Mathematically-Themed Icebreakers 2: Hiding a Favorite Theorem in a Classroom Icebreaker

This week, we discuss an activity that arose by turning a favorite theorem (in this case, a result from Ramsey Theory) into a first-day activity that got students flexing their mathematical muscles right from the start of class.   This Icebreaker was created by one of our post-docs: Kris Hollingsworth.

This activity works well both in the classroom and over Zoom breakout rooms; in fact, it is the first activity we designed from scratch for the remote teaching environment. This activity has been used in an Introduction to Proofs class with 160 students with great success, but would be appropriate in almost any mathematics course.

We'll first describe the activity and how we would expect it to work (and maybe you can play along to try and solve the math as well!).  Then, we'll prove the underlying theorem in a way that can be done in class and will finish with some possible suggestions for follow-up discussion.

1) Start by breaking the class into small groups of 4-5 students each. Give them a short list of the "standard" ice-breaker questions, for example:

• What is your major?
• What is your favorite animal?
• What's an incredibly common thing you've never done?
• What is your favorite movie genre?
• Which is your favorite carnival ride?
• Where are you from?
• Etc, etc, etc ...

So far, this pretty much looks like an ice-breaker that's no different from most others whatsoever.   However, here's where you make some changes. Introduce the following rules:

• Divide students into groups of 4.
• Students should select 3-4 of the Icebreaker questions from the list, and try to use the answers from those questions to and create certain arrangements within their group:
• Two students are "friends" if they have at least two answers in common.
• Two students are "strangers" if they do not.
• Students want to find a set of answers from the Icebreaker Questions so that no three people within the group of 4 are all (mutually) friends with each other, and no three people are all (mutually) strangers to each other.
• If the current list of Icebreaker questions is insufficient, design your own to accomplish the goal.
• As students manage to complete this task, redistribute them to other groups so the groups of 4 become 5, and groups of 5 become 6.

If you're interested, now would be a good time to stop reading and try and work out possible arrangements for groups of 4, 5, or 6 students.

For groups of 4, this is very easy. For groups of 5, it turns out to be a bit challenging as there is only one possible arrangement that works, which involves creating friends which form a "cycle" (if all "friends" joined hands, they'd form a circle of 5 people). Once students get in groups of 6, they may try all they like, it actually turns out to be impossible! After 10-15 minutes, some students may even conjecture this fact, and you can ask them how they might verify whether or not it is possible.

It turns out, this is a special case of the Ramsey Theory, which demonstrates that in a graph on 6 or more vertices (points), there is always guaranteed to be a triangle in either the graph or the graph complement (switching all edges to non-edges and vice-versa). In the common notation of Ramsey Theory, we would say R(3,3)=6. It turns out to have a fairly straightforward proof that you can actually have students act out in a classroom (or talk through over Zoom).   Here is a visual aid to help:

Above from Professor Rich Wolski at UCSB: 
https://sites.cs.ucsb.edu/~rich/class/cs293-cloud/notes/Ramsey/index.html

Proof:

Get six volunteers. Choose one at random to be the "fixed" person. Then what can you say? They must be friends or non-friends with at least three members of their group. To be cheerful, let's suppose we have three friends.

So let's consider what happens among them. Either two of them are friends with each other, and thus form a group of three mutual friends with the original fixed point, or all three of them are strangers, in which case you have found three members of the group who are all mutually strangers. In either case, you are guaranteed three mutual friends or three mutual strangers.   

(If you'd like to test this before your class or Zoom call, you can try this with any of the example people listed below!)

This is such a simple proof, yet the implications are profound. The existence of Ramsey Numbers imply that in any sufficiently large system, there will be guaranteed patterns emerging --- in other words, complete chaos is impossible! To guarantee 4 mutual friends or 4 mutual strangers, it is known that the magic number is 18. However, once you ask this question about 5 mutual friends and 5 mutual strangers, the answer is unknown! We know it is between 43 and 48 (inclusive), but haven't been able to nail that number down any better yet.

Wrap-up:

So let's take a quick inventory of what we've accomplished with this first-day activity. You've gotten students working with different groups of their peers and getting to know each other while productively working on a real mathematical problem. You've done it in a way that hides the mathematics, so when you connect the problem to math at the end, they will be forced to reframe their experience in these terms. You've also managed to introduce students to mathematical problem solving and to the fact that open problems exist and may be simple to understand but difficult to solve. All of these points make for good follow-up discussion and open the door to numerous possible avenues of bridging into course content and material.

We also recommend looking for creative ways to act out your own favorite theorems to make new activities like the one above!  That's how this one came into existence. Let us know if you have some favorite results that you think might make for a good icebreaker in the comments, and check back in a few weeks for the next activity in our series!