Schools' back! That means Corey takes the absolute maximum number of overloaded units and works both as a TA and a supplemental instruction leader. Not really. My current course lineup is - 21-701 - grad discrete math - 15-259 - probability & computing - 21-341 - linalg (with abstract algebra prereq?...) - 21-441 - number theory - 21-295 - putnam seminar - 82-131 - chinese - 57-227 - jazz orchestra I'm thinking about dropping my grad class. Doing so would boot me from the candidacy of attaining my master's degree, on top of my bachelor's in four years (the 'honors' degree program). Why am I thinking about voiding (in the name of another degree) all the hard work over the last two years? I think there is a larger discussion here worth your time if you are someone who constantly feels like you are surrounded by quick thinkers (score 50+ on [zetamac](https://arithmetic.zetamac.com/)) who can summon so much theory/knowledge off the dome. This article also goes out to my mom who offered great wisdom, offering anecdotes to her meetings at work; her other co-workers often seemed to just 'get it' with no questions asked (or the questions asked demonstrated strong, immediate understanding). To that, we wonder, what prevents us from being like this? Within the first week of arriving at college two years ago, I realized that students were roughly partitioned into levels of ability. You had the fishies like me: barely having touched proofs before school and only entering college with a bit of calculus experience. Then, you had the seasoned students - the bluefin tuna: they were placed into the honors version of first-year linear algebra and likely dropped 100s on their concepts of math exams (notorious first-year, intro-to-proofs math class); they probably did some competition math in middle and high school, and maybe even studied proofs or math over the summer for funsies. Then you have the sharks: these are the students who sometimes show up from high schools like Thomas Jefferson, PRISMS, etc who likely got USACO gold+ (google how insane of an achievement this is) or qualified for USAMO (roughly the top 0.6% of kids who would go to math club in high school); they probably did so well on the math maturity exam during orientation week (placement test) that they were able to skip honors linear algebra entirely and head straight to the math studies sequence (some of the most challenging, abstract math courses @ cmu). But there is a final tier. When I was much younger, my sister and I invented 'buggies' (this deserves a whole book tbh), which were essentially little characters you made with your hands to pass the time. We decided that the most powerful, almighty buggy would be called the '**ULTRAMEGASAURUS**.' Channeling this energy in naming conventions, the final tier of problem solver are the **omega whales** - the anomalies that show up to school and make college look like pre-school: some take graduate classes their first semester of school, they tend to place top 500 on the Putnam math exam (if they pursue math or CS), and simply let their analytical operating system let them coast. That is not to say these students don't work hard; I believe persistence & dedication are one of their characterizing traits - they just make solving insanely hard problems look so easy. I've recently thought a lot about the characterizing traits of these upper-level problem solvers. My ultimate goal two, five, or 10 years out is to be able to solve problems at that speed and capacity. That is not to *compare* myself to other people - instead, I observe a genuinely interesting ability that people *can* wield, and I wish to wield it as well. You know what I mean? At this point, I don't take it personally that some people have simply more experience than I do with solving problems; I'm okay being the 'dumb guy' in the classroom if that means I get to learn new things. I'll come back to this 'selfishness' later on. Anyway, I think that these highly able problem solvers obtain a few core traits: - Memory: When I'm in class with these students, they can recall theory as though they were identifying flags of the five most populous countries. What is paired with this level of memory is a logic checker. If they are slightly shaky on a detail within a definition or theorem statement, they can summon context from other neighboring relevant theory to figure out the 'missing puzzle pieces.' In this way, their memory is eternal; there is no way they could forget a theorem or definition because they have the mechanisms to construct them back in full rigor using their massive accumulation of absolutely true statements. This also allows them to correct the professor, who is often not able to simultaneously recall from memory, speak, write on the chalkboard, **and** check themselves. For instance, if we want to recall the axioms of probability, we might be able to recall that $P(\Omega) = 1$ by a recollection that a probability of any event (no matter how likely) can be gt; 1$. My former discrete math professor, Po-Shen Loh, famously says he never remembers how to prove certain theorems or sometimes the raw statements, but he uses definitions along with main ideas to reformulate proofs for himself. That made class totally brilliant; you would see exactly how a strong basis of definitions directly leads to an ability to reason at a higher level. Memory is akin to the literal memory or storage (and backup mechanisms) of a computer. - Abstraction capabilities and hence speed: Not only are these students able to recall theory in a snap, they are also able to be told a brand new, never-before-seen definition and immediately operate on it at the highest level. In Calculus, your professor likely aided the abstraction process: she might define the derivative of a function $f$ at $a \in \text{dom}(f)$ as the limit as $h \to 0$ of $\frac{f(x+h)-f(x)}{h}$ and then provide an emporium of examples: what is the derivative of $e^x$? What is the derivative of $x^2+1$? What is the derivative of $(x+1)^2$? Then, your professor may even show you some non-examples of a definition if you're lucky. Only then will your professor move on to discuss exercises involving this new term: the 'derivative.' In higher math (such as graduate discrete math) the professor (rightly) assumes an insanely strong abstraction ability. We were told "A hypergraph, $\mathcal{F}$, is a collection of subsets over a vertex set $V$. A hyperedge is hence an arbitrary subset of $V$. We call a hypergraph $\mathcal{F}$ $k$-uniform if $|f| = k$ for all $f \in \mathcal{F}$. A proper $\ell$-coloring of a hypergraph $\mathcal{F}$ is an assignment $c: V \to [\ell]$ such that for any $f \in \mathcal{F}$, we have that $|c(f)| \ge 2$. We say that $c$ is a rainbow coloring of $\mathcal{H}$ if $|c(f)| = |f|$ for each $f \in \mathcal{F}$." and then we were immediately asked "What is the least number of edges in a $k$-uniform hypergraph that is not $2$-colorable?" No examples, no discussion about the contents of a hypergraph nor its relationship to incident matrices or bipartite graphs (for what it's worth). Just a full plunge head first into the most famously studied abstract extremal questions. For many of the students around me, they've developed a cognitive hardware to deal with these fast pace environments. They've come to view a mathematical statement or object as nothing other than its definition. Consider, by analogy, your understanding of a car. Now, if I ask you to consider a red car with a white stripe down the middle, you'll likely have zero issues summoning a relatively solid picture or understanding of the topic I'm describing. You've likely seem so many cars and varieties that the mutation of a template car within your head is trivial. The same game is played in math. The professor asked us to consider a $k$-uniform hypergraph (default car), and then one such that the number of number of edges makes it not $2$-colorable (the white stripe). For people who have tons of experience 'considering' variations of different mathematical objects, this question they are asked is well-understood. If you are me, you haven't the experience to understand the question once it is asked; you need to instead ask "wait a second, what is a hypergraph again?" and then ask "what do we mean by minimum number of edges? Do I believe that if we place too many edges then it won't be $2$-colorable?" I ask these questions because I have not yet abstracted away the notions of a hypergraph and $2$-colorability; for me, working with these objects is a lot like booting up a computer: a bunch of programs need to be run first before any high level software can be used. The experienced 'cracked' students are an already-booted-up computer with the software open and ready. For what it's worth, I believe I can and will become the latter. For now though, there are all of these subquestions about the nature of the english used too pose the question and the implications and definitions. I have not yet the ability to speak and think math at 200 wpm. Yet, this is a skill that can be trained (more on this later). Because students are able to abstract an idea so quickly, they are granted a natural speed of thinking as they are not burdened by the questions I mentioned previously: to them, a mathematical object is **nothing** but its definition and a theorem is **nothing** but its statement in terms of the objects involved. Abstraction capabilities and hence speed - essentially how quick you can get to thinking in terms of new material - are akin to the rate it takes to open highly costly computer programs. - Metacognition AND ultrarationality: Now we turn to what these students do when they actually go to do the thing: solving the problem. The first thing they are able to do is formulate the questions they need to answer in full mathematical rigor (in an effort to void any ambiguity that would lead an idea to be inaccurate). If they are asked to prove that the square of any even number is divisible by $4$, they are instead answering "Let $n \in \mathbb{Z}$ such that $n=2k$ for some $k \in \mathbb{Z}$. Prove $(2k)^2 = 4m$ for some $m \in \mathbb{Z}$." The next thing they do is use pure logical reasoning to manipulate the propositions. If they are stuck, they invoke a metacognitive guide; if they are asked to prove a proposition $p(x)$ for any $x \in X$ and they are not convinced it is true, they will logically ask "why can't there exists $x \in X$ for which $\neg p(x)$?" Perhaps that leads them to discover a key property about all $x \in X$ that proves the claim. A strong problem solver will **never** have no ideas. When 'stuck' they will always be able to formulate a logical question that gets them unstuck, and make it specific enough to where the answer is obvious. The rate by which they make small technical errors also tends to zero as their practice ventures onward. They **will not** move on from a mathematical step unless they know for certainty that it is true. In this sense, when they write something on their paper, it **must** be true. Their is also a degree to with which these students are extremely rational. They will never accept a statement without verifying it for themself. There is also no 'want' when it comes to solving a problem. If a problem asks to 'find the minimal $n$ for which $p(n)$ is true' the student will never 'want to find' such an $n$. They will instead **let** $n$ be the minimal $n$ for which $p(n)$ is true and then logically deduce $n$. To these students, there is no 'wishful thinking' and there is no 'hoping.' It's exactly like how the phrase 'there is no hope chess' is printed in every single chess theory book: the idea is that you cannot think of a line in chess dependent on your 'hope' that the opponent plays a certain response. Like chess, there is simply no RNG in math (not even in the probabilistic method 😂🤣😂🤣😂🤣😂🤣😂🤣😂🤣). I think that these students 'do' math with this philosophy manifested within their thinking patterns. In turn, what thinking with these skills does for a student is weight heavily the **most** rational and logical ideas and weight minimally the ideas that are not grounded in purpose. That is to say that one's grand ability in math is present insofar as one's extreme deliberation and focused attention on logical operations, claims, and ideas. Within the computer of the omega whale, metacognition AND ultra-rationality are akin to the technologically advanced features and algorithms your computer can run. Old software is inefficient and operating is slow, and new software has fancy algorithms that can permit more capability. - Experience: When I try to holistically characterize the exceptional problem solvers at this school, I think of the binding glue: experience. If you score 10 on zetamac and then you grind mental math for an hour a day, you have no choice but to compute more quickly the expression $384 \div 16$ because you've already computed $16 \times 24$ so many times. Given you've computed $50^2 =2500$ and $7^2=49$ and $5\cdot 7 = 35$ so many times, you'll have no choice but to compute $57^2 = (50+7)^2 = 50^2+2 \cdot 50 \cdot 7 + 7^2$ with increased confidence and speed. This simple idea scales as we'd expect. When you commit to massive problem-solving, you'll have no choice but to construct rigid thinking models and deploy theory and ideas in a wider range of contexts. If you solve even 10 problems involving the pigeonhole principle, you'll eventually be able to see "The points of the plane are colored by finitely many colors. Prove that one can find a rectangle with vertices of the same color" is screaming at you to use this idea. But that's just scratching the surface of how experience cascades. Imagine you solve 100 pigeonhole problems. Then, your search space of ways to deploy pigeonhole is very likely to contain the method needed to crack this problem. If you solve $500$ pigeonhole problems, your search space will have weights that practically point to the solution of this problem. I want to discuss my further thoughts on experience in a moment, but to motivate how important I think it is, we should consider the seasoned problem solvers' computer again. Imagine that our problem-solving computers begin with a wiring material that is less conductive than copper, in which electrons don't travel at the speed of light, but instead at some resistive rate. This makes **everything** slower and more prone to power issues and shortages. For each problem you solve, perhaps one single wire of the thousands on buses going from component to component becomes copper. Presuming we have all the fancy parts and algorithms and storage abilities as previously mentioned, having all copper wire unify the components within the computer would, in essence, make your problem-solving computer utterly celestial and sorcerous. Before returning to experience, I must mention a recent discovery I had about a secret final characteristic of the omega whale computing system. - Detachment from emotion in the name of progress: those who are truly exceptional problem solvers are the way they are because they trained and gained experience without judging themselves or turning their progress into a statement about their identity. At school, I've been so caught up in the hype of taking the hardest classes I could manage, even if they were more conducive to my ability to 'catch up' and 'cram' than learn deeply. This is because I suffered from an identity problem if I didn't participate at the same level as my peers: "Why are they so good and I'm so bad? Why shouldn't I be able to do what they do? How am I not at the level of everyone else at this point? How come I'm not able to do USAMO problems yet?" But for the most omega of whales, I'd go so far as to conjecture they almost **never** have these thoughts, because they're simply an unnecessary narrative and distracting layer of consciousness on top of their actual goal: improvement. Why do most of us *identify* so closely with our ability and make it more complicated than it is? Why do we try to contextualize our ability in the grand scheme of others? Why do we take our abilities *so* personally? My analogy about exceptional problem solvers in terms of a computer is solely to make this point: a human might care about what their ability says about their membership in society or the social and emotional implications of their skills. For a computer, that is not even remotely a concept. Do you see what I mean? The computer simply works, or it doesn't. There is no judgment, no doubt, no fear, no regret, no anxiety, no depression, no happiness, no motivation, no emotional concept whatsoever distracting it from carrying out computations. As humans, we are not to strive to become emotionless machines. I'm merely saying that to be exceptional at problem-solving and to channel extraordinary progress, we must simply let go of our judgment of the self. There are plenty of domains in life in which emotion is essential: art, love, writing, conversation, food (this deserves a whole article in itself, and specifically about my mom's banana bread/muffins/bars). In problem-solving, voiding your emotional tendencies is a skill, one that permits you to think rationally, operate logically/rigorously, and fail very often without judgment. As I was sitting in 21-701 the other day, my professor said, "I'm sure almost everyone here has seen Hall's Theorem on Bipartite Graphs." I hadn't, so the whole lecture went over my head. At that point, I basically realized that I had been deluding myself for years about the nature of making progress. Is it really worth 50% of the material in class going over my head, resulting in me having to read a textbook and formalize things at my own speed for hours and hours after each lecture? Is it a reasonable use of my time to tackle problems I know I cannot solve without significant help or hints from peers and my professor? Am I to reasonably expect I can amass graduate-level mathematical thinking and problem-solving skills (something that I'd argue takes at least five years of consistent effort) in just two months before the exam? No. No, and that is okay. It is *okay* that I am not as refined as my math+CS bubble in this school. It is *okay* that I need to work on more fundamentals before I tackle even the prerequisites for this class. It is *okay* that I will need to do the work my peers likely did in middle school and high school to get to where they now are (again, not comparing myself, but observing a desirable skill). I bascially realized that my efforts are fundamentally unaligned from my goal: my goal is not to be comfortable with being a fraud, my goal is to be a better problem solver, which means working on my fundamentals right now; the things *just* above my skill level, without judgement such as "I can't believe I'm only as good as my peers were in early high school." I basically realized that it is tremendous that I have recognized that I'm at the problem-solving level of my peers when they were in early high school! That means I know exactly how to improve! I need not be able to solve "Generalize standard $3 \times 3$ Tic-Tac-Toe to three-dimensional $8 \times 8 \times 8$ Tic-Tac-Toe on the board $[8]^3$. In particular, define the underlying hypergraph. How many edges does it have? Prove that the 2nd player has a drawing strategy (a strategy to prevent a 1st player win)" right now. My goal should instead be to work on the problems just above my level, which are honestly some of the upper level AMC (high school competition) problems and middle-level AIME (advanced high school competition) problems (ironically, being around brilliant students I'm decent at teaching advanced problems and how to think of solutions to other students - and this can be possible simultaneously as I struggle to do them consistently). I believe it is necessary to be almost **selfish** when it comes to improvement: selfish in the sense that you do exactly what is just above your level and no one else's, regardless of where that level is. 'Selfish' in the sense that one must not care in the slightest what is 'expected' of them 'at this point,' or how they perform relative to their surrounding world. I believe internalizing this is what people mean when they say 'academic freedom.' This, as a term, is not just to encapsulate the ability to study whichever field, but to study whichever part of whichever field. Yesterday, my roommate taught me how to multiply $3$-digit numbers because I had forgotten, and would always solve a multiplication problems as follows: $(324)(578) = (300+20+4)(500+70+8)$. I don't care that I forgot the regular algorithm to do this, because now I wield a power I didn't use to have. I exercised this carelessness in my number theory class on Friday. When a student offered a clever idea to prove a claim, my professor asked 'Does everyone generally see how we would use this idea?' No one objected, so he moved on. A minute later, I thought to myself, you know what? I don't f$!king get it, and who cares if other people do. I asked my professor to explain how the idea worked. My professor pointed at me and literally yelled, "YES!! People, *this* is bad ass!! Everyone, this is how it is done!!" All I could think was YES, THANK YOU, YOU SEE ME!! Like, this is just me taking my learning into my own hands! Who cares if everyone else knows? The craziest part is that the whole class was stumped for a moment. It was a learning moment/key piece of formalization. For what it's worth, I know I'm probably the least capable person in the class, but I don't care. If it means I get to learn the material by asking something like that, what is the hesitation? Really? There is no social judgment in a classroom, and people who judge others' inquisition will get nowhere! After class, I had a fantastic discussion with my professor about his experience in math and his belief that, outside of two people he has met (Terry Tao and one other wizard), 'everyone is an idiot' and 'no one knows what is going on.' He described how this 'selfish' framework is what led him to come up with 'suprising results' in number theory (he was being very humble, his work has done far more than suprise the community of number theorists), and how this 'selfish' nature is what led James Maynard (my prof's former roomate) to win a fields metal. He asked me, "You know how James Maynard learns? He sits in a lecture and doesn't write anything down for ten minutes, and asks seemingly simple questions at first." I imagine these are the questions that are 'simple' in that maybe everyone else knows the answer, but James didn't care because *he* didn't know. My professor continued: "But then Corey, do you know what happened? He would then scribble something down in his own words, all of a sudden, because the simple questions led to his understanding. Then his questions became less straightforward. He was understanding." He also told me that there was a time in college when he realized it actually just doesn't matter if he is the least capable in the room. Provided the material is at the right level (a little too difficult for it to be comfortable), he was perfectly fine, knowing he was the stupid one.' Ultimately, he started asking questions selfishly, and look where he is now: a professor at Williams College (he is visiting CMU this semester) who has come up with outstanding results and is highly aware of the experience of his students. Consider yourself lucky if you ever get to sit in on a lecture with Leo Goldmakher; his joy radiates among those he teaches. So guess what? If you are a friend or family member reading this, great. If you are a recruiter for a job I applied to, reading this, great. If you are a student of mine, great. If you are a mentor of mine, great. If you know me, great. If you don't, great. I care no longer about my perceived intellectual status, because my becoming a good problem solver is not about that; it's about me. One year out, I'll be thrilled if I can score a 110+ on the AMC12 in the regular time allotted. Two years out, I'll be thrilled if I can get 8+ on the AIME. Five years out, I'll be thrilled if I crack Olympiad problems with rigor and creativity and am able to solve coding problems using an amassed library of mental models. But ultimately, the goal is to improve my problem-solving ability. To do this, - I'll solve problems just above my level to build experience & basic intuition, regardless of whether they are entry-level CSES problems, AMCs, AIMEs, or random problems from a book. - I'll practice reading technical books with focused attention so as to build my abstraction & formalization abilities further. This means practice seeing new definitions and operating on them with little exposure, and developing the cognitive hardware to do this in a fast-paced environment like a meeting with a whiteboard or class. This would force me to become metacognitive if progress is not made. - I won't judge myself when I'm bound to fail thousands of times because... why? Literally, why would I do this? I know what I need to do. Now, it's time just to do that. Thanks for tuning into: "human being realizes they are overcomplicating things." I hope you got something from reading this. Cheers :)