Creative Counting

In college I specialized in a field of math called Combinatorics. Combinatorics is the study of counting. Counting seems like a really simple concept. It's the first thing we teach about math. Let's with a relatively simple counting problem. If I flip a coin 4 times how many possible outcomes are there. Well I can get all heads, all tails, 3 heads and 1 tails, 3 tails and 1 heads, and 2 heads and 2 tails. That's 5 outcomes but what if we cared about the order of the flips and not just the total and result? That gets a little trickier. For each of the 5 outcomes we can count how many each one can happen. Here's a visual representation so we can count this up easier. This process of breaking a task down into smaller more manageable tasks is called partitioning. I'm about to get technical so it's ok if you get a little lost we'll circle back and make things more clear. In higher level math we often deal with proofs. A proof is basically a logical explanation of why some idea is true. All of those formulas you've seen in school like y = mx + b and a² + b² = c² have some rigorous proofs that explain why they work. I use the word rigorous because a proof's argument needs to be iron clad. You have to come up with an argument that explains how there is no loop hole or weird case where it doesn't work. There is a particular combinatorial proof technique that I love.

The set up: you have some equation whit complicated stuff on one side then an equals sign and then some other complicated stuff on the other side. Our goal is to construct an argument that proves those two complicated things are in fact equal. If you can explain how each side of the equation is counting the same scenario then you proved the are equal. Here's a really quick example 1+4+6+4+1 = 2⁴ is our equation. Yes we can put these in a calculator to see that they both equal 16 but let's try to use the combinatorial proof technique.

Let's go back to the previous example with flipping 4 coins. We can partition the outcomes by how many heads they have and add them up like we did earlier. That's the left side of the equation covered. Now our goal is to count how many ways we can flip 4 coins but using a different method. This method is a bit more involved so let's look at a visual aide.

This is called a decision tree. When a new coin is flipped there are two possible outcomes. In probability consecutive outcomes can be multiplied together to count the total. This decision tree helps us to see that every time we flip a coin with 2 options, we double our total number of outcomes. So since another way of saying 2×2×2×2 is 2⁴ we now have our right side of the equation. Now for the left side of the equation. Lets dive in a bit more on the decision tree and look at the end results of each branch.

If we count how many 4 Heads, 3 Heads 1 Tails, 2 Heads 2 Tails, 1 Heads 3 Tails, and 4 Tails outcomes we find there are 1, 4, 6, 4, and 1 respectively. Does that look familiar? If we add them up we have the total number of outcomes and that is our left side of the equations. Since they both count the same thing we know they will be equal.

Okay that's cool and all but the title says creative counting and so far this hasn't been very creative. Now for the fun part. The example with coins is nice to introduce this topic but the fun part comes with more complicated problems. I found I had a knack for seeing numbers and matching them to real life situations so I though I would challenge myself to come up with some fun things to count using real problems I was assigned in class. I asked some friends for themes to use and my favorite response was "Is mayonaise a theme?". The result is a proof that I wrote and presented for a class back in college.

I have a few others themed after recruiting a league of super villains, a mishap in scheduling among the leaders of Atlantis, and a local non profit trying to make a viral video. Back when I was taking this class I was daydreaming in class and I had this vision.

Combinatorics Comics, wacky stories stories that propose 2 way of counting something and then end the comic with the identity the story proves. I've haven't taken this idea much further but it is a project idea that I always have in the back of my head.