Do you want to get on my nerves?

Then litter!

Obviously, this is sarcasm. Littering is not okay, and it continues to baffle me why people think I do it. For those of you who know me in person and wonder why I have problems with a certain demographic of my “peers,” this is why.

Book – Proofs in Competition Math

This is a book which is about proofs in competition math. This blog post is a promotional one for its release.

Authors: Alex Toller, Freya Edholm, Dennis Chen.

Important Info

Preorders start on March 14th, also known as Pi Day. Yes, this was on purpose. (3/14)

Release is on April 5th. The best way to remember this is you will have to run for your life if you don’t get this book on four-five(4/5)

Why you should pre-order (when the time comes)

First, if you want to see my writing… this book doesn’t have much of it at the time of this post. (Oops.) Unfortunately, I’ve been bogged by other obligations, so I haven’t written much. (If you think the issue of splitting the earnings and me potentially getting money for nothing is an issue, I do too. More on that later.)

But if you agree with what I say on this blog, here’s something you’ll definitely agree with when the book comes out (and even if you don’t agree with me most of the time, you will agree here) – this book is high-quality. I will have proof for that (aka samples) soon, if the other two agree to release nontrivial yet non-significant portions. I’m writing Inversion, so if you liked EEG’s Inversion, you are going to like this one even more.

Also, Alex and Freya are very reputable within the math community themselves (though  I suspect most readers of my blog already know this). (In fact, they’re much more reputable than me, but shh!)

Anyway, this is quality, this is hype, and this covers a lot of stuff.

Actual content

Why you should pre-order: Part 2.

Here’s a rough ToC, with wording that is totally inaccurate.

Part 1: Proofs (this is logic in general, also stuff like iff. Good for beginners.)

Part 2: Algebra

A: algebra for noobs

B: Basic Inequalities + Complex Numbers + just stuff you should know for polynomials

C: Really hard stuff (see newton sums)

Part 3: Geometry

A: normal stuff

B: normal, but more advanced stuff

C: really hard normal stuff + bary/polar/cylindrical + inversion (I’ll get spiral similarity and homothety added if I can)

Part 4: Number Theory/Combinatorics

A: intro+interesting but unimportant stuff (its kind of the equivalent to spiral similarity or inversion)

B: More interesting but unimportant stuff.

C: Classic NT (Bases, mod arith)

D: Hard NT (HELP)

E: Even harder NT. (HELP II)

Part 5: Open Problems + for fun

A: Goes over open problems. Makes no progress but defines the backgrounds. (If we could make significant progress, we’d submit that as an article hm?)

B: Pythagorean’s Proofs (my favorite) and Fake-Proofs (my LEAST favorite).

I’m not being paid for this

I forget if I’ve said this to Alex, but I don’t intend to be paid for this simply due to the fact I’ve done an embarrassingly small amount of work on the book. I will still

a) take responsibility for the final product

b) be involved in marketing

c) actually work on the book now

but the little work I did does not warrant payment. (It’s a wonder I’m on the authors list…) I intend to make up for that by working on it now.

In Conclusion…

Buy the book when it comes out!

I’m a lazy bum too, so I deserve all the blame for flaws because I’d be able to fix them if I was paying more attention and none of the credit for success because I didn’t do anything. (Oops.)

Also, I don’t know where Homothety/Spiral Similarity will go. Preferably next to Inversion, with Homothety before Spiral (since Homothety is a special case of Spiral).

Please support this by sharing with your friends or whoever might be interested!

(If this post seems lazy, it’s because I’d rather get onto writing the book.)

Edit: I’m going to put this in every category, so people see this. I also will update my website soon.


The SAT is Garbage

On Thursday I took the PSAT 8/9. Since I’m contractually bound from discussing any questions, I’ll be talking about the SAT instead and using past questions. Regardless, the point is the same; the SAT, and standardized tests in general, are garbage.

Skill Floor and Skill Ceiling

In theory, it would be nice if we could perfectly tell apart how good people are at something. In practice, this is not possible, but we do want to strike a reasonable balance. The three things we desire are accuracy, a low skill floor, and a high skill ceiling. This means that we’d like our evaluations to be correct within a reasonable range.

Consider video games as an example. Two of my favorite games growing up were Super Smash and Super Mario. Despite being in different genres, the underlying philosophy behind them is the same. Both games reward proficiency in a consistent manner, and are enjoyable for everyone, including those who are not so experienced at it.

Now replace “games” with “the AMC series” and you have a good idea of what makes them so unique. While approachable to beginners in problem-solving, the AMC series is able to accurately distinguish between those who are merely the best in their class and those who have invested their time into getting better at math. Just as I invest time into improving my platforming and platform fighter skills by playing Mario and Smash respectively, I invest time into improving my problem solving abilities by taking the AMCs.

The SAT fails at every count here. Its skill ceiling is the same as its unbearably low skill floor, the accuracy of the test is almost as bad as blind guessing (which, of course, makes sense when that perfectly describes the English section of the SAT), and it’s boring as hell. There’s a very good reason we barely care when someone we regard as smart got a 1600, and we don’t care at all when they don’t.


But I think there ends up being something more important than the inaccuracy and non-existent skill range of the SAT. As major as these flaws are, there is a far larger issue at hand. The SAT is completely boring. There’s no reason to want to take it, the only reason to take it and do well is because “college,” and the questions don’t make people think.

When the AMC or MATHCOUNTS/mathleague tests are over, kids immediately begin discussing the questions.

“What did you get for this?”
“How did you do this problem?”
“There’s actually a nicer way to do the problem; just draw this line, and notice the similarity ratio is 2/3.” (Bonus points for whoever knows which problem I’m referencing!)
“Notice that 13-14-15 triangles have an altitude of 12.”
“How did I get this one wrong?”
“Yes! I got it right!”

These quotes are not word for word, but these get the idea across. Let’s see what people had to say about the PSAT.

“It was really boring.”
“I slept so late last night!”
“You were sleeping during the test? How?”
“I was so close to finishing Mario on my calculator, but I died!” (This last quote was said by me.)

Not so flattering an image, is it?

But MATHCOUNTS suffers many of the problems the SAT does, albeit to a far smaller scale. There is an emphasis on speed over deeper thinking (Countdown Round, anyone?) and the entire Team Round can be described as “unlegit.” So what makes MATHCOUNTS any better than the SAT? MATHCOUNTS is responsible for some of the best times of people’s lives, while the SAT is responsible for some of the worst. (The questions from MATHCOUNTS are also miles ahead of the quality of the SAT questions, but that’s like saying the sky is blue.)

I believe the AMC series and MATHCOUNTS are much more enjoyable than the SAT. This is because the AMCs and MATHCOUNTS build futures; the SAT tears them down.

What Next?

I wish I could find some theoretical remedy to fix the SAT. After all, if it gets fixed before I take the real one, I have no reason whatsoever to complain. But the SAT is one of the few things beyond saving. It really doesn’t matter how good the SAT becomes, and it’s not like anyone’s going to be trying to save the SAT. If someone thought of a good math problem, there are plenty of places to put it. People could submit to the AMCs, to MATHCOUNTS, or make their own competition. (This is one of the reasons Revenge of NIMO was so popular!) One of the last places it would be put is the SAT.

So as of now, my stance is that the SAT should be put down like an old dog. It’s far outlived its supposed purpose, and there is no real need or reason for the SAT to exist.

Rant – What TLDs mean

If you’ve went to public school, I’m sure you’ve heard more than once that “.com is company – don’t trust this, .org and .net and .gov are trust-worthy!” when being “advised” about sources. This is blatantly incorrect.

First of all, .com does not mean anything. It used to mean “company” and still “stands for” company, which is true, but it doesn’t mean company. You might as well say it means “this is a website.” This is because it is the most general-purpose domain. There are plenty of examples of .com companies, but that’s because pretty much every company is going to be using .com. Should I now stop trusting google because it’s a .com? (There certainly may be reasons to not trust google, but being a .com is not one of them.)

The .org domain also used to mean organization, but this is no longer a requirement. Also, .net means whoever registered this domain is probably an old grandpa.

The only two that seem to have a connection with their meaning is .edu and .gov, which I will concede ELA teachers have some things correct about this.

So what does a good “research” website look like? They should first be from someone who knows what they’re talking about. I would not listen to someone talk about TLDs if they’ve never registered a domain. The same is true of research; I would not listen to someone talk about something they don’t know. They shouldn’t look like “tHE BEGGEST STATE IN uNITED sTATS OF aMERIKA IS aFRICA” (neither in content nor in grammar!), preferably should be someone who knows how to write well (you will find most smart people can do this), and definitely should not be a website with excessive advertisements. (I find the final is one thing most teachers tend to overlook when “advising” about Wikipedia. I commend their work of making no advertisements on the site.)

The stance of ELA teachers on TLDs is so extreme while being unfounded it’s unbearable; especially when they talk about domain names as if they know anything about them, even though they haven’t even registered one. (And guess who has?) I think it’s safe to say that ELA teachers who still use google sites should not be advising the more “tech-savvy generation” on domain names.

(Disclaimer: I may be overestimating the average intelligence of internet denizens. Perhaps ELA teachers’ advice, though awfully misguided, is still going to be better than the initial habits of students. I personally doubt this.)

School Spirit

When spirit day comes around and you have an opportunity to wear your school shirt/school colors and the like, how many people actually do it? None, unless they accidentally so happened to wear the school colors, which is fairly unlikely.

So why do you think your school does “Pajama Day” or “Cool Shades Day?” Because who doesn’t like being allowed and encouraged to bend the school rules, especially since they have so few chances to? The only more popular day would be the “Chew Gum Day.”

We also have incredibly annoying acronyms. For example, “ARE” stands for “Accountability, Responsibility, Empathy” if I recall correctly. (Either way, it’s something incredibly edgy.) How about we make that more accurately “Actually Contrived, Really Contrived, and Extremely Contrived?”

So why does the school do this, and what could it do to actually get people not to hate the school (and its spirit days)? Well, let’s ask another question: Why does nothing else that actually gets students to care about them (the glaring exception is school sports teams, and that’s a maybe) have “spirit days” or other stupid stuff?

Let’s ask ourselves a question; how would we describe ourselves? We would obviously describe ourselves. Nobody holds the school they go to that dearly until college, and maybe a little bit during high school, but that’s it. But I don’t think the first thing I’d be telling people about is where I go to do math, or which cross-country team I’m on. I will probably mention them fairly early on, but I probably won’t be too worried about the details.

But here’s the thing; we only mention things which we a) really love or b) really hate. Why do you think kids talk about schools and video games so much? Because they hate school, and love video games. I don’t think an interesting conversation could happen on something you’re neutral about; have you seen a blog post from me saying, “I ate an apple. It was alright. I also ate some bread. It was pretty good,” or something to that effect? No! Because I don’t care about apples or bread, and anyone reading this blog probably doesn’t care about apples or bread, and if they do, they’re not interested in mediocre apples and bread!

So spirit days ironically succeed halfway, putting the school in a worse position. Sure, they get us to talk about the school, but I don’t think what kids are saying really lines up with the whole annoying “positivity” attitude that schools have going for them.

School is Broken

Anyone who has attended public school will nearly unanimously tell you that “school sucks.” Or, as I prefer to say (and Evan Chen prefers to say), “school is broken.”  What does that mean? This means there are fundamental roadblocks to the learning of students and the teaching of teachers set up by the system, and incompetency by those in charge is the reason it remains.

To address this obvious issue, we must first ask ourselves what the point of teaching is. I argue that “teaching” in the traditional sense is not even remotely necessary for motivated kids; and for unmotivated kids, they do not learn anything or learn very little from this public schooling system.

Postulate 1:

We accept the point of teaching as to hasten the growth of or develop understanding of connections in a particular subject and to develop specific and general intuition/problem solving skills in motivated students.

There are a variety of methods for this. Books, online classes, handouts, problem sets, and traditional teaching all place different emphasises. This means that only going through handouts/problem sets will be challenging (but still possible!) to develop higher skills in motivated students, only going to traditional classes will be very lackluster, and a union of traditional teaching and going through material may be an efficient method for most motivated children. (The “motivated” condition cannot be dropped!)

So what is a traditional class about? Well, traditional classes are limited resources; they require immense amounts of time and effort to set up, and motivated students will only absorb a fraction of the information. (Better to give too much to absorb than to bore your students with not enough!) Traditional classes are about pushing regions of math students have not explored before; traditional classes make new ideas more accessible for motivated students. What of the unmotivated students? Trying to teach unmotivated students is pointless, because they will not care about absorbing the information. So we should not be focused on getting unmotivated students to learn; first and foremost, we should be getting them to be motivated. Let’s make a note of that.

It doesn’t make much sense to try and teach unmotivated kids, so we should be trying to make them motivated. However, motivation isn’t black-and-white; a student motivated in maths can be unmotivated in science. It is nearly impossible to find someone who cares about all of their classes, especially keeping in mind the existence of PE.

Lemma 1:

The “point of teaching” as described in Postulate 1 is non-trivial to achieve.


We proceed via contradiction. Let us assume that “the point of teaching” is trivially achievable. Then this implies that teaching is meaningless. However, teaching (when done right!) has been shown to nontrivially improve the performance of students. Teaching being pointless and teaching being meaningful contradicts each other, so the premise we started with (“the point of teaching” is trivially achievable) is false, proving Lemma 1.

Theorem 1:

Effective teaching is hard to achieve.


This follows directly from Lemma 1, otherwise it would be trivial to achieve the point of teaching.

But can we really get everyone to be motivated in everything? I doubt it. We would’ve already solved the problem of the lack of girls in STEM if we could’ve done so this easily. This leads us again to the distinction between anyone and everyone. Yes, anyone can become a USAMO winner, but every year there are only 12 USAMO winners. We clearly cannot have everyone be a USAMO winner. We can pique anyone’s interest in maths, but it is ridiculous to assume we could even get close to getting everyone interested in maths.


Theorem 1 applies for students and teachers.

This should make intuitive sense; a teacher needs to teach well and a student needs to learn well. Again, this is different for every subject; I would make a pretty worthless PE teacher, and in all honesty, I am a pretty worthless PE student. This leads me to a very controversial statement: “Math class should be optional.” Lockhart does a better job of saying this than I could, so I’ll be referencing him. In essence, if a student is already motivated for math, then they would be going to math class; if they are not motivated yet, it is a waste of the teacher and student’s time to go to math class. This applies for any class.

So far we’ve laid out the obvious; don’t make unmotivated kids hate the subjects they are unmotivated in. (I feel it’s important to emphasise this isn’t black and white; most kids are unmotivated in some subject or another.) But why are so many kids unmotivated in the first place? This comes from a combination of two factors; kids are penalized for being motivated and are discouraged from pursuing their motivations, instead wasting swaths of time, and kids have been incorrectly taught that hard work equals success.

Lemma 2:

Kids are penalized for being motivated.


Motivation happens in a long period of time (say a couple of hours) rather than sporadic bursts that can be cut off. School frequently cuts off motivation; this is why we have “settle in, kids” for block and “warmups” for math. Additionally, any second working on a personal project (say, my book) is a second that homework piles up.


Students don’t do their homework for this reason exactly; they care more about personal projects, aspirations, and the like to be doing homework.


This is exactly why students are so lazy; they’re used to their work being useless despite the school teaching them “hard work = success.” This is not the case; you have to learn to do hard work despite the fact that you are likely not to succeed.

Lemma 3:

Kids are taught hard work = success, which is false. This sets kids up to not work unless they are guaranteed success (which we all know a more accurate descriptor for that condition; this is called “never.”)


How many times have you heard the mantra “hard work is success?” This is setting kids up to think that they cannot do anything, and that they are just failures, when the case is that success is never guaranteed.


We live in a society where the mantra is, “Get good grades in middle school to create a habit for high school! Get good grades in high school to get into a good college! Get good grades in college to get a good job! Get a good job, and… get a good life?” Where does the mantra end? This is completely false! School is setting up these students for a world of disappointment, and is not “preparing students for the real world” as they claim they do!

Theorem 2:

School is broken. (Refer to the definition of “broken” I gave above!)


Direct result of either Lemma 2 or Lemma 3.

Notice how I was careful to say either. This means that to fix school, we must at least fix the problems that have been referred to in Lemma 2 and Lemma 3. (This result should be very obvious; it is only here to highlight the problems, aka Lemma 2 and Lemma 3.)

How we fix Lemma 2: We make longer classes, and significantly longer classes. (“Longer classes” means longer than 2 hours and certainly no shorter. This could work on a weekly rotation basis.) This also gives us a lot of quality of life changes, particularly in Science; no longer do labs need to be split into a “Part 1” and “Part 2;” this simply emulates actual science. Scientists don’t sit around waiting for the next “work period” when they have found something. Additionally, we can have longer tests that don’t need to be split up; I think this is also a very important quality of life change. (A quick note: Why do the administrators think FLEX has any chance of helping kids make up a test? This basically makes FLEX pointless… because kids can just work in class! Yes, I know a teacher who regularly gives 30 minute tests… what a childish notion!)

How we fix Lemma 3: This should be obvious. Stop telling kids to “try, try again” because if you don’t care enough about something, you will “fail, fail again.”  Even people who do care don’t always succeed, and I argue (this is taken from Evan Chen once again!) that it is more beneficial for your growth if you fail.

“It’s hard to do a really good job on anything you don’t think about in the shower.” These words spoken by Paul Graham ring true; the natural question to ask is, “Why on earth has nobody figured this out yet?!” I refer again to Evan Chen, who does a better job explaining than I could hope to; he talks about this in “Diversity and Neg EMH.” In essence, it is very likely that if you think enough about something, you will get a big “a-ha!” moment that should seem obvious to everyone, once you think of it. This is true for nearly everything; even in maths, we have examples of people making obvious mistakes, or obvious oversights (how did it take people so damn long to think of Gaussian elimination)?

Then I present a question: How many fundraisers have happened? Perhaps you have not been very aware of them, and they only took up a small figment of your thoughts (I know this is the case for me), or you have been constantly aggravated when you heard, “Another fundraiser?” Again, this really depends on what position you’re in; for me, it certainly is not a big thing. But for the administrators? Their job has become running the fundraisers, which is ludicrous to me. How can they expect to fill the dual role of fundraiser and making the school better? For a fundraiser to succeed, you need to think about raising money in the shower, and this means you can’t think of making the school better in the shower.

This does explain why the school is run so childishly; the people in charge are focused on raising money, rather than improving the school.


A huge disclaimer: My tips, advice, and thoughts are only for teachers with motivated students. If your class has a bunch of slackers, this advice will not apply. I fortunately have no experience with slackers (my kids are amazing!) so this was never a problem for me. Although I may share my thoughts on slackers on a later day, I simply do not have the expertise.

A quick note before we get started: I have taught various classes and written handouts. Those are available on my main website. This is the experience I am going off of.

Generally, I think that if you can google something and understand it, it usually isn’t worth teaching. You have motivated students; granted, maybe they don’t know what to search for. But you shouldn’t be teaching random AMC 10 problems; motivated students know where to look for AMC 10 problems, and where to look for good ones if they’re a bit more experienced.

Many great classes, however, do teach AMC 10 problems. (Fill in olympiad problems, AIME problems, whatever you want.) Why is this so? What differentiates this from doing AMC problems?

First, it is very important to note that taking a class is far from necessary. But there are some reasons this may help:

1) Writeups are a chore.

By the time you’ve worked out the details, you might forget to highlight the main ideas and forget what made the problem… well, a problem.

In contrast, your teacher will make sure to point out the main idea of the problem because talking is not as much as a chore. This means you will get the main idea much faster.

2) Connections

I reference Evan Chen once again. Let us take a look at his example:

A1 B8 B1 A4 C11

A9 C22 B27 B64 A25

C44 B125 C55 C33 A36

Of course, it still isn’t very hard to see the connections.

But take a look at this:

A1 A4 A9 A16 A25

B1 B8 B27 B64 B125

C11 C22 C33 C44 C55

Liberties were taken with the exact ordering (and maybe the patterns as well), but the point should still be clear. If we consider the A’s as a certain subject (say inversion) and do something similar for the B’s and C’s, we can think of the patterns as the main ideas; they are all related. Also, false connections will not be formed; nobody will make the dubious claim that the A’s and B’s are the same, because they just saw A1 and B1.

But I think there is something more to consider; the teacher arguably benefits the most. You see, the teacher starts seeing connections and main ideas easier because their goal becomes to see it, once they have a reason to.

I can’t really give much more advice than this (which should be well-known and seems obvious, but many people are informed. See Diversity and Neg EMH) but I do have a few closing remarks.

You can always ask students how they feel about you, but unless you know them really well, they may be a little bit shy to tell you that you suck. This means you can only really look at yourself. Are you reaping the benefits of teaching? Then you should be good to go. Are you not? This is a red flag, but this doesn’t make you bad; everyone makes mistakes.