# Archived Math Circle Material

## Contents

- 1 Previous Math Circle Meetings for Spring 2014
- 2 Previous Math Circle Meetings Fall 2013
- 3 Previous Math Circle Meetings Spring 2013
- 3.1 Infinitely Often
- 3.2 Making taffy with the Golden mean
- 3.3 Guarding an Art Gallery
- 3.4 The Coming of Enigma
- 3.5 Origami
- 3.6 The Game of Nim
- 3.7 Pythagorean Triples: A Personal Interview
- 3.8 Doodling and Graph Theory
- 3.9 A Mathematician's April Fools
- 3.10 String puzzles
- 3.11 How to Win (or not) at Tic-Tac-Toe
- 3.12 Playing with Zomes

- 4 Previous Math Circle Meetings Fall 2012
- 5 Previous Math Circle Meetings Spring 2012

## 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 | Jean-Luc 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! | Jean-Luc 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 Tic-Tac-Toe |

April 22, 2013 Register! | Lalit Jain | Playing with Zomes |

### Infinitely Often

February 4th, 2013, **6pm**, Van Vleck Hall room B231, UW-Madison 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, UW-Madison 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 lesser-known 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, UW-Madison 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**, UW-Madison 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 two-millennia 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, UW-Madison 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, UW-Madison campus

Nim is a two-player 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, UW-Madison 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 less-than-obvious traits.

### Doodling and Graph Theory

March 18th, 2013, **6pm**, Van Vleck Hall room B231, UW-Madison campus

Some of the pencil-and-paper 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, UW-Madison 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, UW-Madison campus

They may be familiar from novelty shops or even your mantlepiece--a 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 moves--do 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 Tic-Tac-Toe

April 15th, 2013, **6pm**, Van Vleck Hall room B231, UW-Madison campus

### Playing with Zomes

April 22nd, 2013, **6pm**, Van Vleck Hall room B231, UW-Madison 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 | non-Euclidean 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, UW-Madison 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 n-k 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, UW-Madison campus

**Presenter: Philip Matchett Wood.** Parity---matching objects up in pairs---is 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, UW-Madison 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, UW-Madison 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!

### Non-Euclidean geometry

October 29th, 2012, **6pm**, Van Vleck Hall room B223, UW-Madison 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 "non-Euclidean" scenarios, like when instead of being flat, our surfaces are curved.

### Rubik's Cubes

November 5th, 2012, **6pm**, Van Vleck Hall room B223, UW-Madison 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, UW-Madison 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, UW-Madison 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, UW-Madison 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 | Jean-Luc 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, UW-Madison 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 X-ray 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, UW-Madison campus

**Presenter: Uri Andrews.** We will talk about important techniques of self-defense
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, UW-Madison 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, UW-Madison campus

**Presenter: Jean-Luc Thiffeault.** The hagfish is a bottom-dwelling, 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, UW-Madison 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, UW-Madison 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, UW-Madison 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.