# Spring 2012 Sessions

Week 1 - January 31, 2012

Host: Dr. Maria Wesslen

Topic: Welcome to Math Circles!

In the first week we will work on a mix of problems and puzzles. Here are a few questions which we will consider:

1. We will try to find fractions between √56 and √58 .

a) Are there any?

b) How many are there?

c) Out of all the fractions between √56 and √58 , what could the denominator be?

E.g. 188/25 is one example, so 25 is one possible denominator. What other small numbers could the denominators be? Are there any denominators that will never show up in a fraction between  √56 and √58 ?

Try to come up with answers to these questions, and try to prove your conjectures.

2. The number 7. 565656565656565656…. is between  √56 and √58 . How do we know if it is a fraction or not? Can we write it as a fraction? How can we go about doing that? Or is it impossible?

We’ll consider these and many related questions.

Week 2 - February 7, 2012

Host: Dr. Maria Wesslén

Topic: Complex Numbers

Consider the following questions:

What are the solution(s) or to x2 - 2x - 1 = 0?

What are the solution(s) to x2 + 5 = 0 ?

To answer these and many others we will need ‘complex numbers’. We will do some arithmetic and work on some puzzles and you’ll see that complex numbers are not as complicated as they sound!

Week 3 - February 14, 2012

Host: Dr. Alexander Grigo

Topic: Probabilities and other counting conundrums

Do you think counting is easy? Well, try these problems and see:

1. There are 3 urns A, B and C each containing a total of 10 marbles of which 2, 4 and 8 respectively are red. A pack of cards is cut and a marble is taken from one of the urns depending on the suit shown - a black suit indicating urn A, a diamond urn B, and a heart urn C. What is the probability a red marble is drawn?

2. M men and W women seat themselves at random in M+W many seats arranged in a row. What is the probability that all women will be sitting next to each other?

Counting is a very important part of mathematics, and sometimes it is not as easy as it sounds. We’ll explore some formulas and methods you might need to answer a variety of counting problems.

Week 4 - February 21, 2012

Host: Dr. Maria Wesslén

Topic: Proof by Induction

Consider the following problem:

There are  n  points on a circle. Each point is joined to each other point, so that no three (or more) line segments intersect in a single point. How many regions will this form inside the circle?

Try drawing some pictures with 1, 2, 3, 4 points, and count the regions. Try to look for a pattern and guess what the formula is.

Once you have guessed a formula, how do you know for sure this formula will work for all values of  n ? What if it doesn’t? We can’t check all infinitely many values of  n , so we need a better method.

In this Math Circle we will discuss a proof method called ‘Proof by Induction’, which will help us with this kind of problem.

By the way, try checking your formula for 5 and 6 points. What do you find?

Week 5 - February 28, 2012

Host: Dr. Alexander Grigo

Topic: Generalized Proof by Inductions

This week continues on last weeks’ topic of Proof by Induction, but this time we will generalize this technique it to be able to prove more types of problems.

Week 6 - March 6, 2012

Host: Jordan Watts

Topic: Hilbert’s Hotel

This week we will explore a very strange hotel called Hilbert’s Hotel. Prof. David Hilbert was a famous mathematician at the last turn of the century, and in his hotel there are infinitely many rooms.

How many guests can Hilbert accommodate in his hotel? We will explore some fascinating problems regarding this hotel with infinitely many rooms. You might be surprised at the results…

Week 7 - March 20, 2012

Host: Dr. Maria Wesslén

Topic: The Pigeonhole Principle

Did you know that in Toronto live at least 2 persons with the same number of hairs on their heads?

How can I be so sure? Do you think I counted the number of hair strands on two people’s heads, and they happened to be equal? No, probably not. The reason is based on the Pigeonhole Principle. The Pigeonhole Principle is a very simple sounding idea, but it can be used to solve surprisingly many problems, as we’ll see in this week’s Math Circle.

Week 8 - March 27, 2012

Host: Dr. Kristin Shaw

Topic: Tropical Geometry

The tropics is a very strange land, where “multiplication” is what we think of as addition, and “addition” is what we think of as taking the maximum of two numbers.

This can make things tricky… and some things that we take for granted do not necessarily make any sense…

Are you ready for exploring the world of tropical mathematics?

Week 9 - April 3, 2012

Host: Shai Cohen

Topic: Game Theory in the Real World

Game Theory is much more than playing mathematical games. This week we will think about how mathematical game theory can be used in the real world. We will put a few of you in jail and think about how you would act if you were in the mafia, or a politician, or bidding at an auction…

Oh, and Shai promises he will win over all of you… do you dare take the challenge?

Week 10 - April 10, 2012

Host: Maria Wesslén

Topic: Final Week of the Year - Invariants

You will get your certificates and we’ll sum up the year.

But we will also do some interesting mathematics. We will pretend to be chameleons and we will try to solve a puzzle by Einstein.

The theme is ‘invariants’. An invariant is something that remains the same even if other things change. If you can find an invariant it might help you solve many problems and it can explain many strange behaviours. But finding one is not always as easy as it sounds…