# Fall 2012 Sessions

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**Week 1 – October 2, 2012**

**Host**: Dr. Maria Wesslén

**Topic**: Algebra and rectangles

Can you draw a rectangle made up of (i.e. completely covered by) perfect squares, so that each of the squares have integer side lengths, and all squares are of ** different** size?

Try it first… if you don’t succeed after trying really hard, does that mean it is impossible?

This will turn out to be a beautiful and interesting application of using algebra to solve a geometrical problem.

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**Week 2 – October 9, 2012**

**Host**: Dr. Maria Wesslén

**Topic**: Organization!

We will look at a number of different problems where organizing your work is extremely important. How you organize the information about a problem can help you see new patterns and help you develop new insights into a problem. Here is one of the problems we will consider:

Four ways to make 3:

3

2+1

1+2

1+1+1

How many ways can we make 10???

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**Week 3 – October 16, 2012**

**Host**: Dr. Maria Wesslén

**Topic**: Graph Theory!

A graph is a collection of dots (vertices) and lines (edges) connecting the vertices. In these graphs it is not important exactly where the vertices are located, only which vertices are connected to which others. Two graphs are called isomorphic if they have the same number of vertices, connected in the same way.

Are any of these graphs isomorphic?

It can be surprisingly difficult to tell..!

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**Week 4 – October 23, 2012**

**Host**: Dr. Maria Wesslén

**Topic**: Finding the greatest common divisor (gcd)

Here is a warm up problem for you to try:

Kyle has 18 green beads and 24 blue beads. He wants to use them to make a number of identical bracelets. What is the largest number of bracelets he can make? (They must be identical, and he wants to use all the beads. He can borrow as many white beads he likes from his mother to fill out.)

You might have learnt how to find the gcd of two numbers by finding their prime factorizations. It turns out that even if we know how to do something in theory, it may not be so easy to do in practice, with large numbers. Factoring numbers is very difficult/time consuming to do. It is much more difficult than multiplying numbers! When the numbers get large, factoring is difficult even for computers. We’ll play a game that should convince you that factoring is much more difficult than multiplying, and we will learn a more efficient method for finding the gcd of two integers.

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**Week 5 – October 30, 2012**

**Host**: Shai Cohen

**Topic**: The mathematics behind elections

Shai Cohen is back! This time he will not attempt to win your money, but instead your vote, as we look at the intricate mathematics behind elections. This is closely related to Game Theory, a branch of mathematics which was the topic of Shai’s last appearance at the UTM Math Circle. But each session is freestanding, so even if you missed the last one, that’s not a problem.

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**Week 6 – November 6, 2012**

**Host**: Dr. Maria Wesslén

**Topic**: Congruences and Modular Arithmetic

Dividing one integer by another will not always give an integer. But we can make the ‘leftover’ (remainder) as small as possible. We will discuss how to do this and come up with some efficient ways of working with remainders and using them to solve all kinds of problems.

By the end of the session we should be able to answer questions like these:

*What is the remainder when 5^{437} is divided by 6?

*What is the unit digit of each of these numbers?

7^{120} 3^{1999} 7^{1947}

* If *n* is any integer, is it true that *n*^{5 }+29*n* is divisible by 30?

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**Week 7 – November 13, 2012**

**Host**: Dr. Vinod Vaikuntanathan

**Topic**: Cryptoghaphy!

We will use some of the modular arithmetic we learnt last week to send secret messages…and attempt to eavesdrop and break the code to hear what the others are saying!!!

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**Week 8 – November 20, 2012**

**Host**: Charelene and Mimy

**Topic**: Counting, counting, counting….

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**Week 9 – November 27, 2012**

**Host**: Dr. Zsuzsanna Dancso

**Topic**: Generating Functions

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**Week 10 – December 4, 2012**

**Host**: Dr. Maria Wesslén

**Topic**: The Mad Veterinarian