# “Finishing” My “Textbook”

## What’s the Deal With the Title of this Blog?

Hello, this is the first post on my new Blog: “The Cheese Maze.” The title comes from a recurring nightmare I had as a wee child. I would be minding my business without a thought in my mind. Then suddenly I would get the feeling that around some corner or behind some door there would be “The Cheese Maze.” I was terrified but for some reason I would always peek around the corner or walk through the door. The moment I did, I would find myself in an infinite maze made of cheese. I’d spend hours wandering around but I knew there was no way out.

When deciding to start a blog, I was worried that I would write a few posts and then get bored and move on to something else. So, against my better judgement in an attempt to avoid this, I told everyone that I was starting a blog and I was sure that I would work hard on it all summer. I made sure to sound really confidant when I explained this. So my hope is that because I made such a big deal about how committed I am to this, it would feel like an embarrassing public failure for me to quit. Thus, I have trapped myself with no way out similar to my recurring nightmare about the Cheese Maze.

Additionally, I wanted to find a title that was both interesting and didn’t make me look like a huge asshole. The task turned out to be harder than I thought.

You’ll notice I didn’t include “how accurately this describes my Blog” in my criteria. That’s mostly because I don’t really know what this blog is going to be about. You’ll also notice that there are no listed titles that score a zero on the asshole scale. This is because just having a blog makes you a little bit of an asshole.

## What is this Blog Going To Be About?

Like I said, I’m not entirely sure what this blog is going to be about. My plan is to write about things I find interesting/feel strongly about with maybe a few rare posts about my life. I want to model my blog off of WaitButWhy, which is a terrific blog btw. I’m also gonna try to copy their style of interspersing diagrams and comics within plain text. I have a drawy thingy that hooks up to my computer but unfortunatly I can’t find the chord. This means for the time being, my diagrams will be drawn using the pad on my laptop which is not ideal but I think will get the job done. (Also, I suck at drawing so don’t expect much improvement once I get the drawy thing working) I would love to write about math but I don’t think anyone who is as interested in math as I am will ever read this blog. Currently, I have plans to write a post about our education system and the Sleeping Beauty problem which is technically a math problem but one I can explain to a general audience and anyone can think about. I’m also 18 which means a lot of what I say is going to be bad and I’m likely going to look back at this when I am older and be really embarrassed.

## My “Textbook”

This first post is one of the, what I plan to be rare, posts about my life. I’ve spent the past eight or so months trying to write a textbook on Combinatorics and I kinda miserably failed.  If you would like to look at the completely unreadable unfinished nonsense it came out as, here it is: Combo Book

My main motivation for writing it was to show everybody how smart I was after failing to do well at math competitions.

It wasn’t a super well thought out plan because 1) nobody I know could/would want to understand anything in the textbook so they wouldn’t really know how difficult the material was and 2) the reason I thought it would make me look smart is that all the proofs and solutions to problems in the textbook were going to be completely my own.  But I didn’t want to ever write that in the textbook or tell anyone because it would come off as me looking like a douche trying to show everyone how smart I was. (The final product has a few proofs that are not my own but that was my idea going in)

To maximize the amount of content that would show how smart I was, I decided to make a third of textbook standard undergraduate combinatorics, a third of it olympiad problem solving, and a third of it graduate level Algebraic Combinatorics that I had read a year ago and still don’t have a full grasp on. This was also a pretty stupid idea, not only for obvious reasons, but also in terms of any goals to ever get it published (A delusion I no longer hold) because the intersection of people that would be interested in even any two of those things is basically no one.

I had the inspiration from Evan Chen. He made it sound like you just have some stuff in your head, you carve some time out each day for writing, and wablam, your left with a well-oiled polished textbook. But I had forgot one thing. Evan Chen is a god and I am a mere mortal. It turns out that cleanly explaining a bunch of super complex things and organizing it in a readable manner is really hard. (This might not be super encouraging to someone reading this blog where I began by stating that this is where I want to explain a bunch of complex ideas but I think hope this will be different). It wasn’t that I didn’t think it would be a lot of work, I was ready to put in the hours, it’s that I thought it would be more straightforward than it was.

It sounded like the perfect project for capstone, a program at my school that allows kids to do projects and then get course credit for them. In what became a recurring theme throughout this project, I made a big mistake. The thing is, the people that run capsone assume everyone in high school is a lazy piece of shit that needs to be told exactly how to approach every step of their project at all times or else no one will ever do anything. In all fairness, this might not be so far from the truth but nonetheless, super annoying.

First, it meant that everyone had to attend a meeting every couple weeks. For some reason, I always looked forward to the meetings (probably because I associated them with my project which I was actually excited about) until I arrived and realized it meant sitting in room and having listen to the people who run capstone talk about nothing for forty-five minutes.

Second, it meant that everyone had to submit a literature review where you have to write about everything known about your project (kind of like a mini textbook which they of course made me write in addition to my project), a prospectus where you outline all the goals of your project, notes on how you will present you project, peer reviews of everyone else’s project, ect. (I understand that these things might be helpful for someone who is unsure about how to approach their project but for me and others I knew in capstone, who had a clear picture of what they wanted their project to be, this was basically all busywork.)

The project started out pretty good. I first wrote the section on counting which I knew most of really well and is objectively the simplest part of my textbook. I had a lot of motivation so I was able to take the time to carefully read what I wrote to make sure it made sense. The parts that I didn’t know were easy figure out proofs to, and I learned that the Catalan numbers which I had unknowingly discovered a long time ago and spent time thinking about actually had a name. After finishing the outline of the chapter, I felt really good. Maybe writing a textbook would prove to be not so hard after all.

It’s amazing how quickly you can go from feeling really good and confident about something to really stupid. Once I started my second chapter on graph theory, it became very clear very fast that I was 1) really bad a graph theory problems and 2) Knew a lot less than I thought I knew about graph theory.  I did end up reading other texts to fill in my enormous gaps in knowledge and was able to prove most theorems (Although to this day, I still haven’t figured out how to prove Menger’s Theorem. If you read the textbook you would see that I never resolved that and just chose not to write anything under the statement of the theorem), but I lacked the big picture perspective to see how the theorems fit together. As a result, the Graph Theory section is just kind of a list of theorems grouped loosley by surface features.

Finally, being severely behind schedule and being done with what I hoped was the hardest part of my textbook, I was ready to write what I was most excited about; Olympiad Problem Solving. First, olympiad problems were objectively the hardest part of my textbook so this was the real place I could let the world know how smart and was and second, I actually did think a lot of the problems were really cool. I wanted to copy the style of Pranav Sriram and have the chapter mostly be a list of problems with exposition on how to approach and solve them. The process of writing it ended up being lengthy because I didn’t actually have a large bank of problems that I had previously solved. This meant that I had to scour the IMO shortlist and other archives of past competitions and actually solve a bunch of new problems. It was okay though because I enjoyed solving the problems and might have done them even if I was not writing a textbook. The problem was, by the end of solving each problem, I wanted to immediately write it in my textbook while it was fresh in my mind. But I was exhausted from solving the problem so I didn’t take much care into making the explanation as clear as possible, resolving to go back and clean it up later. But by the end of writing the rough draft of the chapter, I was kind of burned out and didn’t really feel like going back and fixing things. This meant that in the final draft, there are some heinously bad explanations and a lot of rambling.

My fourth chapter was on generating functions. (Fourth that I wrote. It is the fifth in the actual textbook) I knew about as much about generating functions as I did about graph theory but at least I went in knowing I didn’t know much. I started out explaining the basics which I did know. Then, I didn’t really know what I should put next so I just googled “generating functions in combinatorics” and the first thing that came up (that wasn’t what I put in the introduction) was “Linear Homogeneous Recurrence Relations” which I immediately made a section about. It basically turned out to be about general fibonacci sequences in which there is not too much theory. It also gave me a chance to include a couple olympiad problems which didn’t really fit in the problem solving strategies section but were tangentially related to LHRR’s. In trying to prove basically the only theorem of LHRR’s, I realized I needed the following Lemma in Linear Algebra:

For distinct complex numbers $c_1, c_2, \cdots , c_n$, the vectors, $< 1, c_1, {c_1}^2 , {c_1}^{n-1} >, < 1, c_2, \cdots , {c_2}^{n-1} >, \cdots , < 1, c_n, \cdots , {c_n}^ {n-1} >$ are linearly independent.

I hadn’t done any linear algebra in a hot minute and felt that it should be easy to prove but I couldn’t figure out how. Up until this point, my mentor had been completely useless, but for some reason I thought “Maybe it’s a good idea to ask him about this.” To my surprise, he actually was familiar with that statement and told me it was Vandermonde’s Determinant, a well-known theorem in Linear Algebra. He was even able to sketch out a proof.  The next morning in the shower, I thought of a completely combinatorial proof of Vandermonde’s Determinant and I was much more excited than I thought I should be. So excited, that I tried looking for my proof on the internet in the delusional hope that it was new. It turned out it wasn’t but it also wasn’t discovered until 1979, which was pretty new considering the first proof of Vandermonde’s determinant was found in the mid 1700’s.  This gave me a big confidence boost and renewed my motivation for the textbook. For like a day. I finished the theorem about LHRR’s and started doing the unoriginal example of the fibonacci numbers but stopped two lines in (The example still isn’t finished, it’s on the last page of the textbook) because there is nothing more soul crushing than writing up an example that you know literally everyone who has ever written about the subject has included. I also didn’t write anything else about generating functions even though I am told they are a very rich subject.

Finally, the last chapter of my textbook was was the Algebraic Combinatorics section. The section about the graduate material I had read one source of a year ago and didn’t have a full grasp on. Since I didn’t have the perspective to research other sources on Algebraic Combinatorics and splice different material together, I knew what I wrote would be a strictly worse subset of what I had read. (These notes are really good btw if you’re interested in learning Algebraic Combinatorics). Before writing that section I thought the phrase “I don’t fully understand Algebraic Combinatorics and I know whatever I write will be a strictly worse subset of the one source I read” and then immediately after that I thought “Yes, this is a good idea. I will go ahead and spend the remaining time before this project is due writing this section instead of expanding/fixing my existing sections that are shit.” In a surprising twist, the section turned out to be… shit. I covered 1.25 topics and explained them terribly.

With two months before the project was due, I kinda just gave up. I didn’t think about the project. Then suddenly it was 8:00 and the project was due the next day. I booted up my latex code and put everything in one document and hit execute. I got an error that said the code could not find the file that had all the textbook’s graphics. Making my own graphics had ended up being really difficult so I just went online and looked for pictures that were pretty close to what I wanted to describe. In latex, there is a way for the code to find pictures in folders on your computer. But for some reason, that day, it could not find the folder the pictures were in. I did the only thing a reasonable human being would do in that situation and went through my entire textbook and, one by one, deleted every picture that I had gotten online. After that, I got no errors so everything was good and I printed out three copies of my textbook to hand in the next day.

And that’s the story of how I wrote my Textbook.