# Why all the Subjects in High School are Taught Horribly Part 1: English

It seems that the breaks between posts keep getting longer and longer. I have been legitimately busy though. Actually that’s not even close to true but I’ve done so little this summer that doing literally anything at all makes me feel like I’m super busy. In reality, I’ve done like two things in the past month. I went to college orientation and GP Sacramento. Predictably orientation sucked. It’s one of those rare situations where you are simultaneously bored out of your mind but also not completely comfortable. It’s a nasty combo because usually when I’m bored, I just think about something interesting but thats much harder when I’m not fully relaxed. It’s not that I’m actively nervous, just a low level not relaxed which really has no other effect than making it hard to escape to my mind. And usually when I’m not relaxed its because something exciting is happening. At least I ended up in the by far least enthusiastic group so everyone around me was also miserable.

In the GP, I ended up coming in 16th which was a little disappointing given that I started out 8-0 and was pretty confident going into draft but 16th was still good for $1,000 and 3 pro points. Plus, I was happy to see Richard Liu take town the whole thing. Anyway between being “busy”, starting a couple posts that didn’t go anywhere, and lacking motivation in general, this post is three weeks too late. Its really poorly written but I panicked when I realized how long it had been and then promised myself I would complete it by wednesday. But then I left almost all the work until wednesday so I’m sorry for how terrible this is but actually posting something hopefully will give me motivation to work on this blog more. Because if I don’t finish this today, I will be favored to never finish it. This is my second post that is copying a topic from Evan Chen. I would link his post but apparently there is something called a pingback, which will make a link to this post show up at the bottom of the post I link to. It seems a little stalkery to have two posts on the same topic as his, linking at the bottom of his posts, especially because his posts don’t contain any other pingbacks. But this post has lots of different ideas so it’s not a complete copy. Anyway, one of the main points, and a point central to this post, is that school teaches very dishonest writing. The task in writing is always to pick a side and to try and be as convincing as possible. I think this is very true and very bad. I have never once been told by a teacher to make sure I actually believe my arguments or to write so that the reader has the clearest picture of the actual truth but I have been constantly scolded for not being completely focused on convincing the reader of my claim. I remember when I was I was younger that my take away from school was that is was a good thing to be dishonest in papers. It showed that you had the ability to manipulate facts to your own benefit. But this is a terrible thing. We are literally teaching all of our kids to disregard truth and what you actually think in favor of being ruthlessly dishonest and doing whatever it takes to convince others of whatever. But what causes this? In a sense it can be viewed a simple case of the prisoner’s dilemma. It probably favors a student to learn how to be convincing but it is bad for the rest of the world. So society would be much better if all students learned to argue honestly even though it would be better for any individual student if they learned to just make the most convincing argument. But I think a lot of the reason is actually an accident. It is a cause of what kids learn to write about. ## Persuasive Writing A big part of the English Curriculum is persuasive writing. A student is given a prompt and they are asked to pick a side and argue it. A lot of the time the prompt is of the form “Should X?” Now we can divide “Should X?” prompts into two categories. 1) Where the effects of X are not well known and 2) where the effects of X are well known. A good amount of prompts given in secondary school fall under category 1. The thing is, it is incredibly hard to write anything interesting about a type 1 claim. How can anyone possibly add anything of value? To honestly answer such a question you need to find some moral framework, convert all effects of X into equity based on the moral framework and sum them all up. (It should be noted that while most moral philosophies are not talked about in terms of equity it should be theoretically possible to convert anything to equity by simply comparing states of the universe and ordering them by “better”) Now this is an incredibly difficult, basically impossible task especially for a high schooler. For one, it is hard to separate out different parts of X, you are bound to leave something out. And even if you don’t it is hard to know how to break everything down. But let’s just say you accomplish this task. If you do break down X into all its main effects, your left with the task of assigning them all an equity value. Even if it is an easy moral philosophy to assign equity to, like Utilitarianism, people are terrible at assigning numbers to things. And slight variations on how you assign numbers which depend on lots of different factors have a very good chance of impacting your conclusion. So it is not realistic for you to assign equity to each effect. The only plausible strategy seems to be surveying tons of people and asking them what equity value they would place on each of the effects and then things might average out to an appropriate number. No high schooler has ever done this. But what strategy is left. If you are answering a type 1 prompt what else can you possibly add? The answer it not much. You have no hope of finding some new angle because all the effects are known. So what you usually end up doing is just listing all the effects and saying something like “As we can see the good effects outweigh the bad effects” while your classmate is writing “As we can see the bad effects outweigh the good effects.” All anyone can possibly do is list out the same facts as everyone else and claim that it favors their side. Now, if you want to be a “good” student you need to go a little further and try to explain why to good effects are better than one would think and the bad effects are not as bad as one would think. But how can you do this? You really have two options. 1) Use a statistic favoring yourself even though there are other statistics out there which are contradictory and you have no reason to believe the statistic you used over others which makes the argument dishonest or 2) Use a bullshit emotional appeal/chosen diction that makes it sound like the good effects are better than they actually are or the bad effects are worse than they actually are which is also dishonest. The mere task of writing about type 1 claims will make students dishonest even if there is no intention. Kids want to get that A and so they need to find ways to make their writing stand out. Kids want to make the most convincing argument they can make and since there is no way they can do that honestly, their only option is to craft their writing in a dishonest way and skew facts to favor their side. And if they ask for help, the teacher will likely point them in a dishonest direction without even knowing it. Type 2 claims are incredibly hard to write about and boring to read. This is why getting debates like “should abortion be legal?” never end anywhere. Type 2 claims are in theory easier to make an actual contribution to and for serious writers they are, but if the claim is well known, then it is near impossible for an average high schooler to make a useful contribution. If it is a common debate, like say economic policy, then even though the effects are not well known, a typical high schooler is never going to be able to actually figure out for themselves what the correct answer is and argue about it effectively. So what happens, when a high schooler encounters a common type 2 claim? They can’t offer anything useful so they do what their trained to do and treat it exactly like they would a type 1 claim. I would bet the vast majority of high schoolers have ever thought that type 1 claims and type 2 claims are different at all and just end up using the same formula of finding a statistic that benefits them and trying to make their writing style favor their side of the argument without ever realizing they are being completely dishonest and never saying one useful thing. This is made worse by the fact that if the student actually tries to get into the weeds of the debate and find truth, they will probably get a worse grade than if they just tried being dishonest because the paper will look less clean. I remember in 11th grade, we had a persuasive essay assignment and there was a list of topics that were off limits. That list contained “should abortion be legal?” At the time I don’t think anyone really understood why they were off limits but I now realize that these were all either type 1 prompts or common type 2 prompts that no high schooler could make any contribution to. So what should be done about this? I believe that schools should teach the difference between type 1 and type 2 claims but more importantly give kids more niche prompts. Research projects that have never been explored or there is not easy access to information about. Make them actually either find reasonable data or reason from first principles. I think this would have a huge effect not only on how much kids learn, but also students philosophy towards arguments in general. If kids grew up writing honestly instead of being taught to make the most convincing argument with no regard to its validity, it might have surprisingly good benefits on society. ## Literary Analysis Literary analysis is a strange beast. Its insertion into the curriculum was probably quite natural. Back in the day, they taught kids to read and write. When they realized that could be done quickly and kids started going to school longer, they needed to extend the English curriculum. They realized that you had to be able to read to read books so reading books seemed like a good idea. Moreover many books contain a lot of meaning that kids could legitimately benefit from. But this being the American education system, schools needed a way to test that kids actually understood the meaning behind books. Literary analysis seemed like a perfect fit. There’s was only one problem. Traditionally literary analysis is a daunting task. One needs to analyze the whole body of a work holistically to make conclusions. But since American kids are lazy/stupid the schools could only really expect a few pages of work. So the task simply became to pick a claim, find a couple of passages that support the claim and write it down. It doesn’t matter if the rest of the book supports that claim or not and there is no time to adress alternative theories or counter arguments, after all American kids are very lazy/stupid. Even more troubling, the original goal of literary analysis was to test to see whether kids understood the most important ideas of the book. But in now times, kids are one google search away from reading all the prominent analyses of the book. So a kid can’t just make claims about the most prominent claims in the book because then the teacher would assume they just looked online. Additionally, it is culturally cool to think your better than everyone else and to stand out. So this leads to kids needing to make new non-trivial claims. But to figure some hidden small piece of meaning from a book is a difficult task for american kids who are lazy/stupid. And since a piece of literary analysis is just a few quotes from a book, it is much easier to just pick a random claim with no regard to whether it is true or not, and being trained with all the dishonest persuasive writing, it is not difficult to search through the book until you find a couple of passages tangentially related to your made up bs claim and spend the paper finessing your way through the outrageous assertion that anything in the book is actually related to your bs claim. There’s also no need to bother reading the book. The kick is that because all papers are just a few pages and contain just a few passages, papers arguing real thoughtful claims and papers arguing completely made up bs claims don’t look very different. This is just all on the surface stuff. We haven’t even asked the most basic question: what does it actually mean to do literary analysis? What is the task of literary analysis? Well, it usually involves trying to “prove” a claim about the meaning of a story. And to do this you need to “prove” something about the nature of the story. But there is one problem. The story is fiction. It doesn’t exist. So how can you prove something about something that doesn’t exist? A claim will be something like Rebecca is an angry person. But Rebecca doesn’t exist. Are you trying to prove that the author of the story imagined Rebecca as an angry person? No, because you only use evidence in the book. Evidence that the author probably did not explicitly think “I’m going to write this so that people will think that Rebecca is an angry person” and they may have chosen to include that passage arbitrarily over another. Is the task simply to come up with a universe consistent with everything that happens in the book, in which Rebecca is an angry person? No, because you can make anything consistent as long as there are no explicit contradictions. You could come up with a consistent universe where Rebecca is a robot time traveling dinosaur with IBS living in the matrix. Is the claim that Rebecca is an angry person in the universe consistent with everything in the book that is closest to ours under some well defined metric space> This is probably closer to what is actually going on but of course there is no way to define such a metric space and there may be no one universe “closest” to ours. How about this? Let’s suppose that you are told that everything in the book did happen and that you are to make a wager as to whether Rebecca is an angry person (lets ignore the obvious fact that “is an angry person” is not even well defined). Then the task becomes stating which wager you would make and justify it. This is probably the closest thing as to what the task of literary analysis is. But if this is the case, why do we always act with certainty? What you are really saying is that there is a greater than 50% chance that rebecca is an angry person. But if this is the case, why is every paper on literary analysis always claiming that without a doubt, 100%, Rebecca is an angry person.?There has never been a paper whose claim is that there is a 70% chance that Rebecca is an angry person, but shouldn’t this be common given that you are arguing on probabilities off of incomplete information when there in fact do exist consistent universes where Rebecca is not an angry person? So if this is what a person doing literary analysis is claiming, then what does relevant evidence look like? The information is just the collection of every passage displaying information about Rebecca. So let’s say you collect every bit of information and categorize it (this is also not well defined but it at least can be reasonably completed in theory)? Now, you must go through each bit of information, look at the total population and see what portion is an angry person. Then you can take the product of the portions that are angry, and divide that by itself plus the product of the portions that are not angry. A daunting task. But wait a minute, this is still not sufficient. We are ignoring correlation. For example, suppose we have two pieces of evidence. 1) Rebecca likes apples and 2) rebecca believes that she is an apple tree. Let’s suppose there are a 1000 recorded people who like apples and there are also 1000 recorded people who think they are apple trees. 500 of the people that like apples are angry, 500 people that think they are apple trees, but suppose that these 500 people are the same. To recap, in our data set, we have 500 people that like apples, do not think they are apple trees, and are not angry, 500 people that that don’t like apples, think they are apple trees, and are not angry, and 500 people who like apples, think they are apple trees, and are angry. Under the method mentioned above, we would obtain that there was a 50% chance that Rebecca is an angry person but clearly if we combine the data we see that Rebecca is almost certainly an angry person. So we have to look at the data together. Okay, easy, we just analyze the sample of everyone who satisfies all of our evidence and see what portion is angry. The only problem is that no one is going to satisfy all the data about rebecca, because then they would be rebecca and the book would not be fiction. So how do we analyze the data? (This is an actual question. I am no statistician and am actually very curious about how an analysis would go). Simple techniques don’t work. Even the dumbest technique of partition in largest sets with sufficiently large data doesn’t work. For example, consider the three characteristics, is a millionaire, is an evil genius, and can play the fucking piano. There is 1,000000 people in each category, and the intersection of any two of these contains 1000 people, but the intersection of all three is empty. So how do you group these together to analyze data. My best guess is probably a weighted sum over all subsets based on the sample size, but I can’t figure out any way to make this accurate or provide justifications. Ok, so the task of actually making conjectures about the nature of a charter and providing rigorous evidence to support the conjecture is really difficult. So what do people actually do when they do literary analysis? They pick a few selected passages, point at them and say “look at this, my claim is right.” There could be a hundred other passages that completely go against what they said but as long as they have a couple pieces of textual evidence and declare that it supports their claim, everything is good. So what is my point with all of this? I don’t actually have a point. I just started writing and this is where I ended up. But since I have been writing about dishonesty, I am going to be dishonest and try to find a point and act like it was my intention the whole time. Here is my point. I find it annoying that all you’re told is in school is that you have to do literary analysis and the only way to do that is find quotes which prove your point. They never talk about what literary analysis actually means or why that is the way to prove your point. It is probably the best way to prove a claim but it seems like it would be better to let kids think about what literary analysis actually means and how one would go about supporting a claim about fiction. They never even talk about why one would want to do literary analysis. They might say it strengthens your detective skills. But if this is the point, why don’t they just literally give kids detective work? They could find some account of what really happened and ask kids to make claims and then they could actually check to see if they were correct. Schools probably would say that it is not about whether the claim is correct, but whether you argued about it effectively. (I have been told this many times) But if that’s not dishonesty, I don’t know what is. A better answer might be that you are acting off of incomplete information and a claim might be good but is wrong because of other unknown information. Actually checking could make kids to results oriented. This is probably a good point, but if it is the case, I again ask, why do kids always have to write as though they are 100% certain of their claim. Okay, here is my actual point. It is really dumb to make kids argue as though they are 100% sure of their claim when they are acting with incomplete information. I think this mentality of, “Act like you know you are right when you have no way of actually knowing” is super duper harmful for society. We see this happen all the time with adults and opinions and politics and economics and literary analysis in high school is likely partially responsible for this. Theres lots more to say on this topic but because I really want to get this posted, I’ll finish the post now. # The Blockchain ## Introduction: “Up next. The conversation every parent dreads. How to talk to your kids about blockchain” -Silicon Valley Again I am starting this post by apologizing for how long it took. I really don’t want to separate posts by more than a week. But procrastination. What happened was that I procrastinated the whole week after my last post because I didn’t know what to write about and resolved to do it all on Wednesday. But on Wednesday, I watched the first episode of the second season of “13 Reasons Why” which lead to me binge watching the first half of the second season of “13 Reasons Why” which lead me to binge watching the second half of the second season of “13 Reasons Why” the next day. It wasn’t as good as the first season but the first season is amazing and I would still recommend the second season. The first couple episodes feel kind of forced but it gets a lot better after that. Anyway, I got some of the post done on Friday but didn’t spend enough time on it to finish. Then I was busy on Saturday and Sunday. On Monday, I planned on finishing but instead I rewatched the whole first season of “13 Reasons Why”. On Tuesday, I finished most of it but was like “Is there really any difference between me publishing this today or tomorrow?” and my answer was no, so thats why this took so long. The aim of this post is to explain cryptocurrency and blockchain in medium depth but at a non-technical level. All you really need to know to understand this is the rough definition of a function and a function inverse (If you don’t know what an inverse function is and you read the intro to the wiki page you probably understand it well enough for this post). I only just learned about blockchain so I hope I actually understand it and don’t embarrass myself by writing complete nonsense. I think I understand it though. Anyway, my motivation to try to explain this is that there are not many resources at a non-technical level. There was almost nothing I could find that actually explained what is going on without delving into super advanced computer science details. I’m not sure why this is. The actual idea behind blockchain is not super complicated but most sources I could find were either three sentences long or are full of words I am convinced are made up. If this post contains inaccuracies, I blame that. ## Cryptography: Cryptography is actually probably the hardest part of this post to understand. If you don’t get all of it that’s okay, you should still be able to understand the rest of this post. As you probably know cryptography is a way of encoding and decoding messages. For example, you might want someone to send you a letter but you are worried that the letter may be intercepted and you don’t want whoever intercepts the letter to be able to understand it. This is very relevant in today’s world where you need to send entities things like your social security number, bank account numbers, and credit card numbers. We’ll use the example that is used whenever cryptography is taught. Suppose that Alice wants to send a message to Bob but she is worried that Carl will intercept the message. We need a system where Carl will not be able to understand the message but Alice will. So Alice must possess some information that Carl does not. An easy solution would be for Alice and Bob to agree on some secret language beforehand but we want a system that will work if Alice and Bob have never met or are not able to privately come up with a code together. So to summarize, we want a code such that Alice has more information than Carl but Carl and Bob have the same information. We need a system where anyone can send a message in code but only Alice can decode the message. So Alice must be able to announce to world publicly how to encode a message without them figuring out how to decode a message. Once again mathematics comes to the rescue. What Alice must do is come up with a function $f$ along with its inverse $f^{-1}$ and announce the function $f$ publicly. Then if anyone wants to send Alice a message they simply apply $f$ to the message. When Alice receives the message, she applies $f^{-1}$ which will recover the original message so that Alice can read it. The only catch is that it must be hard to figure out what $f^{-1}$ is from $f$ or else Carl could intercept the message, figure out $f^{-1}$ and decode the message. At first this does not seem hard, there are lots of functions that we don’t know the inverse of. But remember that Alice must be able to figure out $f^{-1}$. Maybe Alice is a math genius and she has found the inverse to some special mathematical function that has confounded the experts for millenia. But let’s suppose that Alice is not special, and is only medium at math. How can Alice construct a function $f$ and find its inverse without others being able to also find $f^{-1}$? Alice does have one advantage and that is that she came up with the function $f$. Perhaps first Alice could first come up with an $f^{-1}$ where it is easy to find $f$ from $f^{-1}$ but it is difficult to find $f^{-1}$ from $f$. This is a reasonable thing to try but it is still a lot of work for Alice and such functions don’t really exist. No, we must find an easier way. Okay, what if instead of just starting with a function $f$, Alice starts with blocks (functions) and smashes them together to obtain the function $f$. And what if it was easy to find $f^{-1}$ from the blocks but was is difficult to recover the blocks from just knowing $f$. Then Alice would know both $f$ and $f^{-1}$, she could announce $f$ publicly and no one else would be able to find $f^{-1}$. Anyone could send Alice a message but only Alice would be able to understand it. Sounds pretty dumb right? Wrong. This is the way cryptography is done and it’s fucking genius. Usually, the function $f$ is refered to as Alice’s “public key” because she releases publicly. The function $f^{-1}$ is refered to as Alice’s “private key” because she keeps it to herself. This is important to remember because in future sections I will refer to public and private keys. This system comes with a nice little corollary that is actually what is important for cryptocurrency and the blockchain. Suppose that Alice learns that Bob’s Mom has cancer and has one day left to live. She wants to alert Bob so that he can have one last chance to see his mom and say goodbye. There’s only one problem and that is that Carl is a known prankster. And one of his classic pranks is sending people letters addressed from their friends telling them that the recipient’s Mom is dying of cancer and they only have one day left to live. Bob knows about this and if he received the letter from Alice, he would assume it was a prank from Carl and would miss the chance to say goodbye to his mom. Alice needs a way to prove the letter is from her. How can she supply this proof with just a letter? Well, Bob knows that Alice is the only person in the world that knows $f^{-1}$. So Alice could write the letter and encode it with $f^{-1}$ and tell Bob to apply $f$ to the letter. Then, Bob would receive the letter, apply $f$ to it, and he would be left with the original message written in English. He would know that the letter came from Alice, because the message would only be able to be decoded with $f$ if the person who wrote the message encrypted it with $f^{-1}$. And Bob knows that Alice is the only one who knows $f^{-1}$ so it must have been from Alice. This example is pretty cool because even though Bob doesn’t know $f^{-1}$ and has the exact information everyone else does, Alice was able to prove to him that the letter was encrypted with $f^{-1}$ and that it was from her. This is referred to as a cryptographic signature and is key to cryptocurrency and the blockchain. ### Example: RSA (Warning Contains Math) First I need to establish a little mathematical background before I explain how RSA works. This should be review if you read my post on groups (I know you didn’t you piece of garbage). The math is built from the ground up but if you are not familiar with group theory or number theory, you are likely best served skipping this example. It was probobly a mistake to even include RSA in this post but I just thought it was too cool to leave out. Definition: We say that $a \equiv b \text{ (mod } n)$ if $a$ leaves a remainder of $b$ when divided by $n$. Theorem: (Multiplicative Inverses) If $p$ is a prime number, then for each integer $0 < a < p$ there exists a unique integer $0 < a^{-1} < p$ such that $a \times a^{-1} \equiv 1 \text{ (mod}p$. proof: Take an arbitrary $0 < a < p$. Consider the list created by taking each $0 < i and writing down the remainder when $ai$ is divided by $p$. The list contains $p-1$ entries and there are only $p-1$ different remainders modulo $p$. Therefore, if we show that no two numbers on the list are the same, this must mean that every possible number occurs exactly once and in particual a $1$ appears somewhere. To see this is the case suppose that for some $0 , we have $ai \equiv aj \text{ (mod }p)$. Then, $a(j -i) \equiv 0 \text{ (mod }p)$. But this means that $p$ divides $a(j-i)$ which means that $p$ divides $j-i$ but $0 < j-i . Theorem: (Fermat’s Little Theorem) $a^p \equiv a \text{ (mod }p)$ Proof: Induction + Binomial Theorem Definition: (Euler’s Totient Function) We define $\phi (n)$ for a positive integer $n$ to be the number of positive integers less $n$ that are relatively prime to it. Theorem: $a^{\phi (n)} \equiv 1 \text{ (mod }n)$ Proof: First we claim that for every $b$ that is relatively prime to $n$, there is unique integer $0 < b^{-1} < n$ that is relatively prime to $n$ satisfying $b \times b^{-1} \equiv 1 \text{ (mod }n)$. This proof is exactly the same as the one above for multiplicative inverses. Let $m$ be the smallest positive integer satisfying $b^{m} \equiv 1 \text{ (mod }n)$ (verify such an $m$ exists). We claim that $m$ divides $\phi (n)$. To see that it does start with the set $A = \{a, a^2, \cdots, a^{m}\}$. Now for each $0 < i < n$ that is relatively prime to $n$ add the set $\{ai, a^2i, \cdots, a^mi\}$. We claim that each step in this process either adds $m$ or zero elements to $A$. Suppose that one is already a duplicate. That is $a^{j_1}i = a^{j_2}k$. Then we have $a^{j_1 +r}i = a^{j_2 + r}k$, for each $r$ and $a$ is cyclic so we can just take its exponent to be the remainder when divided by $m$ and the term won’t change. But these terms were already in $A$. So we see that each time in the process we either add $m$ or zero elements to $A$ and at the end $A$ contains $\phi(n)$ elements, so $m$ divides $\phi (n)$. Theorem: (Chinese Remainder Theorem) Given distinct primes $p_1, p_2, \cdots, p_k$ and congruences, $a_i \equiv b_i \text{ (mod }p_i)$ for $0 < i \leq k$, there is a unique $0 \leq c < p_1p_2\cdots p_k$ satisfying each congruence. To put it into simpler terms, a remainder modulo a product of primes can be described by the remainder divided by each individual prime. Actually, a pretty obvious statement but useful nonetheless. Now we have established the background to get to RSA. Suppose Alice wants to encrypt something. To start Alice takes two large primes of about equal size $p$ and $q$. She chooses them large enough so that for any $a$, it is possible to compute $a^{-1}$ mod $p$ or $q$ in a reasonable amount of time (say about ten minutes). Now, Alice multiline $p$ and $q$ together to get $pq$. She then chooses some $e$ that is relatively prime to both $p$ and$q$. She computes $e^{-1}$ mod $\phi(pq)$. Notice that $\phi(pq) = (p-1)(q-1)$. This means she can easily compute $e^{-1}$ by computing $e^{-1}$ mod $p-1$, $e^{-1}$ mod $q-1$ and then using the chinese remainder theorem. For example, if $\phi(pq)$ factors into $p_1^{d_1}p_2^{d_2}\cdots p_k^{d_l}$, and we let the subscript$latex i$of a term denote that it is considered mod $i$, then $e^{-1}$ mod $\phi (pq)$ can be computed as $\sum_{i=1}^k ({\prod_{j\neq i} p_j^{d_j} {p_j}^{-d_j}} e^{-1})_{p_i^{d_i}}$ Which will not take too long to compute (There are actually faster ways to do this but the point is that even this won’t take an unreasonable amount of time). Alice publicly declares $pq$, $e$ and a way to represent a message as a number (so that each number will be relatively prime to $pq$. An important note is that she only shares the product of $p$ and $q$, she keeps the individual numbers $p$ and $q$ secret). They are big enough so that it is not feasible to find $p$ and $q$ from the product $pq$. She declares that if someone wants to send her a message, they write the message, convert it to numbers using the ways she declared, take each of these numbers to the $e$th power and take the remainder when divided by $pq$. Now lets say Bob does this. So Alice is left with a bunch of numbers. Now to decode the message, Alice simply takes each number to the power $e^{-1}$ and if the original number was $a$, she is left with $(a^{e})^{e^{-1}} \equiv a^{ee^{-1}} \equiv a^{1 + b\phi(pq)} \equiv a \text{ (mod }pq)$ So she has recovered $a$ and then she can simply convert $a$ back into letters and decode the message. Now, say Carl intercepts the message. His only hope to decode the message is to figure out what $e^{-1}$ is. But he doesn’t know what $p$ and $q$ are. The easiest way for him to compute the inverse is to just multiply numbers by $e$ and see if the result is $1$ mod $pq$. But since, it took Alice ten minutes to find the inverse for $p$, and mod $pq$ has $q$ times as many numbers to check as mod $p$, it will take Carl $q$ times 10 minutes to find. And $q$ is like a hundred digits long, so Cal has his work cut out for him. Now, of course there are more sophisticated attacks against RSA but it is still very secure and used all over the place today. ## Bitcoin Okay I’m going to explain to you how bitcoin works which will motivate the idea for the blockchain nicely and then I will be able to tell you how we can apply the blockchain to other areas. First I would like to warn you that during the process of reading this you will have a lot of questions. You’ll be like “Why this?” and “Why not that?” Just stick with it and by the end, I hope all these questions are answered. To understand Bitcoin you need to understand the goal of Bitcoin. That goal it to create a completely decentralized currency. What does this mean? Well the traditional system is to have banks keep track of money but 2008 taught us that banks are fucking stupid. We want a system where no one entity is in charge keeping track of shit. We want all records public so people can know that no funny business is going on. So when we say decentralized, what we mean is just a bunch of people communicating on any old public networks, sharing information, and playing by the rules because it is in everyone’s own self-interest. With this motivation, it is easy to see why cryptographic signatures are hella important. We need people to be able to prove that it is them when they declare publicly that they are making a transaction. Also, this highlights that money really doesn’t mean anything. Everyone just agrees that a person has x dollars and can spend it. Lets try to think what a such a system would like. Okay, a naive idea is to try and look for a system that doesn’t need any party keeping track of records. Lets see where this idea takes us. Okay, so everyone has an account and we need everyone to be able to publicly see that account’s transactions. We have already identified that cryptographic signatures are going to be needed so we can just define an account by its public key. Then, if someone wants to sign money to them, that person publicly announces their intent to give money to that public key and signs it with their private key. An obvious question to ask is what happens if that person doesn’t have enough money in their account? The transaction should not go through. A simple solution is to simply agree to only count transactions where the person has the requisite amount of money. And the receiver of the transaction could check this. They would simply look through all the public records of transactions involving that account since the beginning of time to determine if the account had enough money. But the problem is that they would have to determine if all past transactions were valid or not. And to determine those they would have to look at other transactions and determine if they were valid. This quickly devolves into checking basically all transactions in history. This is too much checking if Bitcoin is going to be a thing. Even more troubling, there would be no records of when transactions happened, so even if Bob checked other transactions, he wouldn’t know when they were which is needed to determine if they were valid. No, we need someone to separate out all valid and invalid transactions and to organize the valid ones together. This makes to task of seeing if someone has enough money much easier. But we are left with still needing someone to track shit. Maybe you still think there is a way of avoiding a trusted party to track shit. What if we have a more stringent definition of money? We have identified the need for tracking transactions so what if instead of defining money as just a number tied to someone’s account, we track each unit of money individually. This will have the corollary of making it virtually impossible to change the number of money you have without actually getting it from somewhere. We start with the smallest unit of money which the creators of bitcoin declare and they sell it openly. Whenever someone makes a transaction that person tags that piece of money with their cryptographic signature along with who they intend to transfer the money to and make it public. So in other words, money is just a chain of signatures and the owner of the piece of money is the last tag. All pieces of money started with the creators of bitcoin signing it as the original money, so people would know it was legitimate. Then to spend money, someone would simply point to money in the public records that show them owning it and sign it all over to the other account. The only problem with this is double spending. For example, suppose that Bob signs his money to Alice and then tries to sign the same money over to Carl. Even after Bob has signed the money over to Alice, the records of Bob being the end of the chain still exist so he could point to those records and try to sign it over to Carl. Carl could look through all the records to see if it was spent but he would also have to look through all points in the chain to see if each of them was valid at every point and this is a lot of searching. Additionally, if Bob attempted this, there would be records of two transactions and no one would be able to differentiate between the legitimate transfer to Alice and the illegitimate transfer to Carl. No one would trust the transaction to Alice and that would leave her fucked. Very bad. Finally, without tracking the time of transactions, the creators of bitcoin could create new money out of thin air which by definition makes the money not decentralized. We still see a need for a party to organize everything, to timestamp transactions, to stop this double spending. Okay, so we have identified the need for people to keep track of everything but how does this work? Lets first understand how this “organizing” would work. The idea of tracking money and defining it as a chain, with the owner being the last spender seemed smart, so let’s continue with that. What we could have, is every time interval, an “organizer” looks at all transactions that took place in that time interval, separates out the valid from the invalid ones and publishes the valid ones into a block. Then it is easy to see how much money someone has by checking through established blocks. If someone wants to make a transaction, they simply point to the blocks that contain their money, and the “organizer” only needs to check blocks forward to see if they did not already spend it. And others can check to make sure the “organizer” indeed verified the transactions were valid. These “organizers” are called “miners” (Don’t ask me why). Just remember, miners are the people in charge of organizing everything, time stamping transactions and making sure everything is legit. We’re left with only one problem, we need the network to be decentralized so we would have to let anyone who wants to become a miner (We couldn’t designate it to anyone in particular because then that person would have power and it wouldn’t be decentralized). But this begs the question: what if the miner is a prankster. If we let anyone organize the block, then a prankster could do it and fuck up the whole system. This is a big problem because it seems like if we just let anyone have access and organize we are bound to get a lot of pranksters. How can we sure that whoever becomes this miner will take it serious without designating a predetermined trusted party? Here’s an idea. What if we made it so that there was a real cost to be the miner. If someone had to say, pay a good deal of money, but in exchange, they were allowed to allocate to themselves some bitcoin if their block is legit, we would not get pranksters because it would not be worth that much money to fuck up the system. And if we got any pranksters, it would be a very few and there would be enough legit people monitoring the blockchain that would see the prankster pranking and could correct it with a serious miner. In other words, there are a bunch of legit people trying to make money off of mining, so if a prankster submitted a prank block, other miners would check it, see that it was horse shit, and simply ignore it and continue trying to make a legit block. You may be asking how the whole system will organize legit blocks from non-legit block. The short answer is that the legit miners will connect their new blocks to old legit blocks and people will only accept old blocks that have new blocks attached to them. But we’ll get into the specifics later. Another clear question that arises from the above explanation is how to charge a cost? There is no central entity to process a payment. What we need is for the miner to prove that they incurred some cost when they publish a block and importantly, make it obvious to anyone who wants to check, that they incurred the cost. Then, the bitcoin community and other miners would only accept the block if they proved this. But how can a miner prove that they incurred the cost? Lets think. What takes money? Energy. More precisely computing power. What if we generated a random number and made the miner guess that number correctly in order to publish a block. It would be a race, and whichever miner guesses the number first would get to publish the block and take the reward. The number would be in a huge range, like between 1 and 1000000000 so the miners would have to use a lot of computing power to have a chance at guessing right. But how would we pick what the right number is, and how the miners know if they got it right without a central entity? Who picks the random number, and who checks to see if they got it right? This is where Cryptographic Hash functions come in. Cryptographic Hash functions are highly technical but you can think of them as a function that transforms an arbitrary number (or string of letters) into an integer in some interval, say between 1 and 10000000000. Cryptographic Hash functions behave very randomly and are unpredictable. They don’t contain patterns. So if you take a number, apply the Hash function to it, you are pretty much equally likely to obtain any number between 1 and 10000000000. Another important thing about Cryptographic Hash functions is that they are easy to compute. In other words, if you have an arbitrary number, a computer program should be able to apply the hash function to the number very quickly. But the inverse must be beyond difficult. There is not easy way to define the inverse to a hash function and if I wanted to find a number that after applying the hash function, would result in say 23435 (this is just a random number), my best strategy would be to just pick a random number, apply the hash function, see if the result was 23435 and if not repeat until I eventually get lucky and picked the right number. So now you might be getting a sense for why Cryptographic Hash Functions are relevant to blockchain. What we do is basically convert all the messages for transactions into a “message number” using the hash function and then the miner has to find a “code number”, so when the”code number” is concatenated with the “message number” and the hash function is applied to it, the result is a really small number, (like it must be less than 1000 or something). For example, if the message number was 2435555 and the miner found a code number 132435, then if the hash function was applied to 2435555132435, the result would be a number less than 1000. This way the successful miner publishes the block which contains all the transactions and the code number (and some other stuff which we’ll get to later) and if the code number works, people will accept the block into the blockchain and build on it. The best strategy for the miners is to just pick a random number, concatenate it with the message number, apply the hash function, and see if the result is less than 1,000. If it is, they are done and they publish the block with that as their code number. If not, they try again. And it is in the best interest of the successful miner to make sure all transactions are valid because if they are not, the block will not be accepted and that block contains a transaction to the successful miner which is a lot of money. This is the basic idea. To recap, say Alice wants to send money to Bob. She would publicly declare that she wants to sign over some money to Bob (or more precisely Bob’s public key), point to where that money is and encrypt it with her private key. Everyone else could decrypt her message with her public key and they would know it was her because she was the only one who could encrypt a message with her private key. Then, the miners would put her transaction together with everyone else’s valid transactions that were declared after the previous block was started, use the hash function on the transaction numbers to get a “message number” and race to find the code number. People accept the first block that is published with all valid transactions and a correct code number. The successful miner is allowed to add a transaction to themselves in the block from the void which everyone accepts as money along with transaction fees from every transaction in that block. Then the process starts all over again. A new block is created about every ten minutes. If there are too many miners, the process becomes faster because more people are trying to find the code number. But if it starts going too fast, the metric is more than a certain threshold of blocks published in a week, the challenge becomes harder. The way they make it harder is simply stipulating that the code number, when concatenated with the message number and hashed, must produce an even smaller number. For example, if they first needed to produce a number less than 1000 and too many blocks were being published, they might now need to make the number less than 500. Side Note on payment to miners: Miners get paid in two parts. The first is the transaction fee. This fee is decided by the person transferring bitcoin. So if Alice wanted to make a transaction to Bob, she would include a small payment to the successful miner of the block her transaction gets put in. Alice gets to choose the transaction fee, but the higher it is, the more likely a miner is to include her transaction in the next block. Currently someone needs to pay about$1.5 if they want to ensure their payment gets into the next block (i.e. will be processed in less than 20 minutes) but can pay about 30 cents if they are fine with it taking a few hours. This threshold for a transaction fee to for sure make it into the next block is very volatile and last year during the bitcoin frenzy, transaction fees rose dramatically. Even when the average fee is relatively small, with their being hundreds of thousands of transactions per day, this ends up being a big payment to the miners.

Side Side note on transaction fees: It is better for everyone if transaction fees are low. If the transaction fee is higher, mining will be more profitable which will attract more miners, which will decrease the profitability of mining. And having more miners doesn’t do anyone any good. Once we have enough miners to ensure the security of bitcoin, getting more doesn’t do anything except lead to a larger pool of people competing for the same fixed resource. This, plus the fact that if people had reasonable attitudes towards bitcoin and understood it, lower transaction fees would help to grow bitcoin, which is in the best interests of miners, means that transaction fees don’t actually impact the profitability of mining as much as one would think  Transaction fees have been artificially high just because bitcoin is new and is being hyped a lot. People don’t mind paying transaction fees because they are not using it as a currency, they plan on sitting on their Bitcoin and don’t mind paying a little extra to get it fast. If bitcoin stabilizes and becomes and actual currency people use in their daily lives, we can expect lower transaction fees.

There is also a second way miners get paid. Miners are actually allowed to include a transaction to themselves in their block. This is the only new bitcoin that can ever be created. At the start of bitcoin, the miners were allowed to include a transaction for themselves for 50 bitcoins. But, the creators of bitcoin stipulated that every time 2,100,000 new blocks were created, this reward would be cut in half. It takes about 4 years for 2,100,000 blocks to be created, so in 2013 this reward was decreased to 25 bitcoins and is now at 12.5 bitcoins. Additionally, once there exists 21 million bitcoins, this type of reward stops altogether. So this means once there are 21 million bitcoins, no more will ever be created.

So now that we understand the basic structure let’s go over the steps to make things a little more precise.

First, what is this message that Alice must send to declare a transaction with Bob? Well, it depends on the money she wants to declare and it is mostly just a bunch of numbers. It has to be precise though. It being something specific that is mostly random is more insurance that it is actually Alice sending the message. For example, if she could just send the same message, then someone could copy one of her old messages and pretend to be her.

What else is in the blockchain? The first thing is the header. The header is the code number along with the message number. I never completely explained how the message number is generated so I’ll do that now. Basically, we start by taking all the transaction numbers, lets just assume the number of transactions is a power of two. If not, we solve this in an analogous way to how single elimination tournaments are done when the number of competitors is not a power of two. But anyway, we start by applying the hash function to each transaction number, pair the resulting numbers up, concatenate them with their partner, and apply the hash function to each which generates a bunch of new numbers. Then we pair up those numbers, concatenate them with their partners and apply the hash function. We do this until we are left with a single number. This generates a tree like structure and we are left with the head. We concatenate that with the header of the previous block just to make sure there is no funny business going on and that the miner has to figure out which block they are linking to before they start trying to find the code number, and that becomes the message number.

One nice thing about this structure is that eventually, once the money corresponding to a transaction in the block is used again, there is no need to store the transaction. For example, if Bob, signs money over to Alice, Alice signs the same money over to Carl, and Carl signs the money over the Dave, there is no need to keep the records of the original transaction with Alice and Bob because future transactions with that money have already been verified. All that is needed is the header of the blocks for people to connect new blocks to it and have a recorded history. This allows people to throw out the records of very old individual transactions in order to save space. This is not a huge thing but worth mentioning.

Additionally, the header from the last block is in the new block. This connects the blocks together and reinforces the validity of the last blocks. It shows definitely what chain people are working off of and which blocks are valid.

The block contains a timestamp which helps easily order the blocks and account for when the block was created and when the transactions took place.

The block contains all the transactions in their tree structure so people can 1) verify transactions and 2) verify the message number is correct.

The block contains a nonce number which is kind of complicated but basically just acts as an id number for the block.

Finally included in the block, is the difficulty rating. Basically, there is a public computer program that when you run, will output a rough difficulty rating for how hard the code number was to come up with.

These are the main components of the block. Now you know.

What happens if two people solve the code number at approximately the same time? Everything is run on a network of communication with a slight delay so it is hard to determine who actually finished first. The tiebreaker ends up being the difficulty rating. The blocks will have different difficulty ratings because the blocks are actually slightly different. The transactions and their order might be slightly different and also the transaction to the miner is going to be different for different miners.

The next successful miner will connect their block to the block with the higher difficulty rating so that will be the block people go off of. You might be asking what stops someone from publishing a block after someone else and hoping/colluding with the others for them to connect their new block to that block. This is where the message number being connected to the old block comes in. The other miners have already started trying to find the code number for the message number based off of the old block immediately after a new block is published. It would be suicide to start over with a the other persons block.

It does actually happen fairly often that two blocks will be created at the same time. And the “wrong” block does get recorded but is not used. People are supposed to rely on the chain of blocks with the largest sum of difficulty. Once a couple new blocks become created on top of an old block, it is basically impossible to reverse the validity of that old block, because it will be in a chain with a large sum of difficulty. This is an important point so take some time to understand. I hope this diagram makes it easier.

Now you might be asking, but can this system be gamed? What’s great about bitcoin is that defrauding the system is really difficult. It is impossible to change records because all the records are publicly available and downloaded on thousands of systems. So once a block gets accepted and a couple of rounds pass there is basically no way to change it. It is written into history. If you are a transactor and want to spend money you don’t have, this is basically impossible. You have to point to actual money that got recorded. The only hope you could have is to double spend. But this is really hard.

Let’s suppose Alice wants to double spend. She has made a transaction to Bob for a new pair of Jordan’s. Bob has waited until the block containing the transaction was accepted before handing over the Jordan’s. So what Alice needs to do is to erase the records of her transaction to Bob if she wants to spend the same money again. Her only hope is to make a new chain of blocks growing out of the block before her transaction with Bob with a greater sum of difficulty than the current chain of blocks coming out. But to do this Alice would need to be able to find the code consistently at a much faster rate than all of the other miners combined. That’s hella hard.

But let’s just say Alice is a serious G. She has found a new technique that is much faster than the competition. She could in theory redo all the blocks except leave her transaction for the Jordan’s out. Then people would accept her new blocks and those records would be used to calculate people’s money. She would have her Jordan’s and the bitcoin she spent them on all to herself, leaving Bob fucked. Having her cake and eating it too. Additionally, she would be the miner for all those old blocks so she would even get all those transaction fees as well. She’d be rolling in it.

There’s only one problem. Everyone else could see what she did. They could still see the legit blocks Alice replaced and see that she rewrote them to not include her transaction. Others would technically be supposed to accept in any way but they would still be like “da fuq?” It would be obvious that bitcoin had been defrauded and this could lead to the value of bitcoin dropping like a fat guy bending over to pick up a Twinkie. But the value of bitcoin dropping fucks Alice.

But instead of defrauding the system, Alice could just become a miner and since she created her hella toit new technique, she would win every time and get hella rich. And since she was honest, bitcoin would still be working perfectly well so it would retain value, which is good for Alice.

Another interesting question is what keeps miners actually checking to make sure the transactions are valid? In theory everyone else is supposed to check to make sure all the transactions in their public block are valid but in reality no one really wants to do this. It takes time and energy that could be spent mining the next block. The network so far has done a pretty good job staying honest and checking each others work but there was an instance in 2015 where a bad block was published and a couple of new blocks were published before anyone realized the mistake. They were able to go back, fix everything, and since then people have been more careful about checking each others work. In general, it is in the best interest of miner’s if someone is checking the validity because they own bitcoin. I only point out this example to show that if mining gets too competitive, there are risks of no one checking and invalid blocks getting published.

Can we just take a moment to appreciate how genius this system is?

### Is the Hype Real?

Bitcoin is a real thing for sure, but it is not without downsides. In some sense bitcoin has been a victim of its own success. It is designed to be a stable currency that people use to pay for everything, but because of how much bitcoin is being hyped together with the fact that most people don’t understand how it works, its price behaves like a schizophrenic lion on crack. This causes people to buy bitcoin, not as a currency but as an investment. This is ironic since being secure is supposed to make it more stable than the U.S. dollar. The U.S. dollar has no rules attached to it and its quantity goes up over time. This causes inflation, but since bitcoin has fixed rules and beyond a point none of it can be created or destroyed, it is supposed to hold its value. The hope is that once all the hype dies down and people become more used to cryptocurrency, the price will stabilize. But no one knows if this will ever happen. Until this happens, bitcoin doesn’t really behave like a currency but rather an investment.

Another downside is that it is slower. People have to usually wait at least ten minutes before their payment is verified whereas a payment with a credit or debit card can be verified in seconds. One interesting thing though is that there is no need for the rate of new blocks coming out to take that long. Originally, it was proposed that shorter times would take up too much energy from miners and it would make it hard to organize everything but this has proven to not be true. Ethereum’s blocks come out every 15 seconds and it works great. This long block time has not hindered bitcoin’s value yet but in the future, other cryptocurrencies that publish blocks faster may prove to be more convenient if cryptocurrencies are to be used as actual currency.

Finally, bitcoin is more expensive. Even taking into account fees for checking accounts, transaction fees would make bitcoin cost more than a bank to use as a day-to-day currency. Now, the hope is that these transaction fees will go down if bitcoin becomes widely used as a currency, but again, no one knows.

So to summarize, bitcoin is revolutionary technology that solves many problems of traditional banking, but it brings its own challenges along with it. So it is not just strictly better than traditional currency.

## Crypto Applications

Bitcoin gave us the blockchain, but this genius idea can be applied to more than just money. It lets a create a whole class decentralized applications.

Let’s take social media for example. How could we use blockchain technology to create a decentralized social network? This is actually easy. To be decentralized we just need no one entity in control. So someone would write the computer program for how the network would operate, make the code public, and let the Miners do the dirty work. Similar to how accounts would work with in bitcoin, everyone account would have a public and private key, and whenever anyone wanted to make an action, say send a message or friend request, they would simply declare their intent. Then, every time interval or so, a miner would solve the code message, run the network code on all “transactions” and publish the updated network with all transactions. Of course we would need a currency attached to the system to reward the miners but the idea is actually pretty basic. And shit could still be private. For example, anyone could create a “friend” private key, and share the key with their friends using their friends public keys. Then, they could send their updates using their “friend” private key so only friends could see what was going on.

To make the above process more explicit, let me explain it in steps. First, Alice would create an account. Tied to the account would be a public and private key. Let’s call these the Alicia keys. Alice would keep the private key secret but publish the public key, well… publicly. Additionally tied to Alice’s account would be a “friend” public and private key completely separate from the Alice’s Key. Alice would release the friend public key publicly but initially keep the friend private key private. But say Bob sends a friend request to Alice and Alice thinks Bob is a cool dude so she accepts. To accept the friend request, Alice would take her friend key, encrypt it using Bob’s public key and publish the resulting number and notifying Bob. Bob could then decrypt the message using his private key and he would know Alice’s friend key. Then whenever Alice posts something, she would encrypt what she was posting with her friend public key, sign it with her private key and make it public. The miners would see this post, use Alice’s public key to verify it was her, and incorporate her post into the next block which would become the new state of the network. Only Bob (And Alice’s other friends (she doesn’t have any)) would be able to understand the post because only they have her friend private key. And people would only accept that state of the network, it the block was contained a correct code message.

Of course the above process could be completed with Alice only having one key. She could simply publish a duplicate copy of her posts encrypted with each of her friend’s public keys. This would make it so people had to keep track of less keys but would take up more storage space.

We can use the same idea with DMing. This is actually just the example we started with in the introduction to cryptography. If Alice sent a message to Bob, she would simply encrypt it with his public key.

This process can be used to create a decentralized version of basically any application.

The major upside to using crypto applications instead of centralized ones is that 1) obviously they are more secure and 2) they cannot be censored. This is super relevant in countries like China, where the government regulates the internet and what people see. In crypto applications, the whole network is just a bunch a different computers storing data and operating a code everyone has access to. This means there is nothing for the government to go after. The miners are completely anonymous and there is no central network to attack.

## Crypto Tokens

Finally, I would like to share one more possible use for blockchain technology.

Crypto Tokens create a whole new business model. The way Crypto Tokens work is that someone starts by creating a cryptocurrency, keeping a portion of it for themselves and selling the rest on the open market. They stipulate that no new currency can ever be created. Simultaneously, they release a decentralized app for some service based industry. It will connect service providers with customers having the price for the service be in their cryptocurrency. Then, as the app grows, the demand for the crypto currency will also grow, and since the amount of the currency on the market is fixed, the price of the currency will increase. Note that the price for the service will remain fixed in U.S. dollars, so if the price of the coin increases, the cost of the service will cost less coins.

For example, at the beginning, the service might cost 1 coin, but once the business grows, the service will only cost .01 coin. Then, eventually the founder will sell their coins which will be worth a lot if the business is a success. The business will continue to operate without anyone in charge.

One downside is that since these business will be decentralized, it can be something illegal and since it is completely decentralized, it is extremely hard to stop. For example,

Eli the Entrepreneur starts by using the blockchain to create 10,000 “sex coins.” He keeps 1,000 for himself and sells the rest to anyone who wants to buy them at 1$per coin. He then designs an app that connects prostitutes to customers. Samuel the Sex Worker uses the app. He starts by declaring what city he lives in which is Miami and lists his services along with how much they cost. He lists them in U.S. dollars but the app converts that to the exchange rate for sex coins. Then anyone can anonymously create a cryptographically secure customer account and all they have to do is declare what city they live in. Then, they can search through sex workers in that city on the app and see what they price their services at. Harold, who also lives in Miami, is in desperate need of a hand job. Luckily, Samuel offers hand jobs for 10 sex coins (he is a pro after all). Then they can use the app to arrange where to meet and do the deed. Harold will buy sex coins on the market and use that to pay Samuel (With an extremely small transaction fee to pay miners. Something like a cent). Soon, the app erupts so hundreds of thousands of people are simultaneously using the app. Since there is only 10,000 sex coins on the market, the demand for them rises. So now, all of a sudden each coin is worth 100$ and Samuel is charging one tenth of a sex coin for a handy. Finally, at the height of the app, each coin is worth is 1,000$. Eli sells his 1,000 coins for a million dollars without ever having to do anything beyond writing the code for the app. The app and sex coins continue to be exchanged in the market with no one controlling them. The thing about this is that it is extremely hard for the police to stop. There is no central entity for them to bust and end the whole thing. The only thing they can do is set of individual stings arresting individual prostitutes. But this requires tons of effort and doesn’t really do anything beyond discouraging people to use the app but strategies like this have never really worked in the past. So there you go. Hopefully you now understand how blockchain works and what some of the ramifications are. If you have been finding yourself wondering “Is there a new post on the Cheese Maze” and the uncertainty has been causing you anxiety, I have a solution. Simply scroll to the top of the page, look in the lower right hand corner of your computer screen and follow this blog to get an email every time a new post comes out. # Some Stanky Thoughts on Political and Economic Systems “Both Capitalism and Democracy kind of suck but no one’s come up with any better ideas”-Every Reasonably Intelligent Person Ever. Sorry for the delay. I have been kind of busy and I spent the last week thinking that becoming good at the piano was something I could do. Things started out good. I was practicing for like five hours a day. But then on friday my sister put a pile of her clothes on the piano bench so that ended that. These are some of my ramblings on politics not really organized any way, mostly just a collection of unrelated topics that I thought of while writing this. It is more of an analysis of politics in general rather than just a liberal rant. ## Models vs Complexity When creating a system, there are two roads you can go down. You can either decide on a simple model for how the world works and base all decisions off of that or you can try to embrace the complexity of the world and make all decisions in isolation based off everything you know. There are pros and cons to both. Model: Pros: Decisions are easy. Everything is cohesive, efficient and working towards the same goals. Cons: The world is not simple and bad things can happen if your model is incomplete or even worse, wrong. Complexity: Pros: All concerns are addressed. Reflects actual events of the world and can be responsive to change. Cons: The world is very complex and no human could understand everything. Often different decisions are inconsistent and can negate each other. Additionally, constant problems require fixes which create other problems which leads to an endless cycle. (If I’ve learned anything from math it is that this never works out well.) So far, every economic and governmental system has, at least in theory, been based on models. For example, Democracy is based on the model that everyone is rational, has complete information and acts in their own best interest. Capitalism is based on of or at least sold using the same model so it is unsurprising that most democracies have a fairly free market economy. (There are other reasons for this too) Communism is based off Marx’s model that human history has been a struggle between the bourgeois and the workers in addition to his model behind human psychology and our desires. Hobbes’s model was that people are inherently selfish and in a state of nature there are many prisoner dilemma scenarios where people will mostly defect. Of course in reality there are no “pure” forms of any of these. What happens is that the people in charge have a model in mind, pick the according system and then change things that don’t work until things actually are quite complex. Really, there is no true system based off a model or based solely on complexity. It is a spectrum but I think we have fallen too far on the model side and not embraced complexity. This model vs complexity applies to all issues and it is something to keep in mind when making decisions. I love models and if you do to it is important to be able look past them. What I propose is to instead of starting with a terribly simplistic model, and then constantly tweaking things that are wrong, we initially start looking for a more complex model and to spend more time finding a system that fits that model. Of course this is much easier said than done. Finding a complex model that accurately describes human behavior and how the world works is no easy task, I am just saying that if one is found, a good model of how things should be run may follow as a corollary. ## Our System The way our government was set of was actually surprisingly good. Lots of thought went into it and the founding fathers are probably accurately rated. This doesn’t mean the system doesn’t have its shortcomings though. And some of the worst aspects of it are coming out now. Whether it’s the system’s, our own, or technologies fault, with all the problems facing humanity, if we don’t address the problems in our system, we are in for some trouble. Many people complain about our democracy being representative and while I do agree it’s stupid, this is not really the root cause of our problems. If we switched to a direct democracy today, our problems would be far from solved. The way I see it, there are two main problems that our political system is facing right now and those are corruption and tribalism. Let’s talk about tribalism first. Tribalism is clearly a problem. If you think everyone thinks rationally and makes decisions based off objective logic, you are wrong. If you think liberals are different, beyond tribalism, you are also wrong. Conservatives of course are no better, my point is just that people from both sides of the political spectrum can behave tribalistically. For example, there is objectively little correlation between social political and economic politics and yet most people who are socially liberal are economically conservative and people who are socially conservative are economically liberal. Additionally, people are very likely to have the same political views as their parents as well as the people living in their community. Tribalism has always been a problem and it is something that humans do naturally but right now we are seeing it like we haven’t in a long time. The main culprit is our two party system but other things escalate tribalism like social media and filter bubbles. People blindly follow their party without thinking a single thought and refuse to listen to anyone who’s opinion differs on anything. This is made worse by social media where people are only exposed to things reinforcing their tribal mentality. Even the smartest, the best thinkers in the world fall prey to tribalism. Even I have tribalisic tendencies. I find myself disappointed if I find out someone I admire is conservative. When Trump says “It’s the democrats fault” I immediately think, “No, it’s the republicans fault.” If someone talks about how there is racism against white straight men I’m immediately like “You fucking bitch ass dumb dumb”. But I don’t actually know. I can imagine that it is probably very different than someone who is a minority but I don’t actually know what is happening in the world. This happens with lots of other issues like climate change. I have never actually seen evidence of global warming or read the most compelling counterpoints but I get angry if anyone dares to question it. Again, given that the figures I have been told are true, and what I know about the world, the scientific community, and how there is money to be made if climate change is false, it seems very likely that it is true. But that’s not what I’m thinking about when someone denies climate change. I go immediately tribal. I feel like the denier is an enemy. Tribalism lets politicians get away with not saying anything of substance. I really wish everyone could come together and demand that politicians not use baseless rhetoric. It is very easy to spot when a politician says something that only appeals to pathos and is a terrible argument but the politicians followers love it which makes it very effective. I think it would greatly benefit everyone if people stopped responding to this type of argument even if it comes from the politician they like. One problem is that the issues of the world are so complex that most people understand basically none of it at a fundamental level. So it would take an exobament amount of time for a politician to make any argument of real substance when that time could be used attacking other canadities or riling up their base. But if we made politicians actually explain at least one issue and explain themselves all the way down to first principles, we would actually get a sense of how they think about stuff which imo is much more important than what they think. Because even if someone has the same surface view as a politician in a broad sense, it might actually deep down be for different reasons and this will greatly affect how the policies are carried out and it might be in a way you fundamentally disagree with. And if people could really get a sense of why they think what they think, then it would inform people about how that politician would approach other areas and what they would be able to get done. For example, Bernie Sanders talked about free college and that was effective because it was something a lot of people were angry about. But I think it would be hugely beneficial if he actually talked all the way through to the pros and cons of it and why it should be done. If he actually explained why college is important and addressed counter arguments. If he specifically explained where the money would come from and talk about why it would be a good trade off and what we would be giving up. And the effects of all that. If he reduced everything down to how it would affect human suffering. I think everyone would more accurately understand his ideas and people from the other side would be less angry. This is probably a pipe dream though. It would not be beneficial to the politician and I don’t see how a movement like that would get started given how tribal we are as a nation. I don’t think most people have even investigated why they truly believe what they believe. Currently, if politicians had to explain down to first principles, they would have to make concessions and talk about all the effects of their policy. It would likely alienate many people without leaving many benefits. Another obvious thought is to get rid of the two party system. This is one of the root causes and without labels, people would be forced to look at the issues objectively and make their own decision. And while I do think this would be great, getting rid of parties is basically impossible. People with the same views will just naturally come together and there is no real way to stop this. And people love to belong to tribes to give them an identity. In a democracy, tribalism is inevitable and it is only made worse by our technology. The only hope seems to be education to teach people how to think but the catch-22 is that as long as we remain tribal, education is not going to get better. This brings us to corruption, the second major problem in our system. Capitalism is half to blame for this but corruption thrives in democracy. Technology has let corporations become huge and small policy changes can have drastic effects on their profits. Luckily for corporations, there is a thing called lobbying. They can talk directly to politicians and as it often happens, after the meeting they like the politician so much that they donate a bunch of money to their campaign. And what a coincidence, once the politician is elected they push policy that directly benefits the corporation. You might say people are not that corrupt. Many people are good, have values and couldn’t be bribed. So why should politicians be any different? The problem is that people that are corrupt are exactly the type of people that do well in politics. I don’t have any exact figures but it seems that to get very far up the latter of politics, some corruption is needed. This might be all dandy if it weren’t for the fact that policies that help large corporations are often directly in opposition to the good the general population. So we are left with corrupt politicians pushing policy for two reasons. It either helps a corporation that contributes to their campaign, or it is to help their future campaign. The second seems good. After all, policies that would help their future should be beneficial to the people right? Well, actually this just incentives politicians to push policies that will only bring short term benefit or only helps the demographic of people whose votes they think they can get. And things that bring short term benefit often are disastrous in the long run. Just look at all of the environmental regulations that are currently being rolled back. Now an obvious idea is to outlaw lobbying. But good people use lobbying and it is the only way people to have a voice and tell politicians about it. Additionally, even if we outlawed lobbying, politicians would still find ways to communicate with large corporations. The other obvious idea is to restrict donations to campaigns. And you’re right this is an obvious idea and if you think corruption doesn’t exist the fact that our campaign finance laws are what they are is evidence that corruption is alive and well. Additionally, once cryptocurrency becomes a thing, we will not be able to track donations so we’re looking at even more corruption. Now, I’m not saying we shouldn’t have a democracy. Like the quote at the beginning said, we don’t really have any better ideas but I think it is important to realize democracies shortcomings and find ways to solve/mitigate them. ## World Government The idea of countries seems really dumb to me. It seems like the only reason countries have existed in the past is that it is impossible to govern the whole world because it is so big. But now with technology, we can send information anywhere across the globe and everything is highly organized. So it makes a lot of sense to have a world government. Word government would allow anyone to travel wherever they wanted. It would provide insurance so that if some area fails, attention can be directed towards that region to help it. Most importantly, it would end all wars. And not only are wars terrible for everyone involved, but technology like nuclear weapons which are only going to get more advanced, pose an existential threat to humanity. It seems like a no brainer to have a world government with its own military to force diplomacy. We could even have a fairly week world government that does not enforce laws for individuals and lets current governments rule. It would just regulate how different countries act toward each other. We have acknowledged contract theory and that we need enforceable laws to keep individuals out of prisoner dilemma scenarios. So why should we not do the same with countries which have the potential to do much more damage than any individual? I am not surprised that we don’t have a world government because countries don’t want to give up power but I am surprised that no one is even talking about it. Also, I want to point out that Einstein was a proponent for world government so take that. # Some Ideas Now, after all that talk about how we need to look at more complexities I’m going to present some ideas for government that are based off incredibly simplistic models. These ideas are all ridden with errors and would never work. Still I think it is important to think about new ideas because we might one day come up with something great. Here is the list. ## 1. Robots Self explanatory. ## 2. Randomness What I am going to talk about is a system of government I came up with a while ago. It’s horribly flawed and not well thought out but I think the key ideas might have merit. The main idea is to get rid of corruption and tribalism. This system is built around choosing a good non-corrupt leader and everything follows from that. The leader has unlimited power under some constitution and appoints his staff which appoint their staff and so on until we get to a low enough level where everyone is just hired and people carry over from one administration to the other. But how do we choose this leader? The key idea is randomness. To start, we want someone who is generally a good person. So we select about 20 million people randomly or about 1/100 of the population. Oh yea and we make sure they are between 35 and 65 in age. We use an interface like Facebook to send a questionnaire to people who know them best. The way this would work would be choosing kind of randomly from their friend list but using data to weight the search towards people that know them really well. I think this is possible given how much data facebook keeps although I’m not sure about the details. It would take into account if you ever tag them in photos, DM with them, and them liking or commenting on your posts. Anyway, you choose about 20 close friends for every person chosen and send them a questionnaire. The questionnaire is one question: Is this person a) One of the best people you know b) a good person c) I don’t know/ average person d) a bad person e) One of the worst people you know An a) is worth 3 points, a b) is 1 point, a c) is 0 points, a d) is -1 points, and an e) is -4 points. I think weighting the d) the worst is the best because you really want to avoid people who are terrible. Although I didn’t put much thought into the exact scores. Now, from the 90th percentile an up of scorers, 10,000 people are randomly chosen. The reason the first pool was so big is so that it would be a normal thing to get the questionnaire. If we started with the search narrowed down, getting the questionnaire would be like “Oh my god, this person might become our leader.” This might cause the receiver of the questionnaire to either say a) one of the best people I know because they think it would be really cool if someone they knew became supreme ruler or e) one of the worst people I know because they are jelly of them. But, if you start out with 20 million people, it is extremely unlikely that getting chosen will lead to you becoming the SR so people will, in theory, be more honest. Now, this is of course not a perfect morality test because lots will depend on who you choose to give the questionnaire to and subjectivity but you will get generally good people. I’m not sure the top 10% is the best percentile to choose from but I think it is important to not just choose “top 10,000 scorers” because this could lead to 1) bad people gaming the system and 2) it would be selecting a very specific homogenous group of people and the point of this is that we do not know the exact type of people who would make the best leaders. We want a variety to choose from in the future tests. We just want “generally good” Ok, so we are left with 10,000 people. What we now do is randomly select 100 universities and task each of them with coming up with a question that measures intellect. I’m biased so I think the task should go to the math department but maybe mathematicians aren’t best at coming up with questions to test intelligence. Anyway, about 30 of the 100 questions that are received are randomly chosen make up the big Test. All 10,000 people take the test and we randomly select 50 people from the 95th percentile. These people become the candidates. Again, I know that IQ tests are not the best measure of intelligence but again, we are just choosing people who do generally well so on the balance we will get “generally intelligent.” I think it is also important to not centralize the creation of the test because this could lead to corruption. We want a bunch of different independently made questions, a lot so any one is not that important or worthwhile to target, made by different independent ethical groups good at testing intelligence. I think universities are good at this. Now, we basically do the same thing for the voters. We start by randomly choosing 10 million people over the age of 18 and send their friends a questionnaire. We take 50,000 randomly selected people, choose 25 other random universities and tell them each to come up with a question which altogether makes up the voter test. Now, we choose 500 people from the 80th percentile of scorers, making sure to include every demographic as it appears in the U.S. So the number of Mexican voters corresponds to the number of Mexicans in the U.S. Same with socioeconomic status, age, gender, religion, etc. It is possible, that to achieve this we need to start with more than 50,000 people but that is my rough estimate as to how many people we need so that after the test, we are able to choose equidistributed from our demographics to make up the voters. Ok, so now we have the voters and the canadities. immediately after they are chosen, everyone is shipped off to two remote islands, one for the candidates and one for the voters where neither will have any contact with the outside world. It is important that they do not have contact with the world so that they are not corruptible. Now, we again use randomness to take 50 history/political science/economic professors from universities. It is also important that immediately after these professors are chosen, they are shipped off to the island so that others cannot influence them. And I think 50 is enough so that we would get professors with different political ideas. For the next four years both the candidates and the voters learn from these professors about our history, our government, and current events happening. Once these four years are up, for the next two years, the voters learn about the candidates. But they never see them, this mitigates bias. The way they learn about the candidates is that the candidates send them letters but the candidates never know anything about the voters. This stops them from just writing what they think the voters will want to hear. The letters are about policy and the set of professors who have taught the voters (they don’t know anything about the candidates) choose the subject of each letter. We could even have some objective computer program check the letters to make sure there are no fallacies or emotion appeals. Though I’m not sure who would get to write the computer program.. At the end of the two years, the voters rank their top 5 choices for who they think should be the leader and also their bottom 3 worse choices for who they think should definitely not be leader. Their top choice gets 5 points, second gets 4, third gets 3, fourth gets 2, 5th gets one. And their third worse gets -1, second worse gets -2, and third worse gets -3. The candidate with the most points becomes supreme ruler. This person then chooses 10 of the remaining candidates to become their cabinet. I think it is important to have bottom 3 choices. This can help prevent really radical candidates from getting elected. For example, if one is from the alt-right, they will get -3 points from basically everyone which will ensure they would not become SR even if they get 5 points from a few voters. I think this is a good way to choose for the cabinet because it will be people who the candidate knows well, understands our system, and is not someone corrupt from the establishment. From there, the cabinet chooses people they know to run different things and it goes down until there are just lower level employees who keep their jobs. We can do the same thing but at a smaller level for lower territories like states. The Supreme Ruler’s terms is 8 years long but every year there is a chance for the general population to vote them out of office. Everyone gets a survey and if 75% of people vote for them to be removed, they are. This is just another safeguard against a dictator. We could have the same system but on a smaller scale for the selection of governors. I think we can keep local politics as they are even though they can be corrupt. This is because 1) I think they are generally less corrupt than big politicians 2) it would be hard to reproduce the presidential election on that small of a scale and 3) I hope this would not lead to a national two-party system. There would be no members of a corrupt party above the local politicians and local issues would dictate where the politicians would stand. So we might get local parties, but this is much less bad than national parties. ## 3. A Data Based Approach This is a method that definitely won’t work but I’ll mention it because I thought of it and this is my blog so I can say anything I want. The idea would be to start with no bias and have trials of different policies and see how well we do. We would have to have some objective measure for how well we are doing. This could be done by surveys of the people asking how happy they are with society. We could have different cities have different laws that are constantly changing due to the overall data. The reason that this is a terrible idea is that policy is very complicated and there doesn’t seem to be a big enough sample size to reach conclusions. ## Economics First lets go over capitalism. Capitalism is really good at motivating people to work, producing lots of stuff, and creating innovation. The downside is that unregulated it leads to a ton of inequality. ## Brief Overview of Economics: I don’t really know much about economics but I took APES and I’m good at math so this is what I guess economics is like. The normal way of explaining things seems to be with the intersection of supply and demand curves but I’m going to explain things using a different model which I’m not sure is accurate but will be more useful for analysis later in the article. The basic idea is to assume that sellers are completely rational and that they will price their products in order to maximize profits. Profits are simply $(\text{Number of units sold})(\text{price}-\text{cost to make one unit})$ And since a seller will make as many units as demanded, we can change “Number of Units Sold” to demand. But demand is a function of price and the higher a seller prices a unit, the lower the demand will be. This can be seen visually in supply demand graphs The Y-axis is the price and the x-axis is quantity. So see how if the price is very low, then the demand is very high but if the price is high then the demand is low. We also have this “Supply” curve which we will ignore for now. So for example, in this case the demand curve is linear so if price is p, then $\text{demand} = c_1-p$. If we let $P = p-\text{cost to make}$ and $c_2 = c_1-p + P$ then we want to maximize the quantity $(c_2 - P)(P)$ Let us rewrite the equation as $(\frac{c_2}{2} + (\frac{c_2}{_2}-P)(\frac{c_2}{2} - (\frac{c_2}{2} -P)) = (\frac{c_2}{2})^2 - (\frac{c_2}{_2}-P)^2$ So we see the terms are maximized when $P = \frac{c_2}{2}$ and in particular when the two terms $c_2 - P$ and $P$ are equal. This is an important example. And we’ll see that products get bigger when the terms are close to equal. So in economics there will always be a pretty even balance between the price of something and the demand for the product. This explains why places price things so that they are reasonable and about what you value it at. Also this implies that if something causes a sudden decrease in demand, then the price will drop because all of a sudden price and demand are not in balance and the seller needs to drop price to increase demand to make them in balance again. So for example, if all of a sudden there was a shift in the mindset of consumers and the demand became $c_2 + c+3 P$ then the optimal price the seller would be $\frac{c_2+c_3}{2}$ so the price would increase by $\frac{c_3}{2}$. ,Similarly this explains why, if all of a sudden something costs more to make, then the seller will increase the price to keep everything in balance. With that out-of-the-way we can look at our own system. We want an economic system that maximizes the well-being of everyone involved. We can think of well-being as the product $(\text{production}) (\text{equality})$ which of course means nothing but the idea being that we want both a productive economy and to have the profits evenly spread amongst society. So we can think of those two things like numbers and we want to maximize the product. We use product because either extremes (complete inequality with great production or complete equality with no production are both terrible.) To maximize well-being we can use regulation. Roughly what regulation does is it will decrease total production but it will also decrease inequality. To see why this is the case take minimum wage for example. Increasing minimum wage decreases the total profit a company makes per unit sold because they have to pay someone more to make that unit. And if the profit is less than all of a sudden demand and price are out of balance so they will increase the price which will decrease the demand which will decrease the total units made. It will decrease inequality for the obvious reasons. Another side effect of increasing the minimum wage is that all industries will be out of balance with respect to price and demand so they will have increase price to put things back in balance, this will actually cause inflation because all of a sudden all products will cost more. So should there be increase the minimum wage? Well, another thing we have to look at is any other externalities that may be happening. With the advent of technology, all of a sudden it is much cheaper to make products and as a result more products will be made. This means that as time goes on and technology makes stuff easier to make, we can actually increase the minimum wage as well to make equality and products produced in balance. In other words suppose that 50 years ago, the minimum wage of 8$. So equality was, say a 10. And at that time the minimum wage was the correct amount to maximize (equality)(production) so production was also a 10. From then, technology has made it easier to produce stuff so now production is a 12 but equality is still a 10. (In real life you could argue that equality has actually gone down for other reasons but for this exercise suppose it is the same). So now (production)(equality) is 120. But if we raised the minimum wage so that equality would be an 11, then production would go down by about 1 so it would be also be an 11. But then (production)(equality) = 121. So we have increased the quantity that we wanted by increasing the minimum wage. This suggests that it may be a good idea to increase the minimum wage.

Now of course the above model is way over simplified but I think the idea holds true. As time passes and we develop better technology, we can also afford to increase the minimum wage without stifling business too much. So minimum wage is not something in a vacuum. The context of how business is already doing affects what the minimum wage should be.

Then, of course there are other types of regulation. I hear a lot of people making arguments about how we need this regulation specifically because of the thing the regulation is supposed to do. For example people might argue we need regulation to make sure companies don’t pollute in the ocean and their argument is entirely based around all the damage it’s doing. But we should look at it like “can we afford the regulation?” Is business doing well enough that we can preserve these things and still have a functioning economy?

## UBI

A universal basic income or UBI is where everybody is just given a livable wage for doing nothing. It is an extreme prospect that would have been outlandish at any other point in history. But with the onset of automation and more efficient technology, we just don’t need everyone to work. And we are producing enough that with the right way, this might be a possibility.

There are of course some concerns. If we do implement such a system it must be with care. First we must ask where the government is to get the money?  One answer is that as more automation takes over, corporations become more profitable which allows the U.S. to be able to tax them higher. It could use those taxes to subsidize a universal basic income. The problem is that automation does not cost nothing. If automation costs $n$ and the company was paying people $m$ before, the government can only tax them an additional $m-n$ without having bad effects on the economy. This means that the UBI would have to be less than what the people would be making if they were working.

And even if they could find a way to get people enough money another problem is that this has the potential to cause inflation. As our poorest class gets more money, companies can raise prices and there will still be the same demand. They can raise prices to a level where the UBI is comparable to welfare now and the sellers wills still get business because these people are forced to buy the products even if it means living in terrible conditions, not eating healthy, and having no security. In other words, sellers of necessities like food and housing can price their products so that the poorest class can barely afford them. The corporations will not lose any business though because demand doesn’t change as a function of price when the thing is a necessity. The people living off of the UBI would technically have the money to pay for the worst product.

Another problem is that it will encourage people to not work. And while automation may make it so that this doesn’t cause our economy to crash, we would be left with a huge population of people without anything to do with their time. This could detriment them because their life might become meaningless and empty. They might also dedicate their time to something like politics which could be dangerous because these will be the people with the least education. This could lead to even more tribalism in our political system.

I’m not saying that a UBI is a bad idea. It may be very good. It is just very dangerous and we can’t just implement it willy nilly.

## Communism and Capitalism Together

This is similar to a UBI but less extreme. It is also probably not feasible or a good idea. The idea is that we keep capitalism around so that we have production and innovation.

But we also have a system that anyone can opt into run by the government that will pay them a livable wage. But in exchange they have to work. The reason this might work now is that we have technology to keep the system organized and hold people accountable. We would be to have someone keep track of all the work needed to be done in order to feed and house everyone in the program. Then a computer could equally distribute the work. So if you were in the program, everyday you would get a message on your phone telling you what jobs you had to do. For example you might be farming, working in a factory, or doing construction. At the end of the day, it can be checked if your jobs were done. If they are not you get a strike, and if you get enough strikes you get kicked out of the system. A computer couldn’t run it alone so they would have to hire people from the capitalism side to oversee everything. This would be inefficient for sure and could easily go wrong, but it would at least gives any poor person a way to get the resources they need.

Ok, I can’t think of anything else I want to say so good-bye. One more thing though. I want you to take ten seconds to center yourself before reading the following sentence and make sure you are 100% present to absorb it…

# 8 Reasons Why the Earth is a Torus

Recently there has been a sudden growth in the flat Earth movement. I was intrigued. I dug deep and what I found surprised me. It turns out they are right. The Earth is not a sphere. So why is it that they have not gotten recognition from the larger community? It’s because there argument is incomplete. They are missing one key piece. They believe the earth is flat, but through my research, I discovered that the Earth is actually… a torus. Here is the correct model of the solar system.

The torus is the Earth and the yellow thing is the sun. The sun orbits in a figure eight. You’ll notice there are no other planets and no moon. That’s because they don’t exist.

Ok, you might not believe me because you’re so indoctorated with the sphere Earth ideology. I’ll show you six reason, each sufficient on its own, that proves Earth is not a sphere.

# 1.

They say earth is a sphere. What Else is a sphere? Balls. What do balls do? Bounce. Has the Earth ever bounced? No. Next.

# 2.

They say earth is a sphere. Spher’s are smoothe. Mountains. Next.

# 3.

They say Earth is a sphere because of gravity. Helium Balloons. Next.

Jesus. Next.

# 5.

The only people who claim to have seen Earth as a sphere are astronauts. But are astronauts trustworthy? Lest look at the best, most trustworthy people in history. Jesus, the Buddha, Gandhi, Martin Luther King Jr, Abraham Lincoln, George Washington, Bill Gates, Steven Spielberg. The list goes on. Notice anything these people have in common? None of them are astronauts. Next.

# 6.

The established model of our Solar System says that the Earth is a sphere but the Earth is not at the center of the Solar System. But you know what does have a center? Spheres. Fishy.

The evidence is clear. The Earth cannot be a sphere. So it seems pretty natural to conclude that the Earth is flat. That what the FE’s did. But they missed one key fact. If the Earth was flat then it would have an edge. But people are naturally not edgy. For example, the PC movement. Objectively the PC movement makes no sense until you realize that the PC movement is against edginess. Towards our nature of smoothness. Smoothness like a torus.

So how did I figure out that the Earth was a torus? It came down to 8 observations.

# 1.

Donuts. Donuts make no sense. Why are they shaped like a torus? It would be much easier to just make them spheres. Why take the effort to cut out the holes and make them into toruses? It’s almost like toruses are in our nature. We have a natural tendency to make things into the shape of a torus. Additionally, people love donuts. But this makes no sense. Donuts are fried sugar bread. That’s gross. If someone offered you a piece of bread you would never say “Sure, but only if you fry it and then marinate it in sugar.” That would be fucking disgusting and everyone would think you were a repulsive moldy creep. Likely a rapist. Maybe even a fedora wearer. (Immediately after I wrote this I felt like I had copied it from some bit by Jim Gaffigan but I looked online for a while and couldn’t find anything. Oh well.) But everyone loves donuts. Why? Because they are torus shaped and we are naturally drawn to toruses. Because our Earth, the thing we come from, is a torus.

# 2.

Continuing along the lines of the first point. People are also naturally drawn to holes. The greatest invention of all time. The wheel. Why is there a hole in it? We could have easily created the wheel as a sphere. But have you ever seen spherical wheels?

No. Why? Because people naturally shape things with holes in them. People try to make things like toruses. Like our earth. Also, wells. For millions of years people have gotten their water, the most essential thing for life, by digging holes. Whys is this the case. Because we love holes. There are countless other examples. I’m not even going to talk about the relationship between holes and sex.

# 3.

There are a fuck ton of asteroids in the solar system. So why is Earth not constantly bombarded by asteroids? It’s almost like there is a giant hole in the middle of earth for asteroids to go through.

# 4.

Stars. They say stars are giant balls of burning gas light years away. But I don’t buy it. If stars are that far away then how come I can see them? I can barely see across the room. They say it is because they are so big. But if there existed balls that big completely made of fire, wouldn’t I be dead? Fishy. No. Actually there are no such thing as stars. What we think are stars are actually the lights of the cities across the world.

The same thing goes for planets. They are just really big cities. You might say no, that can’t be true. The planets and stars move. But actually, it is an illusion caused by the fact the sun moves and reflects differently off of them. You might also point out that stars are mentioned in old books like the bible before we had lighted cities. But are they actually mentioned?

# 5.

The moon. This one stumped me for a while. But then I realized something. The moon is round. What else is round? Dumplings. Where are dumplings from? China. That’s when I figured it out. The moon is China. Or at least what we think is the moon, is China. Like the stars, it’s actually just a place on the other side of the Earth.

# 6.

Underdeveloped countries. We know that all people are equal. So why are some countries more developed than others? This is a seeming contradiction. But it would make sense if some countries only had half as many days in history.

The underdeveloped countries are the countries that are on the outside of the torus. They have only had half as many days as developed countries. The outside parts of the top and bottom of the torus are the North and South poles.

# 7.

The Bermuda Triangle. It’s known that planes and ships are always lost in the Bermuda Triangle. But this doesn’t make any sense at all. Why would things disappear in the ocean? Maybe because the Bermuda Triangle is not ocean at all. The Bermuda triangle is the hole in the center of the Earth. The hole is very big so it’s easy for stuff to get lost in it.

# 8.

Many past civilizations have “figured out” that the Earth was a Sphere. They each measured the circumference of Earth. If the Earth was a sphere, each of these civilizations would have found the exact same circumference. They used math and math is exact so getting any different results already proves that it is not a sphere. But why did they actually get different answers? Because they were measuring the circumference of the torus from different angles.

So there you have it. Proof that the Earth is a torus. Remember to follow this blog. You know what they say, “People who don’t follow blogs are nothing but hollow logs.”

# 4 Things I Don’t Understand About Comment Sections on the Internet

## 1. People Responding to Trolls

In every comment section there will be some troll that posts something inflammatory that clearly serves no purpose other than to get a reaction out of other people. There are two common types of responses. The first is to actually try and argue with them. The funniest part is that what the troll posts is obviously wrong and completely ridiculous but the person who responds does not even address that it is ridiculous and finds some local part of the troll’s “argument” to attack like it is a real debate.

The second type of comment is just yelling at the troll telling them that they are an idiot, which they obviously are. What I don’t understand about this type of comment is that the commenter must know they are a troll that only wants to get a reaction out of people and decides the best course of action is to yell at them. This is the only possible explanation I can give to what people are thinking when they respond to trolls.

## 2. 10 People that Comment the Exact Same Thing.

Inevitably in posts that elicit opinions you will get ten people, often times more or less in a row, that all literally say the exact same thing. Really, do you think you are that important that it is not enough for people to see this opinion. They need to know that you came up with it yourself. Its okay if you acknowledge that your opinion has already been expressed and say something like “I agree with X” or “Like people have been saying..” but people craft the exact same complex argument like it is completely original. Even worse, but happens just as much, is when the post is some puzzle with an objectively correct answer. And you will get 20 people that all post the same correct solution. Why is this socially acceptable? Do you really feel the need to show strangers that you were smart enough to solve a puzzle?

## 3. People Responding to Famous People that they Have Never Met like a Close Friend.

This happens every time anyone remotely famous posts anything. For example a famous person will post their breakfast (Something I don’t condone of btw) and say something like “Better than sex” and then some random person who has never met the poster in their life will respond with “Not that you would know ;)”. Or even worse, the commenter will know something about the famous person’s personal life and use it to craft some “inside” joke. “I hope you didn’t get any ideas about putting those roller blades on. ‘Cause we both know what happens with you and roller blades.” I’m not sure if this happens too often with really famous people but it is constant with kind of famous people.

## 4. People that Disagree with Long Thought out Articles by Experts with Hubris and No Explanation.

Now, I have no problem with people disagreeing with experts if they have a good argument and they act with some humility. But, for example, people will respond to a 10,000 word article about the complexities of nuclear energy and why it could be promising with “Actually, nuclear energy is bad.” WTF do you expect will result in your comment. Will someone finish the article thinking “Wow, nuclear energy actually could end up being really good” and then read your comment and be like “Well, I guess I and the person who wrote this article were wrong. Nuclear is energy is bad”? Now sometimes this is a troll, but more I often I think it’s an incredibly serious person who really does just think their opinion matters that much. Pure narcissism

# Groups

## I’m Sorry

I’m sorry. I know I said I wouldn’t make any posts about math but this is the only one, I promise. Anyway, my motivation for writing this was that Group Theory is considered one of the hardest undergraduate math classes. But the ideas in introductory group theory are actually very simple. And moreover, in a standard undergraduate course, there aren’t even that many ideas. What makes group theory hard is that there is so much new terminology being thrown out that it can be hard to decipher what is even being said.

To the newcomer, learning group theory is kind of like trying to read Jack and the Beanstalk in French with only a French to English dictionary. You would understand each word individually but by the last word in a sentence you likely would have forgotten what the sentence was about in the first place. So even though the story of Jack and the Beanstalk is not that complex, you might finish the story with not idea about what happened at all. Learning group theory, there are so many new terms that you have to constantly be flipping to different pages to your notes to reference different definitions and in the process you lose sight of what is really happening. (Group Theory is definitly not the only area of mathatics like this. I think its just one of the worse offenders)

Really, the only answer to this problem seems to be just doing a lot of excersises until you eventually internalize all the definitions but I would like to try and cheat. It probobly won’t work and trying to not include terminology will just make everything harder to understand but I have nothing better to do with my time.

So these are my sparse notes containing as little terminology as possible and only focusing on the fundamental ideas. Really, there are only a couple ideas needed to understand group theory at this level but these ideas are often lost in traditional texts, shrouded by mountains of terminology and theorems that are actually obvious if you understand the fundamental ideas. So if you can get through these notes and really understand them, then I think you should be able to learn all the rest of undergraduate group theory very quickly.

These notes are written at a basic level so anyone that has taken precalc should be able to understand everything. This does mean I simplified some stuff a little which to a mathematician would probably be heresy but I don’t think it will affect your understanding at all.

I alluded to this above but I wan’t to make explicit these are very short notes and nowhere near all of a standard course in Group Theory. Just what I think is the most important and a very solid foudation for understanding everything else.

## What is a Group?

A group is set together with a binary operation $*$. A binary operation just means that you can combine any two elements to get another element. Addition and Multiplication of numbers are both binary operations. As we’ll see lots of sets of numbers are also groups. Anyway, in order to be a group the binary operation must satisfy the following requirements.

1. Associativity: $(a*b)*c = a*(b*c)$
2. Identity: There exists a unique element of the group called the identity, we will always refer to it as 1, so that for any element of the group $a$, we have $1*a = a*1 = a$
3. Inverse: For any element of the group $a$, there exists a unique element of the group $a^{-1}$ that satisfies $a*a^{-1} = a^{-1}*a = 1$.

This is a lot to digest so let’s spend some time understanding what this actually means. First, from now on we will refer to the binary operation as “Multiplication” This doesn’t mean multiplication in the traditional sense you are used to. We just use it because we want a shorthand for “using the binary operation to compose two elements.” To avoid confusion, I will write traditional multiplication if I mean the traditional multiplication of numbers you are used to.

Associativity just means that the order of how you multiply doesn’t matter. This means we eschew parenthesis so we will write $(a*b)*c$ as $a*b*c$ and we will actually just drop the $*$ because it is annoying to write. So it will just be $abc$.

In addition, $0+ \text{something} = \text{something}$. In traditional multiplication $1\times \text{something} = \text{something}$. The identity is a way to generalize this idea to groups.

In addition, $\text{something} + -\text{something} = 0$. In traditional multiplication, $\text{something} \times \frac{1}{\text{something}} = 1$. (As long as “something” is not zero of course). Again, inverses are a way to generalize this idea to groups.

Notice that communitivity is not one of the requirements for something to be a group. This means that in some groups it is not always true that $ab = ba$.

Lets look at some examples.

Example 1. Real numbers where the “multiplication” is addition. Here, 0 is the identity and negatives serve as inverses. (This might be confusing because we usually denote the identity as $1$ but in this case it is $0$ and $1$ is just some random element with not special importance.

Example 2. Real numbers where the “multiplication” is traditional multiplication. Actually, this not technically a group. (why?) Because, while 1 is an identity and $\frac{1}{a}$ is the inverse for most $a$, notice that there is no inverse for zero. There is no number that we can multiply zero by to get 1. Okay, so the real numbers using traditional multiplication is almost a group but not quite. But if we just removed zero our problems are solved. So the real numbers without zero is a group where the “multiplication” is traditional multiplication.

Example 3. The integers where the “multiplication” is addition.

Okay, these past examples are all things you are probably familiar with and they turn out to be groups. Next were going to get into groups that are really close to numbers but you might not have seen it before, after that we will look at examples of groups that are nothing like numbers. Additionally, from now on, all groups we will examine will be finite. And assume all groups I am talking about are finite. (Being finite just means having finitely many elements).

Example 4. Mod 5 where the “multiplication” is addition. Mod 5 means the elements of the group are the non-negative integers less than 5, and if you would add two numbers and get something greater than 5, you take the remainder modulo 5 instead. So for example,

3+4 =2

2+4=1

3+2 =0

and

2+2 =4

The identity is 0 and the inverse of something is 5-something (unless something is 0, in which case the inverse of 0 is 0). So the inverse of 4 is 1 and the inverse of 3 is 2. Obviously “5” can be replaced by any positive integer and you will be left with a group.

Example 5. Mod 5 where the “multiplication” is traditional multiplication and the elements are 1,2,3,4 (notice 0 is not an element of this group). This is the same as mod 5 with addition except you multiply instead of add. So for example.

$3\times 2 = 1$

$4\times 3 = 2$

$4 \times 2 = 3$

and

$3\times 1 = 3$

The identity is $1$. The existance of invereses might not be obvious but notice that is sufficies to see that for any $a,b,c$

$ab = ac$

implies $b=c$ (why?) But notice that $ab = ac$ is equivilant to

$a(b-c) =0$

which can only happen if $b = c$. Notice that mod n does not always form a group under multiplication though. For example in mod 4, 2 does not have a multaplicative inverse. See if you can find necissary and sufficient conditions for mod n to be a group with traditional multiplication.

Now we’re going to see some groups that don’t look anything like numbers at all.

Example 6. Permutation on 4 elements. The elements of the group are the permutations of $1,2,3,4$ which recall are just, 1234, 1243, 1423, 4123, 1324, 1342, 1432, 4132 ect. (You get the idea). We can interpret a permutation as one-to-one correspondence between the set $\{1,2,3,4\}$ and itself by interpreting the permutation where a $i$ is in the $j$th place as $f(j) = i$. So for example, 1432 would be the function where

$f(1) =1$

$f(2) = 4$

$f(3) = 3$

and

$f(4) = 2$

And 4321 would be the function where

$f(1) = 4$

$f(2) = 3$

$f(3) = 2$

and

$f(4) = 1$

hopefully you get the idea. Multiplication is just function composition so

$(4321)(1432) = 4123$

This is the first example where the multiplication isn’t commutative; notice that

$(1432)(4321) = 2341$

The identity is the identity function 1234 and the inverse is just the inverse function. For example the inverse of 1432 is 1432.

## Playing With Groups

Lets just tinker around a little to get a feeling for how groups behave.

Suppose that in some group

$ab = ac$.

We can multiply both sides of this equation on the right by $a^{-1}$. So we’re left with

$b = c$.

This is an important example of a property of groups and that is that by using inverses, we can essentially cancel two terms that are the same on both sides of an equation.

Subgroups: Start with a random group $G$ and some $a$ that is an element of $G$. What if we multiply $a$ by itself? Then we get another element of $G: aa$. We’ll call this $a^2$. Okay lets multiply that by $a$. Now we’re left with $aaa$ which we’ll call $a^3$. And we can keep doing this generating a sequence $a, a^2 \cdots$. Now, remember earlier we said we will assume that groups are finite. This means that at some point our sequence must start repeating. So we have $a^j = a^k$ for $k > j$. Using the cancellation we saw above, we are left with

$a^{k-j} =1$

Assuming that there is no positive integer less than $k-j$ satisfying $a^n =1$, we see that the elements $1, a, a^2, \cdots a^{k-j-1}$ actually forms a group on its own. And this looks awfully similar to mod $k-j$. This is what we call a subgroup. A subgroup is a subset of a group that itself forms a group. Not all subgroups arise by multiplying an element by itself though. For example, if you start with any subset of elements  $g_1, g_2, \cdots g_n$ of a group, then the set of all elements you can get by only multiplying together elements from $g_1, g_2, \cdots g_n$ will form a subgroup. (If it is not clear I meant with replacement. So ${g_1}^2$ or even $g_2g_1{g_2}^4$ would be in the subgroup)

Using the notation above, what is $(a^3)^4$. Well, this is just $a^3 a^3 a^3 a^3$ and that is just

$aaaaaaaaaaaa = a^{12} = a^{3\times 4}$

so we see that the exponent-like notation for groups works the same way we are used to with numbers. Similarly, $(a^n)^{-1}a^n = 1$ so multiplying both sides on the right by $(a^{-1})^n$ yields

$(a^{-1})^n = (a^n)^{-1}$

which we will write as $a^{-n}$ like we are used to with numbers.

If we’re given two groups, can we smash them together to make one big group? Of course we can. For two groups $G$ and $H$, we define the catesisan product of $G$ and $H$ to be the group $G \times H$. In other words for any $g$ in $G$ and $h$ in $H$, we can put them together to make an element $g \times h$ of $G \times H$. See if you can figure out what the multiplcation is before I tell you. Okay, here it is, for $g_1, g_2$ in $G$ and $h_1, h_2$ in $H$,

$(g_1 \times h_1)(g_2 \times h_2) = (g_1g_2 \times h_1h_2)$

Verify that this is in fact a group.

## Understanding the Homomorphism

So far we have looked at individual groups. Now, we turn our attention to examining how we can find similarities between different groups. To do this we use functions.

We could study any old function between groups but if we don’t put any contraint on them then we’re not using the fact that these are groups. They might as well just be sets.

Instead we focus on Homomorphisms. A Homomorphism is a function between groups that preserves the group structure. What does this mean? Well since we wanted to study similarites between different groups, we want to look at functions where you can multiply stuff and then apply the function, or apply the function and then multiply stuff and end up with the same result.

Fomrally, a Homomorphism is function $f$ between two groups satisfying.

$f(a)f(b) = f(ab)$

Just this constraint makes sure that the function preserves a lot of stucture. For example notice that

$f(1)f(a) = f(1a) = 1f(a)$

And we can use the cancellation property we saw above to deduce that

$f(1) =1$

$f(a)f(a)^{-1} = 1 = f(aa^{-1}) = f(a)f(a^{-1})$

so

$f(a)^{-1} = f(a^{-1})$

Next, lets start with an observation. If $f(a) = 1$ and $f(b) = 1$, then $f(ab) =1$. Aditionally, if $f(a) = 1$ then $f(a^{-1}) = 1$. Therefore the set of all elements of a group that get mapped to $1$ by a homomorphism form a sub-group.

Now, suppose that this subgroup is $a_1, a_2, \cdots, a_n$. Notice that for any element $b$ in the group,

$f(a_1b) = f(a_1)f(b) = f(b)$

And the same is true for any of the $a_i$. The converse is also true. Suppose that $f(b) = f(c) = d$. Then we have

$f(b) = f(c)= f(bb^{-1}c) = f(b)f(b^{-1}c)$

Therefore, $f(b^{-1}c) =1$. In particular this mean that exactly $n$ elements of our group get mapped to $d$. Since $d$ was arbitrary, this means that for any element that can get mapped to by our homomorphism, there are exactly $n$ elements of the group that map to it.

We can also partition the elements of a group into Equivalence Classes by what they map to for some homomorphism. What we just saw above is that each of these equivalence classes are the same size and that two elements $a \text{ and } b$ are in the same equivalence class if and only if

$f(ab^{-1}) =1$

Or to put it into perhaps simpler terms, there exists a $c$ such that $f(c) =1$ and $ac =b$. We call these equivilance classes the Quotient of the group by the subgroup. This is written $G/H$ for a group $G$ and subgroup $H$.

So really what a Homomorphism does, is take a subgroup and wraps it up into one element. Then other elements of the group also get wrapped into one thing if they “differ” by some element of the subgroup.

For example, take the homomorphism that maps mod 10 into mod 5 by $f(x) = x$ for $x < 5$, and $f(x) x-5$ for $x 5 \leq x < 10$. Or put more simply, just takes mod 5. The subgroup that maps to $0$ is $\latex \{0, 5\}$, and the equivilance classes are $\{0, 5\}, \{1, 6\}, \{2,7\}, \{3, 8\}, \{4, 9\}$. See how the homomorphism is taking mod 10 and squishing it down to mod 5.

What happens if no squishing is taking place? What if, for some homomorphism, $f$ the only element that maps to $1$ is $1$. And moreover ever element of the group $f$ maps to can get mapped to using $f$. Then nothing is really happening at all. The group getting mapped to and the group getting mapped from are basically the same. $f$ is just renaming elements. We’re just taking an element $a$ and disguising it as $f(a)$ but it still behaves exactly like $a$. If you put glasses on a cat, its still a cat. In this case we say the homomorphism is an Isomorphism. And if there exists an isomorphism between two groups, we say the groups are Isomorphic.

Always rememebr, Isomorphic basically just means the same. No squishing.

Wouldn’t it be nice if the quotient of a group by a subgroup was a group? Unfortunately. If the subgroup is $H$, then we would like there to exist an $h_3$ in $H$ so that

$ah_1bh_2 = abh_3$

for any $a$ and $b$ in $G$ and $h_1, h_2 in$latex H$. Certaintly it would be true if the group were communative but remember that groups don’t necissarily have to be. We don’t even need communativity, we just need to slide that $b$ to the left of the $h_1$ and we don’t mind if in the process we turn the $h_1$ into a different element of $H$. What we need is there to exist and $h_4$ in $H$ so that $h_1b = bh_4$ or more concisly, for any $h \in H$ and $b$ in $G$, $b^{-1}hb \in H$ This allows us to slide that $b$ right past the $h$. In this case we say the subgroup $H$ is Normal in $G$. Just like in real life, being normal is a desirable property. ## Actions How can we use group theory in our everyday lives? We can’t, but it can be useful in other areas of mathematics. One of these ways is with actions. We can think of an action as a way to multiply elements of a group by elements of a structureless set which will in turn give structure to the set. It satisfies some properties to make sure being a group matters: An Action of a group $G$ on a set $S$ is a function $*$ from $G \times S$ into $S$ (which will be written $g*s$ instead of $*(g,s)$ because that’s annoying to write) which satisfies for any $g$ and $h$ in $G$ and $a$ in $S$. 1. $g(h*a) = (gh)*a$ 2. $1*a = a$ Actions are used to solve the Rubik’s Cube. See math is valuable in real life. I won’t talks anymore about actions beyond stating their definition because most of their introductory theory is very similar to what we have seen already with Homomorphisms. The excersises will provide a way to get an understanding of actions. ## Excersises These excersises might be too hard but I couldn’t find easier ones so deal with it. 1. Show that the intersection of two subgroups is a subgroup. 2. Find necissary and sufficient conditions for mod n with traditional multiplication to be a group. 3. (Legrend’s Theorem) For a group $G$ and subgroup of $G$, $H$. Show that the number of equivilance classes in the quotient of $G$ by $H$ divides the number of elements in $G$. 4. The order of an element $a$ of a group is the smallest positive integer $n$ such that $a^n =1$. Show that the order of each element divides the number of elements in the group. 5. If a group has an even number of elements show that the number of elements of order 2 is even. 6. Show that if $f$ is a homomorphism from $G_1$ to $G_2$ and $g$ is a homomorphism from $G_2$ to $G_3$ then the composition $g of$ is a homomorphism. 7. A group $G$ is abelian if its elements commute i.e. $ab = ba$ for all $a,b$ in $G$. If $G$ is an abelian group and its elements are $a_1,a_2, \cdots a_n$, show that $(a_1a_2\cdots a_n)^2 =1$ 8. Prove the Second Isomorphism Theorem: Let $G$ be a group. Let be a subgroup of , and let $N$ be a normal subgroup of . Then the following hold: 1. The product $SN$ is a subgroup of , 2. The intersection $S \cap N$ is a normal subgroup of , and 3. The quotient groups $(SN) /N$ and $S/(S\cap N)$ are isomorphic. 9. Suppose we have a group $G$, a set $S$, and an action $*$ of the group on the set. Show for any $x \in S$ that $|G| = |\{g \in G: g*x =x\}| |\{g*x: g\in G\}|$ (Note that |A| denotes the number of elements in a set $A$) 10. Suppose we have a group $G$, a set $S$, and an action $*$ of the group on the set. An orbit of $S$ under the action $*$ is a set of the form $|\{g \in G: g*x =x\}$|where $x \in S$. If $O$ is the number of distinct orbits of $S$, Show that $|O| = \frac{1}{|G|} \sum_{g \in G} |\{x \in S: g*x = x\}|$ 11. (Hard) Prove Cauchy’s Theorem: If $p$ is a prime number and $p$ divides the number of elements in a group $G$, then there exists a subgroup of $G$ with exactly $p$ elements. 12. (Hard) Prove Sylows 1st theorem: If $p$ is a prime number and$p^k\$ divides the number of elements of group $G$, then the number of elements.

13. (Hard) Show that all Abelian groups are isomorphic to a unique product of cyclic groups.

14. (Really Really Really Hard) Show that if $G, H_1,H_2$ are finite groups and $G \times H_1$ is isomorphic to $G \times H_2$ then $H_1$ and $H_2$ are isomorphic.

(Funny/Infuriating Story: When I took this course last year, the teacher was trying to prove a theorem. I sketched a proof in my head using this claim (I thought is should be simple to prove and I had a sketch of the claim in my mind) and went back to doing something else. Like 30 minuets later I looked up and the professor was still doing the same proof. He was struggling and looked confused I raised my hand and offered my outline. The professor told me it wouldn’t work because this claim wasn’t true. I was puzzled because it seemed so intuitivly obvious. I tried to actually prove it but realized that the obvious technique did not work. After class I asked the professor if he could explain to me why it wasn’t true or offer a counterexample. He told me it wasn’t true because the obvious method to try to prove it didn’t work! He even went through the work of doing the obvious method which I told him I had already tried and when it didn’t work he exclained “See this doesn’t work. The claim isn’t true.” This was a a professor of mathematics. I went home and looked up the claim online and sure enough it was true.)

# MY TOP 10!!! FAVORITE MUSIC ARTISTS OF ALL TIME!!!!!!!!!!!!!!!!!!!!

I do want to limmit the number of trashy posts like this and get to something with real substance but I have such great tastes in music that it would be unfair to the readers to withold this information. Now I’m not saying my impecible taste in music makes me better than everyone else becasue there are lots of things that can make one human have more value than another. All I can say is that I am confidant that by the end of reading this post, your life will be forever changed for the better.

## 10. Lorde

If this is your first time reading this blog, no, I am not a forteen year old girl that likes to wear dark makeup, Lorde is just a legitimatly good artist. She is a great singer and her songs contain a lot of real emotion and substance.

Most Famous Song: Team

Most Catchy Song: Yellow Flicker Beat

Song that gets Worse evertime you listen to it: Tennis Court

Most Underrated Song: Glory and Gore

## 9. The Gorillaz

I know this is a boring but the Gorillaz were the first group that once I found, I immedeatly listened to all their songs. I hadn’t listened to them in a while before writing this post but I can tell you their music is still good.

Most Famous Song: Feel Good Incorperated

Most Catchy Song: Up On Meloncholy Hill

Most Overrated Song: Sleeping Powder

Best Song: Clint Eastwood

## 8.  The Swingrowers

The Swing Growers are an up and coming group that play electro swing. Their songs are extreamly catchy and they are establishing a distinct style. Their second ablbum is infinitly better than their first so if they continue the trend, they could rise further up this list.

Most Famous Song: That’s Right

Most Catchy Song: Butterfly

Most Overrated Song: That’s Right

Most Underrated Song: Frank

Song that Gets Better Every Time you Listen to it: Stay Swing

Best Song: Midnight

## 7. Old Jazz

This is where if my blog had a larger following, I would get a barrage of comments telling me how different all these musisans are. But the reason I put them all together is that I really like listening to old jazz music but often I don’t pay attention who does what and aditonally I didn’t want to take up too much room on this list. You probobly hate old jazz music and I used to as well. But the thing is, once you get used to it and listen to it enough ,it gets really good.

Most Famouse Song According to the Internet: So What

Most Catchy Song: Yardbird Suite

Most Overrated Song: Autumn Leaves

Best Song: ‘Round Midnight

## 6. Mikly Chance

I’ll admit I havn’t actually listened to Milky Chance that much, but making this list, I realized how much I like all of their songs. Their music is not very diverse so maybe if I listened to it more I wouldn’t like it as much but whatever.

Most Famous Song: Stolen Dance

Most Underrated Song: Flashed Junk Mind

Worst Song: Down by the River

Best Song: Stolen Dance

## 5. Caravan Palace

Caravan Palace is probobly the most polular electro swing group. Like the Swingrows, there first album was kind of trash but their newest ablum, <|°_°|>, is really good. Their music is pretty diverse and all of it manages to be great. (At least all their new stuff)

Most Famous Song: Lone Digger

Most Catchy Song: Dramaphone

Song that Gets Worse Every Time you Listen to it: Wonderland

Song that Gets Better Everytime you Listen to it: Comics

Most Underrated Song: Midnight

Creepiest Song: Aftermath

Best Song: Human Leather Shoes for Crocadile Dandies? (This was really hard. Carvan Palace has a ton of songs that really good and there is no clear best one.)

## 4. Klingande

Klingande is the first of two on this list in the genre of Tropical House. Tropical house is a subgenre of Deep House which is a subgenre of House. House music is a type of EDM which is very repetitive and has a strong 4/4 beat. Deep House is much more complex than regular House and borrows ideas from jazz and soul. It is usually very relaxing and the chord progressions are often dissonant. Tropical house is more uplifting, less dissonant, and usually a little slower than Deep House.

Most Famous Song: Jubel

Worst Song: Riva

Best Remake of an Old Pop Song: Pumped Up

Best Song: Punga

## 3. Eminem

I like most of the big late 90’s early 2000’s rappers but Eminem is the one I’ve listened to most. Eminem had great albums as late as like 2010. Even “The Marshall Mathers LP 2” was pretty good. His songs are incredibly clever even though the lyrics can be pretty offensive. Although that’s kinda true of most rappers and something you just have to get passed. In 30 years people will probably be appalled that rap music like this was popular but hey, it’s socially acceptable to listen to it now so whatever.

Most Famous Song: Lose Yourself

Song that Gets Worse Every Time you Listen to it: Monster

Song that Gets Better Every Time you Listen to it: Groundhog Day

Most Catchy Song: Amityville

Most Underrated Song: Untitled

Best Song: Stan

## 2. Bakermat

Bakermat is just a strictly better version of Klingande. He pretty much invented Tropical House but also does some shit in straight deep house. His songs just make you feel good when you listen to them.

Most Famous Song: One Day

Song that I’m Embarrassed How Much I like: Ballade

Song that Gets Worse Every Time You Listen to it: Games Continued

Song that Gets Better Every Time You Listen to it: Zomer, Gone

Best Song: Uitzicht

## 1. Parov Stelar

Finally, we are left with the G.O.A.T. The average Parov Stelar song might be worse than the average Bakermat song but the thing that makes Parov Stelar number one is how prolific he is. He has so many songs across so many genres, from pop, to electro swing, to Jazz, to I don’t even know what, and all his songs are amazing.

Most Famous Song: Booty Swing (Not sure why this is his most famous. It’s good but kinda boring tbh.)

Song that gets worse every time you listen to it: Walk Away

Song that gets better every time you listen to it: Literally every other song.

Most Catchy Song: Chambermaid Swing

Song Featuring the Most Insane Dance Video: Catgroove (Watch This)

Best Song: There are too many good ones. Some of of the best include Heaven’s Radio, Autumn Song, Menage A Trois, Summertime, and The Lonely Trumpet but there are way more than I can list.

Okay thats my list. You’re welcome. There were way more artists I couldn’t fit on this list  I’ll leave you with one last song that is on a whole different level. Remember, if you don’t follow this blog, I hate you.

# The Kinda Growth Mindset

There are two ways people view Intelligence. The first is with a Growth Mindset. A growth mindset is when you believe people’s abilities are mostly due to how much they worked at something. Not innate intelligence. People with a growth mindset believe that people’s abilities can change and that hard work will always beat talent.

On the other side, there is the Fixed Mindset. A person with a fixed mindset believes that talent and intelligence are mostly innate. Sure, dumb people can work hard and achieve something, but it won’t be as good as someone who was born gifted.

Research has shown that the growth mindset is probably more accurate although we still don’t know exactly how much genetics plays into intelligence. But as long you don’t have some mental disability, you can, for the most part, get as good at something as anyone else as long as you work hard in the right way.  (Obviously this is not true of some things requiring certain genetics like being in the WNBA)

Maybe more importantly, research has shown that people that have a growth mindset achieve much more than people with a fixed mindset. People that believe they can work hard and get better at something, do, and they end up much better off than the people that think everything is innate and intelligence is fixed.

Many people know about this and claim to have a growth mindset. Hard work is admired and praised, but at the same time hard work is also stigmatized. Despite everyone claiming to have a growth mindset, everyone also believes that hard work is for people whose raw abilities can’t cut it. Hard work is looked at paternalistically. People view hard work the same way suburban moms view vaccinations, the lesser people should do it and if no one did, everyone would probably be dead, but Baby Billy is just a little to good for it.

Pervasive in my high school seemed to be the secret equation everyone knew but no one would explicitly say

$\text{Smartness in Subject} = \text{Grade}-\text{Time Studying}$

I certainly found myself getting sucked into it. I would find sligh ways to humblebrag about how little time I spent studying for some test and then immediately hate myself for it. This is a big problem in our culture.

So why, when we know better, when we know how toxic it can be to think this way, does this idea still progate? Let’s first look at what we actually believe. I think a lot of people think they have a growth mindset but what they actually have is a “Kinda Growth Mindset.”

Let us turn our imagination on for a second and use it to imagine Tamatha, we’ll call her Tammy for short. Tammy was an average gal. But, Tammy watched all five season of Breaking Bad three times in high school. As a result, Tammy was inspired to major in Chemistry in College. She fell in love with Chemistry, worked really hard, and graduated with honors. Currently Tamatha, sorry Tammy, is in the middle of a PhD program for organic chemistry at MIT. And she’s killing it.

Now, we can rate all possible beings with consciousness on a scale from one to ten on how good they are at chemistry.

Currently Tammy is a 6.4.

So the question is, how much work will Tammy have to put in to get to the next level?

If you have a kinda growth mindset your first reaction is probably that it depends. How quickly does Tammy learn? How smart is Tammy? There is currently not enough information to tell how hard Tammy will have to work. But this is where the fallacy lies. Someone with a true growth mindset would tell you that how hard Tammy has to work to get to the next the level depends only on Tammy’s current level. Any person who has a rating of 6.4 has to work the amount to get to reach that big 7.

So someone with a Kinda Growth Mindset believes that people can work hard to increase their intelligence but they also believe how hard that person has to work depends on their innate intelligence. No matter how much someone has grown, a person with a Kinda Growth Mindset will think that the person who started out dumb will have to work much harder than someone who started out smart to progress further.

Now, I do want to clear something up. Being a 6.4 in chemistry doesn’t just mean that Tammy knows a lot of chemistry facts. That rating takes into account how well Tammy can reason, how fast she can think, and how much she truly understands chemistry at a deeper level. Another misconception about the growth mindset is that it says everyone starts off at the same level. (You are an obvious counterexample to this) Two six year olds both may have never learned any chemistry but one might be a 1.2 while the other is a 1.4. Whether this is due to genetics, environment as a baby or a combination of the two is unknown but the point is that some people can start out at a higher level. What makes someone have a growth mindset is that they believe anyone can become great at anything and how hard they have to work to get to the next level depends only on their current level.

The fact that some people can start at higher levels might seem discouraging to someone who is not naturally gifted. But actually, if you think this you are wrong. The thing is, it’s very difficult to go from good at chemistry to great at chemistry and even harder to go from great to extremely great, but to go from bad to good, not that much work. And no one’s innate abilities will really get them past being good at something. The above statement might not seem true. You tell yourself you have witnessed close friends, family, lovers, maybe even yourself, all try really hard to get something. Spend hours focusing and still end up feeling the a Mentally Retarded Walrus trying to memorize the periodic table. But this is because they/you are not working hard int the right way.  You might also point out that most geniuses started out gifted from an early age but 1) this is not always true and 2) them being good at it early encouraged them to work hard and get great.

## 1. It’s an excuse for dumb people.

It’s hard to be someone with no accomplishments surrounded by a bunch of assholes projecting a genius vibe. People’s ego’s naturally can’t accept that others are better than them so they make up excuses and “I don’t work hard” is an easy one. You are a really smart person bursting with potential but you chose to relax instead.

At first glance this view actually looks like the growth mindset. People are admitting that they are not smart because they didn’t work hard. But the people dicks who hold this view, believe that they are innately intelligent and hard work is just the vehicle that can express it. Of course, they are secure enough that they don’t feel the need to actually prove their potential.

## 2. It’s Verified by Empirical Evidence

Jane spends hours studying everyday while Tommy spends his free time licking ice cream. At the end, they have the same intelligence. This happens all the time in high school. Kids notice that while their classmates who study sometimes do better on tests they aren’t actually any smarter than their classmates who don’t study. Moreover, a lot of times the kids who don’t study end up doing better on tests. So what else can the kids conclude other than hard work is worthless and all intelligence is innate.

So what’s the real reason study students don’t make smart student? The answer lies in the fact that to increase your intelligence, you have to work hard in the right way. And studying for high school classes is not the right way. You memorize a bunch of facts without thinking a single thought. Our education system is so bad that upwards of 90%  of the material will never benefit anyone in any way. (This statistic was made up by me on the spot.) More on this in my upcoming post on our education system.

So not only does our education system apple fudge every student in America in all the ways we were aware of, it also helps propagate an unhealthy view of intelligence.

## What’s This right way?

A lot depends on what the thing you are trying to get good at is. But for basically anything, you need to challenge yourself. But not too much. For example, if you want to get good at math, find problems that are difficult and make you think. A problem is the right difficulty if you stand approximately a 50% chance of solving it. A good litmus test is if you can’t accurately predict whether or not you will be able to solve it after looking at the problem for one minute it is a good problem.

In general you need to think about what you are weak at and deliberately design exercises to try to fix that. However you are practicing, always be aware of how that practice is helping you. Set goals and keep track of your progress. You’d be amazed at how often people spend so much of their time on something and never really improve. Okay, I’ll stop sounding like a self-help book and wrap this post up.

Whether or not you enjoyed this post, remember to follow this blog. If you scroll to the top of the page, there will be a floaty thing in the bottom right hand corner which you can click on to follow. If you disagree or have any thoughts at all about this post please tell me in the comments.

# “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.