Past Session Descriptions & Resources
Summer 2026 Sessions - Resource Folder
Session 1: Listen, Share, Play: Lessons from Preschool for Problem Solving
Description:
This session will largely be a workshop on problem solving, where we will work on some (hopefully) unfamiliar questions. Part of the purpose is to practice working on difficult math - our students need to learn new things all the time, and so should we! But we will also discuss mindsets and attitudes help with problem solving.
The title of the session comes from something my son said when he was 4: I had told him I was teaching problem solving, and he said "ya, I'm learning that too!" It turned out his preschool was teaching "problem solving," meaning "avoid fighting with the other preschoolers." This is not exactly what we usually mean in a math class, but when I thought about it many of the things the preschool was working on (be patient; listen to what others have to say; be open to compromise; accept that you won't always get your way) are in fact very important for mathematical problem solving as well. SO, we will explore that idea.
Facilitator Bio: Peter Tingley is a professor and the chair of the department of Mathematics and Statistics at Loyola University. He teaches a wide range of math classes from precalculus through game theory and abstract algebra. He has been involved in math circles since 2006 and co-founded the Chicago Math Teachers' circle in 2015. Most of all, he loves to solve problems!
Session 2: The Math of Spot-It
Description: Spot-It is a matching game for kids, but the mathematics behind the game is both deep and accessible. This session is an open-ended investigation of the game, where participants discover the hidden structures, patterns, and formulas that make the game special. This is an opportunity to see how mathematics can be used to model the real world.
Facilitator Bios: Angela Antonou is an assistant professor at the College of DuPage. She earned her doctorate in 2014 from Northern Illinois University and has been involved on the leadership team of the Southwest Chicago Math Teachers' Circle since its inception. She loves working with the community to share the joy of mathematics. Christina Jamroz earned her doctorate from Purdue University in 2018 and has been involved on the leadership team of the Southwest Chicago Math Teachers' Circle since 2019. Her commitment to faculty professional development is reflected in her work on the SWMTC and as the Section Next Coordinator for the Illinois Section of the Mathematical Association of America.
Session 3: Diophantine approximation and bikes
Description: Bicycle gears are based on the ratio of the number of gear teeth on the front gear to the number of gear teeth on the back gear. Which gear combination should we choose to make our ride as smooth as possible? We will try to approximate our favorite ratios with limited choices in the number of teeth (aka without making the denominator of our fraction too big).
Facilitator Bio: Karl Liechty is a math professor at DePaul University. His own research is in the areas of probability and analysis, but he likes to think about all sorts of mathematical problems. Karl has been involved in Math circles since 2012, and with Chicago Math Teachers' Circles since 2016.
Session 1: Listen, Share, Play: Lessons from Preschool for Problem Solving
Description:
This session will largely be a workshop on problem solving, where we will work on some (hopefully) unfamiliar questions. Part of the purpose is to practice working on difficult math - our students need to learn new things all the time, and so should we! But we will also discuss mindsets and attitudes help with problem solving.
The title of the session comes from something my son said when he was 4: I had told him I was teaching problem solving, and he said "ya, I'm learning that too!" It turned out his preschool was teaching "problem solving," meaning "avoid fighting with the other preschoolers." This is not exactly what we usually mean in a math class, but when I thought about it many of the things the preschool was working on (be patient; listen to what others have to say; be open to compromise; accept that you won't always get your way) are in fact very important for mathematical problem solving as well. SO, we will explore that idea.
Facilitator Bio: Peter Tingley is a professor and the chair of the department of Mathematics and Statistics at Loyola University. He teaches a wide range of math classes from precalculus through game theory and abstract algebra. He has been involved in math circles since 2006 and co-founded the Chicago Math Teachers' circle in 2015. Most of all, he loves to solve problems!
Session 2: The Math of Spot-It
Description: Spot-It is a matching game for kids, but the mathematics behind the game is both deep and accessible. This session is an open-ended investigation of the game, where participants discover the hidden structures, patterns, and formulas that make the game special. This is an opportunity to see how mathematics can be used to model the real world.
Facilitator Bios: Angela Antonou is an assistant professor at the College of DuPage. She earned her doctorate in 2014 from Northern Illinois University and has been involved on the leadership team of the Southwest Chicago Math Teachers' Circle since its inception. She loves working with the community to share the joy of mathematics. Christina Jamroz earned her doctorate from Purdue University in 2018 and has been involved on the leadership team of the Southwest Chicago Math Teachers' Circle since 2019. Her commitment to faculty professional development is reflected in her work on the SWMTC and as the Section Next Coordinator for the Illinois Section of the Mathematical Association of America.
Session 3: Diophantine approximation and bikes
Description: Bicycle gears are based on the ratio of the number of gear teeth on the front gear to the number of gear teeth on the back gear. Which gear combination should we choose to make our ride as smooth as possible? We will try to approximate our favorite ratios with limited choices in the number of teeth (aka without making the denominator of our fraction too big).
Facilitator Bio: Karl Liechty is a math professor at DePaul University. His own research is in the areas of probability and analysis, but he likes to think about all sorts of mathematical problems. Karl has been involved in Math circles since 2012, and with Chicago Math Teachers' Circles since 2016.
Zombie growth: A discussion of Mathematical modeling vs. Mathematical problem posing - December 2025
Facilitator: Yvonne Lai (U Nebraska Lincoln)
Zombie Growth Problem
Facilitator: Yvonne Lai (U Nebraska Lincoln)
Zombie Growth Problem
Can the voice of the people be heard? - February 2025
Facilitator: Karl Liechty
MTC article: "We the People"
Video (Arrow's theorem)
Article (complete proof)
We talked about what makes a voting system “fair,” and looked at several different systems for conducting elections. Are some more fair than others? I’m attaching a short article from the Math Circles magazine which describes most of what we did. For those that might be interested in the how to prove Arrow’s theorem, this Youtube video does a pretty good job of explaining the basic idea, though it’s not a complete proof. If people are interested in the complete proof, this article is relatively easy to follow.
Facilitator: Karl Liechty
MTC article: "We the People"
Video (Arrow's theorem)
Article (complete proof)
We talked about what makes a voting system “fair,” and looked at several different systems for conducting elections. Are some more fair than others? I’m attaching a short article from the Math Circles magazine which describes most of what we did. For those that might be interested in the how to prove Arrow’s theorem, this Youtube video does a pretty good job of explaining the basic idea, though it’s not a complete proof. If people are interested in the complete proof, this article is relatively easy to follow.
RSA Cryptography - December 2024
Facilitator: Peter Tingley
Warm-Up (code)
RSA Worksheet
Also, two mathematical comments:
1] As Chris pointed out afterwards, it is not actually always true that m^{(p-1)(q-1)}=1 mod pq - there is a strange case if m is divisible by p or q. Which is rare, but can happen. However, even in that case it is true that m^{(p-1)(q-1)+1}=m mod pq, which is what you actually need.
2]The facts we worked out on the worksheet:
Facilitator: Peter Tingley
Warm-Up (code)
RSA Worksheet
Also, two mathematical comments:
1] As Chris pointed out afterwards, it is not actually always true that m^{(p-1)(q-1)}=1 mod pq - there is a strange case if m is divisible by p or q. Which is rare, but can happen. However, even in that case it is true that m^{(p-1)(q-1)+1}=m mod pq, which is what you actually need.
2]The facts we worked out on the worksheet:
- M^p = p mod p
- M^{(p-1)(q-1)} is usually 1 mod pq
Sequences and Juggling - October 2023
Facilitators: Peter Tingley and Karl Liechty
Handouts
We made conjectures about the regions formed in a circle that has 10 random points on it connected by lines, and counted triangles in an image. How did this connect to juggling, you ask? We showed you how.
Facilitators: Peter Tingley and Karl Liechty
Handouts
We made conjectures about the regions formed in a circle that has 10 random points on it connected by lines, and counted triangles in an image. How did this connect to juggling, you ask? We showed you how.
Conway's Game of Life - December 2020
Facilitator: Karl Liechty
Session slides
Wiki on Game of Life
We looked at two famous creations of the mathematician John H. Conway: the Look-and-Say sequence, and Conway's Game of Life. The Game of Life provides simple rules by which configurations of squares on a (infinitely large) checker board evolve, stagnate, or go extinct. We played around with some of these configurations to see where Life would take them, learning about still lifes, oscillators, and spaceships.
Facilitator: Karl Liechty
Session slides
Wiki on Game of Life
We looked at two famous creations of the mathematician John H. Conway: the Look-and-Say sequence, and Conway's Game of Life. The Game of Life provides simple rules by which configurations of squares on a (infinitely large) checker board evolve, stagnate, or go extinct. We played around with some of these configurations to see where Life would take them, learning about still lifes, oscillators, and spaceships.
Chameleon Mayhem! - July 2020
Facilitator: Peter Tingley
Chameleons on an island come in three colors. They wander and meet in pairs. When two chameleons of different colors meet, they both change to the third color. Given initial amounts of the lizards of each color are 13, 15, and 17, may this happen that, after a while, all of them acquire the same color?
Solutions can be found here: https://www.cut-the-knot.org/blue/Chameleons.shtml#Solution
Additional Resources for this Problem:
https://mindyourdecisions.com/blog/2016/04/24/the-chameleon-riddle-sunday-puzzle/
https://nrich.maths.org/651
Facilitator: Peter Tingley
Chameleons on an island come in three colors. They wander and meet in pairs. When two chameleons of different colors meet, they both change to the third color. Given initial amounts of the lizards of each color are 13, 15, and 17, may this happen that, after a while, all of them acquire the same color?
Solutions can be found here: https://www.cut-the-knot.org/blue/Chameleons.shtml#Solution
Additional Resources for this Problem:
https://mindyourdecisions.com/blog/2016/04/24/the-chameleon-riddle-sunday-puzzle/
https://nrich.maths.org/651
Conway's Rational Tangles - December 2019
Facilitators: Tim Stoelinga (North) and Sarah Bockting-Conrad (South)
We played with pieces of rope, and also talk about mathematical operations. Somehow these ideas came together and we (hopefully) understand some things.
Facilitators: Tim Stoelinga (North) and Sarah Bockting-Conrad (South)
We played with pieces of rope, and also talk about mathematical operations. Somehow these ideas came together and we (hopefully) understand some things.
Welcome to Cataland! - October 2019
Facilitators: Peter Tingley (North) and Emily Barnard (South)
We explored some famous counting problems, such as counting ways to walk in a grid. Amazingly, lots (but not all) of these problems have the same answer: the famous Catalan numbers!
If you want to see even more things counted by Catalan numbers, you can look here.
Facilitators: Peter Tingley (North) and Emily Barnard (South)
We explored some famous counting problems, such as counting ways to walk in a grid. Amazingly, lots (but not all) of these problems have the same answer: the famous Catalan numbers!
If you want to see even more things counted by Catalan numbers, you can look here.
Summer Jamboree
July 2019
July 2019
- Facilitator: Peter Tingley
- Topic: Frogs and Toads
- Facilitator: Karl Liechty
- Topic: Flipping coins and turning the tables
- Facilitator: Nina White
- Topic: Pentagonal Tilings
- Facilitator: Amanda Harsy
- Topic: Prime Climb
Games! (but not game theory) - April 2019
Facilitator: Karl Liechty
We played a couple of two player games and thought of ways to win them. The first game was like this: draw 10 -’s on a paper. On each turn, you are allowed to either change one - to a +, or two consecutive -’s to two +’s. There are two players who alternate turns, and the winner is the one who changes the last - to a +. We explored the game and found “winning configurations” and “losing configurations. Eventually we settled on a strategy that could win the game no matter what the initial size of the game was.
Facilitator: Karl Liechty
We played a couple of two player games and thought of ways to win them. The first game was like this: draw 10 -’s on a paper. On each turn, you are allowed to either change one - to a +, or two consecutive -’s to two +’s. There are two players who alternate turns, and the winner is the one who changes the last - to a +. We explored the game and found “winning configurations” and “losing configurations. Eventually we settled on a strategy that could win the game no matter what the initial size of the game was.
More Games! - April 2019
Facilitator: Peter Tingley
We are going to think about how to win more games! Don’t worry, this is going to be pretty different from the poker/football stuff, so if you’ve had enough of that or weren’t there, don’t let that stop you from coming!
Facilitator: Peter Tingley
We are going to think about how to win more games! Don’t worry, this is going to be pretty different from the poker/football stuff, so if you’ve had enough of that or weren’t there, don’t let that stop you from coming!
Football and Airplanes - March 2019
Facilitator: Peter Tingley
We will do some more thinking like with poker, this time to understand football. Yes, it is possible! It won’t even take the whole time, so we are also going to think about people boarding planes.
Facilitator: Peter Tingley
We will do some more thinking like with poker, this time to understand football. Yes, it is possible! It won’t even take the whole time, so we are also going to think about people boarding planes.
When to hold 'em - February 2019
Facilitators: Peter Tingley (North) and Karl Liechty (South)
We will try to answer the age old question: how do I win my fortune playing poker. Or, at least, we will make some progress understanding if/when bluffing is a good idea.
Facilitators: Peter Tingley (North) and Karl Liechty (South)
We will try to answer the age old question: how do I win my fortune playing poker. Or, at least, we will make some progress understanding if/when bluffing is a good idea.
Number Machine - November 2018
Facilitator: Brian Seguin
Start with a 2 digit number. If the number is a palindrome, then stop. If not, reverse the digits, and add your original 2 digit number with its reversal. If the result is a palindrome, then stop. If not, then repeat. Keep going until you get a palindrome. For some numbers (like 11), this doesn’t take long at all, but for others (like 89), it takes quite a while. We discussed this, and some patterns that come up. We also thought about it in other bases.
Facilitator: Brian Seguin
Start with a 2 digit number. If the number is a palindrome, then stop. If not, reverse the digits, and add your original 2 digit number with its reversal. If the result is a palindrome, then stop. If not, then repeat. Keep going until you get a palindrome. For some numbers (like 11), this doesn’t take long at all, but for others (like 89), it takes quite a while. We discussed this, and some patterns that come up. We also thought about it in other bases.
The Palindrome Game - November 2018
Facilitator: John Boller (University of Chicago)
Start with a 2 digit number. If the number is a palindrome, then stop. If not, reverse the digits, and add your original 2 digit number with its reversal. If the result is a palindrome, then stop. If not, then repeat. Keep going until you get a palindrome. For some numbers (like 11), this doesn’t take long at all, but for others (like 89), it takes quite a while. We discussed this, and some patterns that come up. We also thought about it in other bases.
Facilitator: John Boller (University of Chicago)
Start with a 2 digit number. If the number is a palindrome, then stop. If not, reverse the digits, and add your original 2 digit number with its reversal. If the result is a palindrome, then stop. If not, then repeat. Keep going until you get a palindrome. For some numbers (like 11), this doesn’t take long at all, but for others (like 89), it takes quite a while. We discussed this, and some patterns that come up. We also thought about it in other bases.
Path Counting and Hats - October 2018
Facilitator: Karl Liechty
We started with the question: if you walk 5 blocks north and 10 blocks east, at each intersection going north or east randomly, how likely are you to pass your friends house, which is 3 blocks north and 10 blocks east of where you start? It turned out that this question was more interested than expected, since you can interpret it in two different ways: either you assume all paths from the start to the end are equally likely, in which case the answer turns out to be about 37%...and depending on how you do it, you might see part of Pascal’s triangle! Or you can take the question literally, and say you flip a coin at each intersection, but once you are 5 blocks north or 15 blocks east, you just walk straight to your destination. This interpretation gives a much smaller answer, around 3%. And you might see an amazingly altered Pascal’s triangle!
We then thought about another probability question: if a teacher passes 100 quizzes out to 100 students randomly, totally forgetting to check names, what is the probability that no one gets their own quiz back? The answer is again awesome, but I’m not going to give this one away...
Facilitator: Karl Liechty
We started with the question: if you walk 5 blocks north and 10 blocks east, at each intersection going north or east randomly, how likely are you to pass your friends house, which is 3 blocks north and 10 blocks east of where you start? It turned out that this question was more interested than expected, since you can interpret it in two different ways: either you assume all paths from the start to the end are equally likely, in which case the answer turns out to be about 37%...and depending on how you do it, you might see part of Pascal’s triangle! Or you can take the question literally, and say you flip a coin at each intersection, but once you are 5 blocks north or 15 blocks east, you just walk straight to your destination. This interpretation gives a much smaller answer, around 3%. And you might see an amazingly altered Pascal’s triangle!
We then thought about another probability question: if a teacher passes 100 quizzes out to 100 students randomly, totally forgetting to check names, what is the probability that no one gets their own quiz back? The answer is again awesome, but I’m not going to give this one away...
Fibonacci Surprise - October 2018
Facilitator: Angela Antonou (University of St. Francis and the South-west Chicago Math Teachers’ Circle)
We counted paths in a grid of hexagons and, surprise, out popped Fibonacci numbers. Well, it wouldn’t have been so surprising if we had read the title of the session, but fortunately most of us hadn’t. Then we did some other things like count paths missing a particular point and, surprise, we found some amazing identities of Fibonacci numbers. We also did some things like counting paths in a bigger grid.
Facilitator: Angela Antonou (University of St. Francis and the South-west Chicago Math Teachers’ Circle)
We counted paths in a grid of hexagons and, surprise, out popped Fibonacci numbers. Well, it wouldn’t have been so surprising if we had read the title of the session, but fortunately most of us hadn’t. Then we did some other things like count paths missing a particular point and, surprise, we found some amazing identities of Fibonacci numbers. We also did some things like counting paths in a bigger grid.
Variations on the Exploding Dots - April 2018
Facilitators: Karin Lange and Emily Peters
Facilitators: Karin Lange and Emily Peters
Liar's Bingo - February 2018
Facilitators: Amanda Harsy (Lewis University) and Rita Patel (College of DuPage)
Facilitators: Amanda Harsy (Lewis University) and Rita Patel (College of DuPage)
Nim and other related games - December 2017
Facilitators: Brian Seguin and Peter Tingley
Facilitators: Brian Seguin and Peter Tingley
Problem Solving - October 2017
Faciltator: Peter Tingley
It has been three years since we really explained what our math circle is about, and most of you weren't at that session. Also, our opinions have evolved. So we are going have another go at introducing problem solving! But don't worry, I'm going to use all new problems, so if you were at the previous session it should seem completely different.
Faciltator: Peter Tingley
It has been three years since we really explained what our math circle is about, and most of you weren't at that session. Also, our opinions have evolved. So we are going have another go at introducing problem solving! But don't worry, I'm going to use all new problems, so if you were at the previous session it should seem completely different.
Taxicab Geometry - October 2017
Facilitator: Yvonne Lai (U. of Nebraska, Lincoln)
The idea of distance seems pretty straightforward, but sometimes the usual definition isn't actually what you want. For instance, if you are driving a taxi around a grid of city streets, you don't actually care about the straight line distance between points, but instead you c are about how far you need to drive to get from one to the other. This is a new and different notion of distance, and you can use it to build up a new and different idea of what geometry should be. We will explore that, asking questions like, what is a line in this geometry? What is a circle? How about a parabola?
Here are links to some stuff we used in this session:
Facilitator: Yvonne Lai (U. of Nebraska, Lincoln)
The idea of distance seems pretty straightforward, but sometimes the usual definition isn't actually what you want. For instance, if you are driving a taxi around a grid of city streets, you don't actually care about the straight line distance between points, but instead you c are about how far you need to drive to get from one to the other. This is a new and different notion of distance, and you can use it to build up a new and different idea of what geometry should be. We will explore that, asking questions like, what is a line in this geometry? What is a circle? How about a parabola?
Here are links to some stuff we used in this session:
Perilous Probability - December 2017
Facilitators: Peter Tingley and Adriano Zambom
We discussed the Monte Hall problem. Then we worked on the question:
Facilitators: Peter Tingley and Adriano Zambom
We discussed the Monte Hall problem. Then we worked on the question:
- 100 people go to a party. They all throw their hats in a big pile.
- When they leave, they each grab a random hat.
- What are the chances that no one gets their own hat?
Controlling Growth. Or is growth controlling you? - April 2017
Facilitator: Peter Tingley
I enjoyed this so much, I celebrated by writing a full lesson plan! If you would like to see, it is here. Note: this version is intended for outreach with pretty bright and enthusiastic high school students.
Facilitator: Peter Tingley
I enjoyed this so much, I celebrated by writing a full lesson plan! If you would like to see, it is here. Note: this version is intended for outreach with pretty bright and enthusiastic high school students.
The game SET, and some geometry - May 2016
Facilitators: Anne Agostinelli and Casey McLeod
Facilitators: Anne Agostinelli and Casey McLeod
The watermelon problem - January 2016
Facilitator: Peter Tingley
How many pieces can you cut a watermelon into using 10 cuts? This was intended to mean the most pieces possible.
It is most interesting if you assume the watermelon pieces don't move between cuts (so it all stays together as you cut it 10 times, then you take it all apart).
The hint here is: find simpler versions of the problem...but maybe not the most obvious simpler version.
If you want to have more fun, try doing this with a donut instead of a watermelon.
Facilitator: Peter Tingley
How many pieces can you cut a watermelon into using 10 cuts? This was intended to mean the most pieces possible.
It is most interesting if you assume the watermelon pieces don't move between cuts (so it all stays together as you cut it 10 times, then you take it all apart).
The hint here is: find simpler versions of the problem...but maybe not the most obvious simpler version.
If you want to have more fun, try doing this with a donut instead of a watermelon.
Counting squares - December 2015
Facilitator: Peter Tingley
How many squares are there in a ten by ten piece of graph paper?
This only becomes interesting when you allow squares that are bigger than 1 by 1. It became really interesting when we allowed squares that were crooked, like the one shown below. That is, squares where the corners are at corners of the graph paper, but the edges do not need to follow the edges.
Facilitator: Peter Tingley
How many squares are there in a ten by ten piece of graph paper?
This only becomes interesting when you allow squares that are bigger than 1 by 1. It became really interesting when we allowed squares that were crooked, like the one shown below. That is, squares where the corners are at corners of the graph paper, but the edges do not need to follow the edges.
Bluffing at poker - September 2015
Facilitator: Peter Tingley
Is bluffing a good idea in poker? Somewhat more precisely, assuming the other player knows your strategy (but not your hand), is it a good idea to use a strategy that involves bluffing?
Perhaps surprisingly, the answer is an unequivocal yes! You should sometimes bluff, even against a player who knows your strategy and plays perfectly. To work this out, we considered the following simplified game:
Facilitator: Peter Tingley
Is bluffing a good idea in poker? Somewhat more precisely, assuming the other player knows your strategy (but not your hand), is it a good idea to use a strategy that involves bluffing?
Perhaps surprisingly, the answer is an unequivocal yes! You should sometimes bluff, even against a player who knows your strategy and plays perfectly. To work this out, we considered the following simplified game:
- Each player's "hand" is determined by the role of a die.
- There is no drawing, etc.
- Betting is as follows: each player antes $1. Then player 1 may either bet $1, or pass. If player 1 bet, player 2 can either call by putting in $1, or fold. If player 1 passed, there is no more betting.
- If player 2 folded, player 1 gets the pot. Otherwise, the players compare hands; the higher hand gets the pot or, if there is a tie, the players split the pot.
Frogs and toads - March 2015
Facilitator: Peter Tingley
This problem comes from the summer 2013 issue summer 2013 issue of MTCircular, the magazine of the Math Teachers' Circle Network.
Frogs and toads are arranged on a 5 by 5 board as shown below. They would like to switch places. That is, they want to move so that in the end there is a frog on each square where there was a toad, and vise-versa. Frogs can move one space right or down, and Toads can move one space left or up. They can also jump over exactly one of the other kind of animal (in a direction they are allowed to go) to land in an empty space. Can this be done? How long will it take?
Facilitator: Peter Tingley
This problem comes from the summer 2013 issue summer 2013 issue of MTCircular, the magazine of the Math Teachers' Circle Network.
Frogs and toads are arranged on a 5 by 5 board as shown below. They would like to switch places. That is, they want to move so that in the end there is a frog on each square where there was a toad, and vise-versa. Frogs can move one space right or down, and Toads can move one space left or up. They can also jump over exactly one of the other kind of animal (in a direction they are allowed to go) to land in an empty space. Can this be done? How long will it take?
