# Archived Math Circle Material

## Previous Talks Fall 2012

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 Talks Spring 2012

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

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.