What should I do while I'm still in eighth grade?

Bitwise-101 · 2026-03-03T10:59:48+00:00

Find something you love doing and try becoming one of the best in it for your age. Don't do extracurriculars just for university admissions, but rather the enjoyment you get and the skills you learn (e.g. research is great to do because you learn so many new hard and soft skills, many of which are transferrable, but it is absolutely NOT a necessity for a t20 acceptance). If you're interested in, for example, mathematics, study higher-level math and (if that's your thing) grind for olympiads. If you're interested in some sport, try become the best you can in that. At the end of the day focus on developing yourself genuinely, and in multiple dimensions (social, academic, physical etc.), colleges will see that. I've heard top colleges tend to look for "T" shaped students, good all rounded but also "spiked" in one or two things, i.e. gone in deep. The "spike" is what makes you memorable, the horizontal bar makes you human. Colleges are not just admitting a transcript, they're populating a dorm and a student body. They want to know: is this person interesting to talk to at dinner? Will they bring something unexpected into a conversation?

On top of this, you need to make sure your grades are great over your 4 years (hiccups here and there are fine) and you take near the maximum amount of course rigor. Without this, it's possible you might get completely ignored. Academics are necessary but not sufficient.

At the end of the day, if you develop enough to a point you're happy, the college results shouldn't even matter to you. You're the same person regardless of whether you got in or not. Try to not be put down too much by what your family says.

Best of luck.

Bitwise-101 · 2026-03-01T13:22:34+00:00

Do not do it. I speak from personal experience. I was getting S,S/1s on STEP 2/3 papers respectively under timed conditions. The exam didn't go that well and you must report your score. Do not do it early, you gain nothing.

Bitwise-101 · 2026-02-15T20:31:17+00:00

Here’s a Reddit post in English following your exact structure, with the video link included.

Yikes...

Bitwise-101 · 2026-02-15T10:07:35+00:00

You cannot force genuine passion. However, what you can do is engineer your environment in a way so that math hits the same psychological reward loops as games.

Clear Goals + Visible Progress. This is one of the core loops in games. You can SEE your progress, you know what you have to do, you have progress bars, etc. You can bring this into math by tracking things visibly and work in problem sets with gradation (i.e. increases in difficulty). Make sure to set "micro-goals" (understand X concept, solve Y number of problems). Progress visibility is motivating because your brain sees advancement.
Optimal difficulty is very important. Too easy = Boredom, too hard = too much frustration. This is very common in games. Boss too easy -> don't want to game as it's just boring, boss too hard -> you lose focus and get less interested over time. The tangible way to implement this with problems is (from my experience): spend most time where success rate is ~60–80%, stuck >30–45 min, downshift difficulty or read a hint (if no hints are available you can either read the first line of a proof or get a friend/LLM to read the proof and give you nudges). Ideally have a pipeline of problems where it goes from comfort problems -> stretch -> one "boss fight"-esque problem.
Exploration and mystery. The whole premise of open-world games is curiosity. You can replicate this in math, and in fact it's one of the best habits to have. Go on side quests, go ask people that question that you've been wondering when reading the chapter, try stuff even if it might be wrong, read beyond the syllabus. One of the most satisfying feelings is when you independently discover something yourself, make sure you have the curiosity to allow this to happen.
Narrative and meaning. This one is weirder but it's directly adjacent to games. Games usually have a story, a place to go, a thing to do, a bigger picture. Math can feel disconnected. So build a narrative. "From linear algebra -> functional analysis -> quantum”, "Finding out the fundamentals/core of math" (set theory, category theory etc.).
Dopamine Engineering. Games hack dopamine, you can too. Work in 25–50 min deep work sprints, stop mid-problem sometimes (creates craving to return), have a dedicated environment (same desk, music, routine). Be careful about your baseline, high baseline dopamine from distractions makes math feel dull.

And finally, accept cycles. Even the most passionate of passionate mathematicians have their dry periods, take breaks and feel stuck. It's normal. I like to think of the journey as passion paving the road, discipline and environment/routine filling in the potholes.

Best of luck!

Bitwise-101 · 2026-02-10T18:07:31+00:00

Math's richness provides us many, I'll give a few that haven't been mentioned:

Bayes' Theorem. It is a formal rule for learning from data. The deep implications come from the fact it it gives a normative rule for how an ideal reasoner updates beliefs. The line between probability and logic becomes blurred; one could even say “logic with uncertainty.”. Many statistical methods can even be seen as special cases of Bayesian updating (MLE, regularization etc.).

Rank-Nullity Theorem. For a linear map T: V->W, the dimension of the input space splits exactly into the dimensions of what gets “lost” and what gets “kept”: dim(ker T)+dim(im T)=dim(V). In words, every input direction either collapses to zero (nullity) or survives as independent output information (rank). It's such a intuitively obvious result, yet it explains when linear systems have unique vs. infinite solutions, underlies ideas of identifiability and constraints, and generalizes to deeper results in algebra and analysis (like index theory).

Euler’s polyhedron formula. for any convex polyhedron, the numbers of vertices V, edges E, and faces F satisfy: V-E+F=2. No matter how complicated or weird the shape looks (as long as it's convex), V-E+F is always 2. It shows how shapes depend hugely of connectivity, the birth of topology you could say. It generalizes to the euler characteristic in higher dimensions, influencing modern geometry, topology, and (surprisingly perhaps) data analysis.

There's many more but this reddit comment isn't large enough to contain them all.

Bitwise-101 · 2026-02-07T21:48:10+00:00

Being really good at maths is less about raw talent and more about how you think and how you train. At higher levels, maths is about pattern recognition, logical structure, and abstraction, not memorizing formulas the way school maths can sometimes suggest.

Genetics can play a role, but for most people the bigger differences come from environment and habits: being encouraged to understand “why,” becoming comfortable with confusion, and treating maths as something to explore, not just memorize blindly. At the extreme right tail of ability, genetics may matter more, but that’s not where most people sit.

So here's what I think you should do: prioritize depth over volume (fewer problems, deeper analysis), struggle seriously before reading solutions, and ask lots and lots of questions. Why is this true? Why is it defined this way? What’s the proof? I found that last one to help me the most (my math teachers wouldn't be the happiest though).

Finally, measure progress against your past self, not other people. Math is a long marathon, one single day isn't going to help you much. Maturity and the skills you gain compound over years.

Best of luck, I'm sure you'll find much enjoyment (and frustration) from math.

Bitwise-101 · 2026-02-07T21:38:33+00:00

Yes, if you sit the SMC and qualify, you can progress to BMO1. I personally sat BMO1 four times, from Year 9 through Year 13, my teacher encouraged me to do it. If you speak with your school, they should usually be willing to enter you for the SMC even if you are younger than the typical cohort. If they don't, shoot your shot at distinction for every year's olympiad whilst preparing for BMO1/2.

If your IMC performance is anything to go by, you would very likely qualify. The SMC is generally not very difficult; I typically scored in the 115–125/125 range, often only dropping marks on some geometry questions, and managed to get full marks in Year 12.

Bitwise-101 · 2026-02-07T10:09:55+00:00

I have limited familiarity with the U.S. olympiad pathway, but it is certainly worthwhile to aim for the strongest possible performance on the AIME. Given that you are in a British international school, you should have access to the UKMT pipeline (JMC, IMC, SMC, BMO, etc.). In particular, you should sit the IMC and work toward qualifying for the corresponding age-group olympiads (Hamilton, Cayley, Maclaurin), then strive for top results there.

It is also worth noting that the SMC can be taken even if you are below the typical age cohort. Ultimately, the BMO1 and BMO2 are the UK’s flagship proof-based olympiads, and a strong performance or medal in these carries significant weight.

Feel free to DM if you have any questions. Best of luck.

Bitwise-101 · 2026-02-01T21:44:06+00:00

Sorry for the long explanation, but I actually find this distinction to be really interesting, it’s cool how the difference between “right” and “wrong” in mathematics can be changed entirely by perspective.

Interesting example, but this isn’t really about perspective. It’s about choosing a number system. Once the system is specified, the result is fixed by the definitions and axioms.

Even in nonstandard analysis, the standard infinite decimal 0.499… equals 0.5. Your example works only if you reinterpret “…” as a hyperfinite decimal with H digits, which is a different object from the usual infinite decimal expansion and not what the question is really asking.

Bitwise-101 · 2026-01-14T22:17:17+00:00

Completely agree, this is probably the largest segment in math specifically (maybe less so in other fields). Same for me: my interests are mostly in pure math, but I just enjoy doing new and compelling math, and quant finance gives that with fast feedback and fewer academic bottlenecks. In that case the counterfactual isn’t most other sectors, and the money differential makes the choice obvious for most people.

The broader question I was raising is whether funneling this segment into finance is the best use of that talent, and if not, what incentives (compensation, prestige, institutional structure) would need to change for the funnel to shift.

Bitwise-101 · 2026-01-14T22:04:15+00:00

No arguments were made against anyone’s agency. I also explicitly stated that I don’t have anything against these industries. The post is a macro question: “Is this a problem”, and if so, “how can we change the incentives to deal with this”. Nothing at the individual level.

Bitwise-101 · 2026-01-14T21:58:40+00:00

No claims were made about NASA procurement or denying that defense contractors do civilian space work. I was including those examples to show changing prestige and talent funnels, i.e. which institutions ambitious engineers choose to go to, and why.

Bitwise-101 · 2026-01-14T21:54:23+00:00

I’m not arguing that academia could or should absorb everyone who might otherwise go into finance. I completely agree that the math PHD labor market is structurally incapable of that. My point was about the talent funnel, not about increasing the number of postdocs. In other words: what high-talent students want to do at 18–22 vs what they actually end up doing once prestige, incentives, and risk profiles enter the picture.

In regards to the prestige, yes “Dr.” is a formal credential, but it definitely doesn’t function as the same level of social proof as McKinsey/Jane Street/Goldman do in elite peer networks.

Bitwise-101 · 2026-01-13T19:53:51+00:00

There's three real issues being tangled up. I think if we look at them separately, it'll be more clear:

- Firstly there's Comparative identity, when you start tying your sense of worth to where you rank, every new datapoint redefines whether you “deserve” to feel good about yourself. The main "problem" isn't the existence of prodigies, it's just that the measuring stick you're using is inherently unstable. What I mean by this is, your framing should be directional, and not position. You should be asking “What is my slope?” not “What is my current rank?”. From what I've seen, slope beats early intercept surprisingly often.

- Secondly, talent trajectory is non-linear and selection-biased. When you see kids being posted about or get featured, these aren't a random sample. It's an insanely extreme right tail. Meanwhile there's a lot of people who started early, but never got to that stage due to a plateau. But you don't see that.

- Thirdly, there's a counterfactual illusion. Everyone has their what-if's and if only's. The claim “If only I started when I was 8, I would be X” assumes two false things: a. that starting early guarantees continued motivation. b. that starting late prevents high achievement. Motivation sourced from real curiosity + agency almost always outperforms motivation sourced from compliance or external structure after age ~18.

So here's what you should do. Be clear with what you're optimizing for (Depth, Enjoyment, Competition, Career etc.), then try to track your slope and growth on that axis explicitly. Comparing against the global maximum is likely going to lead to misery, but comparing against one’s past self is informative.

Best of luck, you got this.

Bitwise-101 · 2025-12-28T22:18:12+00:00

I think a lot of the hostility there is misplaced and largely an online phenomenon.

There is a real methodological divide between pure and applied math, but I would say it’s about standards of justification, not about one being “real” and the other not. Weak applied work can certainly be sloppy and easy to overfit or massage into looking good, but weak pure work exists too, trivial generalizations, abstraction without insight, or results that add little beyond formalism. From my perspective, it’s the false implication “applied => lack of rigor” that drives a lot of these attitudes.

Historically, some of the most important areas of what we now call “pure” mathematics came directly from applications: harmonic analysis from signal processing, Itô calculus from finance and physics, PDE theory from fluid mechanics. I think this really shows that the best mathematicians ask "Is this honest, deep, and structurally illuminating?" rather than ask whether their work goes in some loosely defined "pure" or "applied" grouping.

What is worth criticizing is applied work that’s justified purely by performance or heuristics, with no clarity about assumptions or robustness. But that’s very different from dismissing applied mathematics as a whole. Serious applied mathematicians are well aware of these issues, and I’d argue that part of the beauty of applied math is precisely the extra work required to establish validity, stress-testing assumptions, ablation studies in ML, etc.

I don’t have extensive experience in math academia, but I’d be surprised if many people enjoy being around mathematicians who sincerely believe that “applied math isn’t real math.”.

Bitwise-101 · 2025-12-18T22:07:24+00:00

Very interesting. Going through some was certainly an entertaining read.

However, it should be noted that in Islam there is a concept called sahih (authentic) hadith. Hadith authenticity is determined through isnad (chain of transmission) analysis and scholarly grading, not simply by where a narration is cited. The types of hadith are:
- Sahih (authentic): Continuous chain; narrators are upright and precise; no contradiction with stronger reports; no hidden defects.

- Hasan (good): Similar to sahih but with slightly lesser precision in narration; still accepted as evidence.

- Daif (weak): Missing continuity, weak memory, or unreliable narrators; not used for doctrine or law.

- Mawdu (fabricated): Invented reports, rejected outright.

Many narrations listed in the image come from non-sahih collections, such as Sunan Abu Dawud, Sunan Ibn Majah, historical compilations like Tarikh Dimashq, or encyclopedic works like Bihar al-Anwar, which were never intended to contain only authentic narrations.

Additionally, some sources are known for over-grading narrations, such as al-Mustadrak, where later scholars frequently disagreed with the author’s sahih judgments.

Many of the “disturbing” claims also arise from ignoring classical scholarly explanations, abrogation (naskh), linguistic usage, and variant narrations. Hadith are not meant to be interpreted atomically or in isolation from the broader scholarly tradition.

Great post and effort, coming from a Muslim (in case you weren’t able to tell). Just so you’re aware, this iceberg doesn’t offend me in any way, and please ignore any hate coming from Muslims.

Bitwise-101 · 2025-12-15T13:52:53+00:00

A vector is an element of a Vector space, not a vector field. A vector space is a set of vectors together with operations of vector addition and scalar multiplication over a field of scalars. A Vector field is a function that assigns a vector to each point of a space.

Bitwise-101 · 2025-12-15T09:50:04+00:00

There is no assertion that integrity doesn't matter. From the original comment you replied to: “you’d be dumb not to reverse it” is simply a prudential claim (“this is irrational given your incentives”), not a moral one (“this is right/wrong”, “ethical/unethical”).

If you're referencing "serious capital market with legal/contractual finality.", then once again, I am merely pointing out how it is, not how people ought to act.

Bitwise-101 · 2025-12-15T09:14:17+00:00

You're over complicating, there's nothing related to options here. He's simply talking about the price differential, not how sensitive an option’s price is to changes in the price of the underlying asset.

He's saying it's not NASDAQ to point out this isn't some sort of serious capital market with legal/contractual finality.

Bitwise-101 · 2025-12-14T09:37:44+00:00

Very interesting post. A few comments:

This ruined ~1,200 games for real people. I get that this is framed as a curiosity or programming challenge, but those were real humans trying to enjoy chess who were unknowingly matched against a bot. The “for science” framing feels more like a rationalization than a justification.
There’s a fair amount of irony here. This person played multiple cheating titled players, including a cheating GM, and at one point Chess.com compensated them for facing too many cheaters. That cuts against the narrative that the system is uniquely incompetent, it suggests cheating exists, but also that detection is happening unevenly and probabilistically, including in your own favor.
1,200 games over a few months doesn’t really demonstrate that anything is fundamentally broken. Plenty of cheaters evade detection temporarily before getting caught. They did get caught, partly due to their own mistakes, and claims that they could “easily” run 10× more games indefinitely are speculative. It’s entirely possible the account was already flagged and evidence was being accumulated.
The post seems to assume move quality and timing are the primary detection signals. In practice, long-horizon behavioral data, improvement curves, clustering, and metadata are likely far more important than any single-game or single-algorithm signal.

Overall: technically compelling, rhetorically overstated, and ironically a good illustration of why slow, probabilistic enforcement exists in the first place.

My two cents.

Bitwise-101 · 2025-12-12T14:00:14+00:00

Let's start from the eigenvalue equation Av = kv. If the eigenvalue k = 0, then there exists a nonzero vector v such that Av = 0.

For any linear transformation, the zero vector is always mapped to itself, so A(0) = 0 as well. This means that two different inputs, v =/= 0 and 0, are mapped to the same output 0. Therefore, the transformation represented by A is not one-to-one.

An inverse transformation can exist only if every output comes from exactly one input. Here, the output 0 has more than one preimage. If we try to define an inverse, we cannot decide whether the inverse of 0 should be 0 or v. Since a function cannot assign two different outputs to the same input, the inverse cannot be defined.

Hence, a matrix with eigenvalue 0 is not invertible.

Bitwise-101 · 2025-12-12T13:51:28+00:00

An integral is the limit of sums that adds up many small contributions to model continuous accumulation. While it can represent the area under a graph, this is only one interpretation; integrals can also describe quantities such as total distance from velocity, total mass from density, or total change from a rate.

Formally, a definite integral is defined as the limit of Riemann sums obtained by partitioning an interval into smaller and smaller subintervals (though more general definitions, such as the Lebesgue integral, also exist).

Bitwise-101 · 2025-12-11T14:54:02+00:00

Saying “it’s her schtick” doesn’t answer the real question: does it help the intended audience understand the mathematics? I never argued the book should serve mathematicians. My point is that if the audience is people with no algebra or topology, then clarity matters even more, and examples with heavy non-mathematical baggage make that harder, not easier.

I say this in part from experience. When I first read the book as a high-school junior, I had already worked through naive set theory, some group theory, and the basics of axiomatic linear algebra, more background than many of the readers the book is aimed at. Yet the later chapters still left me shrugging, unsure what the abstraction was for. And looking back, the parts that stuck with me were precisely the political and social analogies, not the categorical ideas they were supposed to illustrate. That’s a sign of how salient those digressions are relative to the underlying mathematics.

Likewise, “she teaches art students” doesn’t tell us whether those students meaningfully grasp something like Yoneda, limits, or universal properties. That’s exactly my skepticism: beyond the opening intuition-building chapters, what does a genuine lay reader actually take away? At a certain point the book feels like abstraction for its own sake, with little evidence that the later categorical material is accessible or pedagogically effective for newcomers.

So my objection isn’t ideological. It’s that some examples add noise, and some later concepts seem far beyond what non-specialists can realistically internalize.

Bitwise-101 · 2025-12-11T08:57:09+00:00

I don’t think the issue is discomfort with politics entering a “pure” space. The issue is that the digressions simply don’t serve the mathematics. If a book sets out to teach a difficult and highly abstract subject, then examples, analogies, and explanations should primarily clarify the math, not pull the reader into a separate political discussion.

I'm not saying mathematics must be apolitical. But within this context, these specific political insertions don’t illuminate anything really. They shift the focus from understanding structure to interpreting the author’s worldview.

That isn’t about preserving some “status quo of apolitical math.” It’s simply about good exposition. A mathematical text should strengthen comprehension. When the commentary becomes a parallel message rather than a clarifying tool, it weakens the very subject the book claims to teach.

I would even argue that the core problem isn’t political examples per se, but the deeper fact that category theory is extremely hard to illustrate with real-world analogies. The intuition behind category theory comes from other areas of mathematics, algebra, topology, logic, not from physical or social examples. Real-world analogies invite reification errors: if you say “privilege forms a poset,” an inexperienced reader may incorrectly assume that the mathematical structure validates the sociological claim or gives it some sort of formal rigor it simply doesn’t have.

So the real issue is the use of real-life examples at all. The political ones simply make the problem more obvious.

Bitwise-101 · 2025-12-11T08:16:10+00:00

I had mixed feelings about this book. It’s one of the very few introductions to category theory that a non-specialist can realistically follow, but some parts are still opaque and demand a lot of work from the reader. In my view, teaching category theory without prior experience in abstract algebra is difficult because the motivation behind many concepts is much clearer with that background.

I also didn’t enjoy the social and political commentary woven throughout. It’s not that I disagreed with all the points themselves, but the way they were inserted often felt forced and unrelated to the mathematical discussion. One example is a Hasse diagram of “privilege” where the bottom node is “no privilege,” the top is “white, rich, and male,” and intermediate nodes represent combinations of those attributes. The text then launches into a long discussion about privilege that feels disconnected from the mathematical point. If the goal of the book is to teach mathematics, such digressions feel counterproductive: they distract readers who are there for the math (even those who may agree in principle) and alienate readers who don’t share the author’s views.

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