# Spring 2011 Sessions

**February 1, 2011
Host:** Dr. Maria Wesslen

**Topic:**Getting Started

In the first week we will work on a mixed bag of shorter problems and puzzles, to get things started.

Here are a couple of teasers, which you can try before the first meeting, if you like:

1. A 10 x 29 x 42 cm box lies on a floor with one of its 29 x 42 cm sides facing down. An ant sits on the floor in front of one of the corners. What is the shortest path the ant can walk/crawl to get to the opposite corner of the box (also on the floor)? The ant can walk/crawl along the outside of any side of the box, but not under the box. It can crawl along an edge of the box and it can walk around the box on the floor.

2. A farmer grows apples and has 3000 apples when it is time to take them to the market, which is 1000 km away along a deserted country road. He has a donkey and cart to transport the apples, with the following restrictions:

i) The donkey can only pull at most 1000 apples at a time

ii) The donkey will only walk if munching on an apple. He eats one apple for every km he walks.

What is the maximum number of apples the farmer can get to the market using only the donkey and cart to transport them? The farmer may bring apples part-way, drop off a supply of applies, walk back (make sure the donkey still has enough apples to do this!), re-boost his supply, and so on. How many apples can the farmer get to the market this way? (Note: This is not an easy problem! Don’t be discouraged if you find it difficult to get even a single apple to the market, at first. But keep trying!)

**About the host:**

Most sessions will be hosted by Dr. Maria Wesslen. Originally from Sweden, Dr. Wesslen is a course instructor in the Department of Mathematical and Computational Sciences at UofT Mississauga as well as a course instructor at University of Waterloo. She holds a teaching certificate from King’s College London, completed undergraduate studies at Imperial College London and graduate studies at University of Toronto. Her area of interest is Lie Algebras and especially Representation Theory of Lie Algebras.

**February 8, 2011**

Host: Dr. Maria Wesslen

**Topic:** Is your cup half full or half empty?

**Problem:** You are standing next to a fountain with a 3 liter bucket and a 5 liter bucket. You want measure exactly 4 liters of water. Is this possible? If so, how?

You may have seen similar problems before. (Did anyone see the movie Die Hard 3?!) We will be thinking about more general questions, such as ‘if you have two buckets with x and y liter capacities, what are the amounts you can/can’t measure?’ and related questions.

**February 15, 2011
Host:** Dr. Maria Wesslen

**Topic:**Not your regular numbers

What is the pattern in this sequence? Find the next 10 terms in the sequence.

1, 10, 11, 100, 101, 110, 111, 1000, ….

The answer is a hint for what this weeks’ topic will be.

**February 22, 2011
Guest Hosts:** Abdul Ayoub and Jaspreet Sahota

**Topic:**Probabilities

**About the hosts: **

Abdul Ayoub is a third year mathematics student and teaching assistant at UofT Mississauga. Apart from taking mathematics and statistics courses, he is working on a research project on Lie Algebra with Dr. Shay Fuchs and a physics research project on Radon and Radioctivity with Dr. Wagih Ghobriel. Abdul also volunteers at the Robert Gillespie Academic Skills Centre, where he is involved with the Peer Facilitated Group Study program.

Jaspreet Sahota is in the final year of his H.BSc (double major: Math and Physics) at UofT Mississauga. This year he is taking several mathematics and physics courses including Quantum Mechanics, Relativistic Electrodynamics, Real Analysis, Topology, Differential Geometry and Classical Mechanics as well as a Physics Research Project. He is also working as a teaching assistant at UofT Mississauga, for mathematics and physics courses.

In the summer of 2010 Abdul and Jaspreet worked together on a research project in differential geometry and relativity theory under the supervision of Prof. Yael Karshon and Dr. Shay Fuchs. They also went to a MathFest conference in Pittsburgh, where they (among many other things) participated in a session by the National Association of Math Circles.

**March 1, 2011 **

**Guest Host:** Dr. Shay Fuchs

**Topic:** Graph Theory

Can you trace out the following figures without taking your pen off the paper and without tracing any line twice? Can you prove your answers?

This problem leads to important and interesting theorems in a field called ‘Graph Theory’, which will be the topic for this Math Circles session. We will discuss this problem, explore other figures, and try to generalize. We will also talk about ‘crossing bridges’ and ‘coloring maps’, which are also related to graph theory.

**About the host:**

Dr. Shay Fuchs is a Lecturer in the Department of Mathematical and Computational Sciences at UofT Mississauga. His area of interest is Symplectic Geometry and Topology. In 2010-2011 he is teaching Introduction to Mathematical Proofs, Calculus for Life Sciences and Introduction to Topology. Dr. Fuchs previously taught Mathematics at High Schools in his home country of Israel.

At UofT Mississauga, Dr. Fuchs is also involved in helping interested students prepare for the Putnam Competition. This is an annual competition where University and College students from across North America compete to solve challenging mathematics questions.

**March 8, 2011
Guest Hosts:** Jaspreet Sahota

**Topic:**Groups

**Note: There will be no session on March 15, 2011 because of March Break.**

**March 22, 2011
Host:** Dr. Maria Wesslen

**Topic:**Congruences

In this week's Math Circle we will solve various problems using congruences. Here are a couple of problems that you can try already. Note: Even if you don't know what a congruence is, you can still try these problems.

1. What is the remainder when 9^157 is divided by 20?

2. A baker has several eggs. His assistant asks how many there are, but the baker responds by saying:

- If I arrange them in pairs, there is 1 left over.
- If I arrange them in 3's, there is 1 left over.
- If I arrange them in 5's, there is also 1 left over

Can the assistant use this information to figure out how many eggs there are? Is there more than one possibility?

**March 29, 2011
Guest Host:** Dr. Stephen Tanny

**Topic:**Various puzzling problems

We will work on a few interesting problems, many with very surprising results. Here is one for you to get started on:

Step 1: Choose a four digit number between 0001 and 9998 (leading 0's are allowed) but NOT 1111, 2222, 3333 etc.

Step 2: Rearrange the digits in your number in ascending order and then in descending order.

Step 3: Subtract the smaller from the larger of these two numbers, to get a new number.

Repeat steps 2-3 with you new number. Keep repeating these steps several times.

Here is an example:

Step 1: Choose for example 4391.

Step 2: Descending 9431, Ascending 1349

Step 3: 9431 - 1349 = 8082

Now repeat with 8082:

Step 2: Descending 8820, Ascending 0288

Step 3: 8820 - 0288 = 8532

Now repeat with 8532.....

Keep repeating these steps several times. What happens? Try different starting numbers to see what happens!

**About the host:**

Dr. Stephen Tanny is an Associate Professor in the Department of Mathematical and Computational Sceinces at UofT Mississauga and in the Department of Mathematics at the St. George Campus of UofT. His field of research is Enumerative Combinatorics; most recently he has focused on the study of nested recursions. In 2010-11 he teaches Introduction to Combinatorics, Advanced Combinatorics, and Calculus for Life Sciences.

**April 5, 2011
Host:** Dr. Eugene Kritchevski

**Topic:**Numerical Methods

**April 12, 2011**

**Host:** Dr. Maria Wesslen

**Topic:** Are you a winner?

We will be working on probelms in a field of mathematics called Game Theory. Is there a particular strategy that will ensure a win? We will play and analyze a number of simple games and discuss how best to approach them. Here is one example for you to try. You will need two players.

Write 15 minus signs in a row of a piece of paper. The players will take turns to change these minus signs into plus signs. You can choose to change 1, 2 or 3 of the minuses into plus. The winner is the person who does NOT turn the last minus into plus.

Try playing this several times. What is your strategy going to be? Is it better to go first or second?