyond these two. How many ways can k zeros and nk ones be put into n places? How many inversions are there in the n! ways of ordering the numbers 1,2,...,n. [123 has no inversions, 132 has one, 312 has two, 321 has three]. These will lead us to what has been called "The world of q".
Proofs with Parity
October 8th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Philip Matchett Wood. Paritymatching objects up in pairsis a surprisingly useful tool for answering math questions. Bring a pencil and notebook, and we will explore many different situations where parity plays a role.
Fun Flipping Coins
October 15th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Philip Matchett Wood. Flip a coin many times, and what happens? A whole mess of cool probability, that what! Bring a notebook, pencil, and some sharp common sense.
Random walks: how gamblers lose and microbes diffuse
October 22nd, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Saverio Spagnolie. We will explore one of the most famous mathematical models of random activity, the random walk. After an introduction to some basic ideas from probability, we will see that the same mathematical tools can be used to study completely different types of problems. In particular, we will find that there are no gambling strategies that can be used to beat the casino, and that tiny microorganisms can't stop moving even if they want to!
NonEuclidean geometry
October 29th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Beth Skubak. Most of the geometry we see in school is based on the ideas of the Greek mathematician Euclid, who lived around 300 BC. While his ideas are pretty useful, we want to consider geometry in some "nonEuclidean" scenarios, like when instead of being flat, our surfaces are curved.
Rubik's Cubes
November 5th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Mihai Stoiciu. Rubik's Cubes. Some people describe mathematics as the science of patterns. We will explore patterns, permutations, orientations, and counting with the famous Rubik's Cube.
Curious Catalan Numbers
November 12th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Alison Gordon. The Catalan numbers are a sequence that shows up as solutions to all sorts of problems in mathematics. Join us as we count handshakes, match parentheses, and build mountains in order to understand these interesting numbers!
Tiling Problems
November 19th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Greg Shinault. Remember tangrams? You know, given some tiles build a specific shape using them. That is an example of a tiling problem, and to some mathematicians they are serious business. We are going to play with a variety of these puzzles, and talk about some of the things that have been figured out about them.
Binary Numbers
November 26th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Claire Blackman. We're all used to doing arithmetic with the 10 digits 0 to 9. But there's no reason why we shouldn't use just two digits, 0 and 1, instead. We'll be exploring the world of binary arithmetic, which is based on powers of two.
Previous Math Circle Meetings Spring 2012
Date  Speaker  Talk (click for more info) 

February 13, 2012  Patrick LaVictoire  Transforms: Pictures in Disguise 
February 20, 2012  Uri Andrews  Hercules and the Hydra 
February 27, 2012  Peter Orlik  Madison Math Circles 
March 5, 2012  JeanLuc Thiffeault  The hagfish: the slimiest fish in the sea 
March 12, 2012  Cathi Shaughnessy  Archimedes' method 
March 19, 2012  Andrei Caldararu  Games with the binary number system 
March 26, 2012  Laurentiu Maxim  How many pentagons and hexagons does it take to make a soccer ball? 
Transforms: Pictures in Disguise
February 13th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Patrick LaVictoire. How are computer graphics like a massive game of Sudoku? How does a CAT scan get a 3D picture from a bunch of 2D Xray images? How can you make the same image look like different people when viewed from close up and far away? I'll discuss all these and more, with some neat illustrations and quick games.
Hercules and the Hydra
February 20th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Uri Andrews. We will talk about important techniques of selfdefense against an invading Hydra. The following, from Pausanias (Description of Greece, 2.37.4) describes the beginning of the battle of Hercules against the Lernaean hydra:
"As a second labour he ordered him to kill the Lernaean hydra. That creature, bred in the swamp of Lerna, used to go forth into the plain and ravage both the cattle and the country. Now the hydra had a huge body, with nine heads, eight mortal, but the middle one immortal. . . . By pelting it with fiery shafts he forced it to come out, and in the act of doing so he seized and held it fast. But the hydra wound itself about one of his feet and clung to him. Nor could he effect anything by smashing its heads with his club, for as fast as one head was smashed there grew up two..."
For more information on some of the conjectures discussed during this talk see http://en.wikipedia.org/wiki/Collatz_conjecture and http://mathworld.wolfram.com/CollatzProblem.html
Madison Math Circles
February 27th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Peter Orlik. A short introduction to elementary and middle school activities in Madison like Mathematical Olympiad and Mathcounts will be followed by some challenging problems. Please bring your favorite pencils and be prepared to work!
The hagfish: the slimiest fish in the sea
March 5th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: JeanLuc Thiffeault. The hagfish is a bottomdwelling, scavenger fish that resembles an eel. It has some interesting peculiarities: first, it sometimes willingly ties itself in a knot. Second, it secretes a spectacular amount of slime, which is used in the cosmetics industry. For a long time the purpose of this slime was unknown, but recently scientists have filmed live hagfish using it. (I'll keep this purpose a secret until the talk...) I'll then discuss how we can apply mathematical tools to study hagfish slime.
Archimedes' method
March 12th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Cathi Shaughnessy. Students will use Archimedes' classical method to determine bounds for the value of the number pi. Please BRING A CALCULATOR with you for this presentation. The presenter will provide compass, protractor, straightedge and worksheet for each student.
Games with the binary number system
March 19th, 2012, 6pm, Van Vleck Hall room B223, UWMadison campus
Presenter: Andrei Caldararu. I will present a few games and tricks which use the binary number system. For more information about binary numbers please see http://en.wikipedia.org/wiki/Binary_numeral_system
How many pentagons and hexagons does it take to make a soccer ball?
March 26th, 2012, 6:30pm (note special time!!!), Van Vleck Hall room B223, UWMadison campus
Presenter: Laurentiu Maxim. I will first introduce the concept of Euler characteristic of a polyhedral surface. As an application, I will show how one can find the number of pentagons on a soccer ball without actually counting them.