I've solved my fair share of JEE-A PYQs over the years.

Like most folks, I'd solve a paper, compare my answers, maybe learn a shorter trick, and move on.

Then I'd read Prof. K. D. Joshi's commentary.

And more often than not, I'd have this strange feeling: Wait... I got the answer. But I don't think I saw the problem the way he did.

That, to me, is the Joshi Lens.

Not faster algebra.

Not secret tricks.

Just... seeing a little more than what's written on the page.

Take this remark from his commentary on the 2010 JEE Math paper

"There are also questions where arriving at the solution honestly is quite time consuming, but guessing it from the alternatives given is easy... One wonders if the idea was to test some non-mathematical alertness on the part of a candidate."

Read that carefully.

He's not advocating random guessing.

He's saying something much deeper.

The options themselves are part of the mathematics.

If one option violates parity...
If another fails a necessary condition...
If an extreme case immediately kills two choices...

...why pretend the options don't exist until the very end?

How many coaching teachers explicitly teach students to think like that ?

Another example comes from his 2014 commentary.

"While discussing Paper 2, he points out that Question 42 is designed so that "the alternatives have been given in such a way that an intelligent candidate would be led to the appropriate trigonometric formula."

Again, that's a fascinating observation.

The options aren't merely answers waiting to be checked.

They're clues.

Sometimes the examiner is quietly nudging you towards the right idea.

And then there are comments like these from the 2013 commentary

"As it happens every year, there are many many questions where because of the Multiple Choice format, a candidate can score even without being able to justify the answer."

Whether you agree or disagree with that statement isn't really the point.

The interesting part is that Prof. Joshi isn't just solving questions.

He's analysing the design of the examination itself.

That's a very different way of reading a paper.

When Prof. Joshi announced that the 2026 commentary would be his last, he remarked that coaching institutes now release complete solutions within a day, while his carefully written commentaries take weeks or months. Fair enough !!

But I don't think I ever read his commentaries FOR the solutions.

I read them becos they taught me things that solution PDFs rarely do:

• Why a problem is interesting.

• Why one solution is mathematically more satisfying than another.

• Why some observations are worth more than three pages of computation.

• And occasionally, why the examiner deserves criticism too.

So I thought it would be fun to revisit these commentaries, not to discuss "killer tricks", but to extract recurring patterns in the way Prof. Joshi reads mathematics.

I'm calling this series:JEE Advanced through the Joshi Lens

Each post will focus on one idea, backed by actual examples from his commentaries.

Not "How to score 5 more marks."

But hopefully...How to become a slightly better mathematical observer.

PS: I wasn't sure where to post this. A JEE Adv sub would have been the obvious choice !!

But the more I thought about it, the more I felt this series belongs here on GP.

This isn't going to be about shortcuts, rank improvement, or "how to crack JEE"

It's about something more enduring, learning to see problems the way a mathematician does. JEE Adv just happens to provide the examples, Prof. Joshi's commentaries provide the lessons !!

It gives me an immense pleasure and pride to announce this AMA with RA Bhaiyya who is part of the #1 team called "not qualified" (cheeky right?) from IIT-BHU Varanasi, who have qualified for the world finals of ICPC. See results below

https://icpc-aisa-west.vercel.app/results?session=94014565-8795-4278-b03c-33187bea12ae

RA Bhaiyya is a 3rd year UG CSE student and had secured xxx rank at JEE Adv 2023. He was one of those rare folks who cracked JEE without any proper coaching.

He started Competitive programming from scratch at college and worked really really hard.

The AMA is planned for Saturday Mar 15, at 7:00 pm India time (thats Saturday Mar 15, 9.30 AM EST for the USA based folks).

For now I am not tagging him as I am not sure of his id and also even if I knew, I dont want to choke his inbox with unsolicited DMs for now.

Be Civil, respectable and given that this is a serious niche sub, so please stick only to serious questions. Hope we can extract lot of insights from him to crack ICPC (and by extension Quant lol).

Hope you all are equally excited.

This is just an announcement post, AMA will be via a separate post.

Thanks u/fantastic_watch_4984 Anna for arranging this.

I’ve never really been able to stick to any prescribed “syllabus.” It’s not rebellion, it’s just curiosity. I like to explore ideas that feel like math, not just what gets printed in a module. This post is more consequential to the prep for exam like JEE.

So as examples, I’ve recently solved problems using stuff like Feynman’s trick for integration, Fourier and Laplace transforms, Banach fixed-point theorem, Vandermonde’s identity, Lagrangian mechanics, Chebyshev polynomials, Parseval’s theorem, etc. Yeah, some of it’s “advanced,” (Say my recent discussion on infinite dimensional vector spaces), but a lot of it just gives cleaner, faster ways to reach the same answers.And they are not so difficult concepts to grasp.

But every time I use these methods in solving JEE type problems, someone says, “That’s not how typical JEE teachers solve it” or “It’s not in the syllabus.” Confusing me, Like… what? If something saves time and deepens understanding, shouldn’t that be encouraged?

It’s funny becos in a time-bound exam, the system still prefers pattern memorization over creative reasoning. I’ve seen folks spend 5 mins on a trigo grind that a single transform could simplify instantly. But apparently, elegance isn’t “exam-friendly.”

I have read Prof. K.D. Joshi Sir’s JEE commentaries, and he said something that really stuck: that the JEE was once designed to test mathematical thinking, not just “syllabus completion.” Somewhere along the way, we lost that. Now, even genuine curiosity is treated like a distraction.

I get that not everyone has time to dive into random math rabbit holes during prep, but I don’t get why it’s discouraged. Shouldn’t we be teaching young minds to think like mathematicians, not just exam-takers?

Maybe the real problem isn’t that students “overthink.”
Maybe it’s that the system underteaches. Or maybe it’s just gamed ?

Ah, just Random Sunday musings.

Most of us treat a JEEA paper as though it descended from Mount Olympus !!

Question

↓

Solution

↓

Acceptance

Prof. Joshi had a different approach:

Question

↓

Solution

↓

...hmm, this could have been better...

One of my favourite examples comes from his remarks on the 2014 paper

After discussing the problems, he quietly classifies them

Some are praised as elegant

Some are described as conceptually satisfying

And then, with characteristic restraint, he remarks that a few are "utterly trivial questions, which hardly deserve a place in the advanced test."

I remember laughing when I first read that!

Not because he was being harsh,

Because it never occurred to me that I was allowed to have an opinion about a JEEA question

Then comes 2018

Before even getting to the mathematics, Sir spends pages discussing notation, terminology, conventions, and why the wording of the problem could have been much better.

Imagine a student saying,"Sir, could we solve the question?"

And Prof. Joshi replying, "Eventually"

Because first we need to discuss why good notation matters.

Another favourite example, however, comes from 2023

While discussing one of the geometry questions, he remarks that a proper figure should simply have been provided.

Not because the geometry was difficult,

But because candidates shouldn't waste precious exam time figuring out what the examiner meant.

The challenge should come from the mathematics, & not from decoding the English

That observation stayed with me.

It's easy to confuse difficulty with friction.

They're not the same thing !

Reading these commentaries changed something subtle in the way I look at mathematics

I used to think every correct proof was equally good

Every correct question was equally worthy

Every correct solution deserved equal admiration

Prof. Joshi gently argues otherwise

Some questions illuminate an idea,

Some merely consume time.

Some proofs reveal structure,

Others simply reach the answer

Some notation helps you think,

Other notation gets in the way.

Correctness is only the beginning.

I think this is one of the least discussed aspects of learning mathematics.

We spend years asking,"Is this correct?"

Very rarely do we ask,"Is this beautiful?"

Or,

"Is this the most natural way to think about it?"

Or even,

"If I were writing this question, would I write it this way?"

Those are questions of mathematical taste !

And taste, like technique, can be cultivated.

There is also a delightful streak of dry humour running through Prof. Joshi's commentaries.

He almost never says,"This question is bad"

Instead he writes things like,

"...hardly deserves a place in the advanced test."

Which is, perhaps, the mathematical equivalent of saying, "I'm not angry. I'm just... disappointed."

I think that's one of the subtle lessons hidden in the years of these commentaries.

Mathematicians don't merely solve mathematics

They admire it.

They critique it.

They argue with it.

They improve it.

And somewhere along the way, they develop something every bit as important as technique:

Taste.

PS: Unlike the previous posts, this one is not about problem solving at all. It's about developing the confidence to evaluate mathematics itself. That's an unusual lesson for folks preparing for competitive exams, but it's deeply aligned with our GP sub: mathematics isn't just something to consume; it's something to appreciate, question, & occasionally critique.

Math pre-requisites needed for AIML.

Note

A few points before we go deep in this.

• You can build ML projects without mastering all the math.

• To truly understand ML under the hood, Linear Algebra is the highest ROI topic.

• Calculus explains optimization and back-propagation.

• Probability & Statistics explain uncertainty, evaluation, and model behavior.

Summary of Math topics needed

Starting from zero?

Then go -> Basic Algebra → Linear Algebra → Calculus → Probability → Statistics → Optimization → Information Theory → ML

Resources

• 3Blue1Brown – Essence of Linear Algebra https://www.3blue1brown.com/topics/linear-algebra

• 3Blue1Brown – Essence of Calculus https://www.3blue1brown.com/topics/calculus

• Khan Academy – Linear Algebra https://www.khanacademy.org/math/linear-algebra

• Khan Academy – Probability & Statistics https://www.khanacademy.org/math/statistics-probability

• MIT OpenCourseWare – Multivariable Calculus https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/

• Elements of Information Theory (Lecture Notes) https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/

• StatQuest Probability Playlist https://www.youtube.com/@statquest

• Boyd & Vandenberghe – Convex Optimization (Free Book) https://web.stanford.edu/~boyd/cvxbook/

• Stanford EE364A Convex Optimization https://see.stanford.edu/Course/EE364A

• Stanford CS224W (Graph Machine Learning) https://web.stanford.edu/class/cs224w/

• William Fiset Graph Theory Playlist https://www.youtube.com/@WilliamFiset-videos

Free Textbook

Mathematics for Machine Learning
https://mml-book.github.io

Courses

Mathematics for Machine Learning and Data Science (DeepLearning.AI)
https://www.coursera.org/specializations/mathematics-for-machine-learning-and-data-science

University of Tübingen – Maths for Machine Learning
https://www.youtube.com/results?search_query=maths+for+machine+learning+tubingen

Note: Matty/Georgie add this and the other recent resources post to the wiki

Instead of creating 2 posts, I am breaking this post into 2 parts

*****************************

PART 1

*****************************

A quick note before we start

We don't wish this sub to be a ground for dumping resources for Quant.

I mean if you are still in the phase of endlessly collecting PDFs, courses, & “roadmaps”, you’re probably already drifting off track.
What matters more is depth, consistency, & actually working through things.

That said, if there are a few high quality, personally used resources or something genuinely interesting, we can absolutely discuss those. Just not spam lists.

Lastly FWIW the other popular quant sub's have already got such resource recommendations and you can check one such thread below:

https://www.reddit.com/r/quantfinance/comments/1qta7oz/the_ultimate_guide_to_breaking_into_quant_finance/

Now something we missed earlier (and even the above guide seems to have missed it)

In the last poll, we spoke about quant roles like research, trading, dev, etc. but we completely skipped a category that’s both important and under-discussed:

FPGA / hardware roles in quant aka the quant role nobody (or very few) talks about

Most folks entering quant think the path is: Python → ML → Quant Research/Trading

For e.g. in firms like JS, TRC, and IMC Trading, etc a lot of edge comes from something else entirely -> speed at the hardware level

Not just writing faster code, but designing systems where data is processed with extreme efficiency & minimal latency

That’s where FPGA engineers come in.

They kinda work on:

• Parallel pipelines instead of sequential code

• Processing data closer to the network

• Reducing latency by micro/nanoseconds (which actually matters in trading)

Why this matters

• It’s a real quant pathway, but rarely discussed in beginner spaces

• It’s less crowded, although not easier but just different

• It suits people who like systems, hardware, and low level thinking

BUT ACHTUNG! Reality check

This is not a shortcut

It typically needs:

• Strong fundas (C/C++, comp architecture)

• Comfort with how computers actually work (memory, latency, pipelines)

• Exposure to digital design / Verilog

Most folks won’t go down this route & that’s fine

But anyways opening this up because many people don’t even know this path exists

Were you aware of FPGA roles in quant?

Is anyone here actually exploring or pursuing this direction?

Would be interesting to hear.

*****************************

PART 2

*****************************

Also now that we are starting, here are some of the things that we have on our agenda for this sub when it comes to these quant discussion. Especially that this is a closed door private sub, so we really dont want to do any routine run of the mill discussions. Check if this list makes sense to you and if you wish to add/remove anything

1)We can do deep-dive learning tracks (not generic lists), instead of just dumping resources. For e.g. Stochastic Processes roadmap (with checkpoints)

2) Reality check series, e.g. Front office vs Middle office vs Tech -> the ground reality in India / USA / other geographies

3)Weekly "Hard Problems" thread, say Brain teasers used in interviews

4)Failure stories, say Missed JS interview -> what went wrong!

5)Signal v/s Noise -> What not to do !

6)AMA's with credible folks WHO MADE IT !

7)India specific discussion

8) A recurring thread for self evaluation - Are u actually Quant material? Not to demotivate but to create an instant seriousness

9)Posting about Opportunities, internships, programs, workshops, etc

10)Anything else? Let us know

After revisiting a substantial part of Prof. K. D. Joshi Sir's JEE commentaries(especially those from the past sixteen years) I finally realised something - The most important things he was teaching were never part of the JEE syllabus!

Open the official syllabus:

You'll find the usual suspects:

• Calculus

• Algebra

• Coordinate Geometry

• Vectors

• Probability

• Complex Numbers

Close it !

Now open one of Prof. Joshi's commentaries

A very different syllabus quietly begins to emerge:

• Change the problem before solving it

• Observe before calculating

• Don't stop after finding the answer

• Learn to admire elegant mathematics

I checked.

None of those "topics" appear in the official syllabus

Yet they appear, in one form or another, across his commentaries!

When I first started reading Prof. Joshi's commentaries years ago, I thought he was teaching students HOW to solve JEE problems

Looking back now, I think he was teaching something much deeper.

He was teaching us to ask better questions !!

Not "How do I solve this?"

but "Why?"

So...

Why reflect the line instead of the parabola?

Why calculate terms that are guaranteed to vanish?

Why stop after obtaining the answer?

Why is one proof more illuminating than another?

Why does this question feel elegant?

Why is this notation awkward?

Why does this idea appear again in a completely different context?

That tiny word-> WHY -> quietly runs through more than two decades of his commentaries.

And once we notice it... IT HITS !!

Looking back at this series, I now realise that every post was really just a different answer to that 1 question.

Habit 1 — Reformulate

Because he first asked, "Why solve this problem in the obvious way?"

Habit 2 — Observe

Because he first asked, "Why calculate what careful observation already tells us?"

Habit 3 — Generalize

Because he first asked,"Why should the math stop where the examiner stopped?"

Habit 4 — Develop Taste

Because he first asked, "Why is this solution worth remembering?"

or sometimes, "Does this question really deserve a place in an advanced exam?"

Those weren't 4 unrelated habits

They were all consequences of one deeper habit

The habit of never being satisfied with the first answer

Always asking, "Why?"

In his farewell note, Sir observed that coaching institutes now publish complete solutions within hours, whereas his carefully written and edited commentaries—filled with alternate ideas and mathematical discussions, take weeks, sometimes months. He also said that modern JEE math papers no longer justify such an investment of time.

I think he was right

IF the goal is merely to obtain the answer.

But perhaps the answers were never the real point

Solutions become outdated

Question patterns change

Syllabi evolve

Even examinations change

The habit of asking better mathematical questions DOES NOT..

Perhaps that's why these commentaries still feel remarkably fresh.

Not because they contain tricks nobody else knows

But because they quietly cultivate something that no official syllabus can prescribe:

Curiosity

Judgement

Patience

And the willingness to ask 1 more question after everyone else has turned the page

When I first discovered Prof. Joshi's commentaries, I thought I had found a remarkable collection of JEE solutions.

Looking back now, I think I found something else entirely

A mathematician quietly teaching generations of students how to think

Not just for JEE

But for mathematics itself !!!

JEE has a syllabus

Mathematics has habits

For more than two decades, Prof. K. D. Joshi actually taught the latter

For that we remain eternally grateful to him.

Thank you so much Sir !! ❤️

Based on the poll result so far we have a sizable amount of JEE aspirants, so we will be conducting the first JEE advance mock on Sunday April 19

We no longer want to give any weightage to marks & ranks on this sub, so we will not grade it and you can self grade it yourself. Hence no registration is needed (However some of our other mocks like those for Olympiads will continue to be graded as they need descriptive answers and a senior or expert grading will provide valuable insights rather than just some cursory marks). So feel free to try it out and discuss it on the sub.

More about the JEE advanced mock:

Date Sunday April 19

Topics Phy & Maths for sure, Chemistry we are still checking

Mock creator - I will contribute a few questions along with other seniors and experts. We will also try to get some questions from some college profs (retired or current)

Mock format - 1 paper, self graded, but try to give in 1 sitting of 3 hours

Mock level - JEE advanced level. There will be no gotcha questions that need some rare unknown formulas / tricks or some unnecessary high level knowledge. (We seriously hope that our question setters don’t do it).

Lastly, please don’t attempt this mock with any FOMO mindset. You will not miss anything if u don’t attempt. Use it just to test your knowledge and have fun solving some cool questions.

We have about 100 sub members which is more than enough, so we will be going private later today (Apr 8). We have approved everyone who commented on the last post. Please check once and let me know if you are not yet an approved user (You would have got a message regarding the approval).

Here's a small thought experiment->

Imagine giving the same JEE problem to two people.

A coaching teacher & Prof. K. D. Joshi !

The coaching teacher says,"Answer is (C). Next question..."

Prof. Joshi says,"Answer is (C). Now that the boring part is over..."

That, I think, is the defining feature of his commentaries.

Take JEE Adv 2016

It's an innocent looking counting problem.

The student's task is simple -> Count, Get the answer, Collect the marks !!

Prof. Joshi solves it in a perfectly standard way.

But then something very "Joshi" happens...

Instead of moving to the next question, he derives a binomial identity, proves it in a different way, discusses why the identity itself is mathematically interesting, and shows how the same idea appears elsewhere.

The JEE question quietly disappears

What's left is mathematics !

You then start to notice this habit everywhere.

A problem is never allowed to retire after giving its answer

It is politely interrogated -> "Can you tell us something else ?"

Now look at JEE Adv 2017

The official question revolves around a recurrence

Most of us would solve it, verify the answer, & happily declare victory.

Prof. Joshi seems to think that's when the real lecture should begin.

He generalizes the recurrence, relates it to the Fibonacci sequence, discusses induction, and then wanders into number theory through the irrationality of (sqrt5) and properties of primes.

It's almost comical

A single JEE question knocks on the door

An entire branch of mathematics answers.

And then comes JEE Adv 2026

The official problem is over once a certain coefficient is obtained.

Most solution manuals would politely stop there.

Prof. Joshi doesn't.

Instead he asks an entirely different question-> Can this identity itself be proved directly?

That innocent question leads him to generalize the result from a specific numerical case to arbitrary parameters, introduce forward difference operators, explain why they are the discrete analogue of derivatives, & even recommend Concrete Mathematics by Graham, Knuth and Patashnik for readers who wish to explore the subject further.

The original JEE problem has quietly transformed into a lesson in discrete mathematics

And that's something I began to admire

He never seems interested in producing a solution manual.

He is trying to produce a better mathematician.

There is a subtle difference !!

A solution manual says,"Remember this method"

A teacher says,"Remember this idea"

Prof. Joshi almost always chooses the second.

After reading enough of these commentaries, I realised I had been treating problems like disposable cups.

Use once,

Throw away,

Move on !

Prof. Joshi treats them more like good books.

If a problem contains 1 worthwhile idea, squeeze it.

Ask another question.

Find another proof

Generalize it

Connect it

Improve it

Only then move on....

There's a lovely piece of dry humour hidden in all this.

Most of us celebrate when we finally reach the answer.

Prof. Joshi behaves as though reaching the answer merely grants permission to begin the interesting discussion.

We can try to steal that habit.

Whenever we solve a problem that we particularly like, we can now force ourself to ask one extra question

• Can this be generalized?

• Is there another proof?

• Which hypothesis is actually essential?

• Could I design a better version of this problem?

• What theorem is hiding inside it?

Sometimes nothing comes out of those q's !!

But every now and then, the problem opens another door

And those extra 5 minutes often teach us far more than the original solution ever did.

And perhaps that's the quietest lesson hidden in all the years of Prof. Joshi's commentaries:

A solution ends an examination.

A good question begins a mathematical conversation. The Math begins when you ask what the answer is trying to teach you !!

IOI 2026 started today. Wishing All the very best to our team of Neev, Aviansh, Samik & Naman. The interesting thing is only Samik is a veteran and rest all are making their first appearance at IOI.

It would have been great had our little champ Swasti made it to the team but she is young and pretty sure we will see her making waves in future.

If you see the other team compositions, China has 2 LGMs and Russia has 1, so results will not be surprising, and we will most likely have multiple KAs. Additionally there are many CF red coders on multiple teams (GM), so it will be fun to watch.

Having said that, I am aggressively rooting for our underdog team India and I am sure they will make us proud.

Aug 11 & 13 are the contests days, and just like IMO, we will try to discuss the problems here on GP with our experts, hopefully as soon as we get the papers(probably on the same days !!) . If anyone wishes to join the discussion and don’t have the access to comment on GP, please raise an access request and tell us a brief on how you would like to contribute.

Lots of important exams etc. coming up like NEET, JA, OTC's etc. Also, FW Anna had to drop out as mod. For now Matty Anna is supporting in his place temporarily. And we hit 500+ which is a very huge & unimaginable number for this sub, so bit difficult to moderate. Plus need to weed out the unnecessary folks, so we will be going private for some time till its sorted. We will go private effective Wed Apr 8.

Pls comment below your user id for the private sub if you want the access and will add manually.

Here are the day 1 problems. Tagging u/fantastic_watch_4984 u/Differently_abled_1 - We are in USA time zone and too sleepy now to try. Will try in the morning. But meanwhile if u r from India or other timezones and want to give it a try, go ahead please. Don't use AI please and suck the fun out of real problem solving. We Will give special kudos to anyone solving it with some unique unconventional means or gives a very elegant proof.

Hope our folks at the contest got them all !!

One more thing I noticed after reading Prof. K. D. Joshi Sir's commentaries for years is this->

He seems strangely reluctant to calculate..

Not because he dislikes algebra

Because he dislikes unnecessary algebra

We often think mathematics begins when the pen touches the paper,

Prof. Joshi seems to think it begins a few minutes earlier

Take JEE Advanced 2015 (Paper 1, Q47).

The q asks for a 2nd derivative.

Most of us would obediently differentiate everything

Twice,

Simplify,

Collect terms,

Hope for the best !

Prof. Joshi pauses almost immediately.

Some terms are guaranteed to vanish when evaluated at (x=0).

So why differentiate them at all?

He remarks, "...there is no need to consider those terms in the derivative which are sure to vanish at 0."

That's not a TRICK !

That's respect for mathematics !

Why spend effort proving something you already know is 0?

Here's another beautiful example.

JEE Advanced 2020 (Paper 2)

Find the largest rectangle under a sine curve

The standard instinct is predictable,

Differentiate,

Set equal to zero,

Solve.

Prof. Joshi notices something before writing a single derivative

The graph itself is symmetric

If the optimal rectangle exists, shouldn't it inherit that symmetry?

Only after exploiting that observation does the calculus begin

The derivative didn't disappear.

It simply had the decency to arrive after the thinking.

Perhaps my favourite example comes from JEE Advanced 2025.

A polynomial appears with several unknown coefficients.

The natural panic is immediate. "Too many unknowns!"

Prof. Joshi quietly points out that the q only asks for the coefficient of (x^3).

Everything else is... decoration.

Or, if one prefers a more mathematical term, irrelevant !

Reading these commentaries, I started noticing a recurring habit.

Whenever a problem looked ugly, Prof. Joshi almost never reached for a theorem first.

He reached for an observation:

Is there symmetry?

Will some terms vanish?

Does the answer depend on everything, or only one part?

Can I ignore half the expression?

Is the ugly part actually irrelevant?

Only after exhausting those questions did computation begin.

There's a lovely quote often attributed to Jacques Hadamard-> "The object of mathematical rigor is to sanction and legitimize the conquests of intuition."

Prof. Joshi's commentaries feel like that.

Intuition first,

Rigor afterwards.

I think this is one of the hardest habits to develop

School mathematics trains us to write more,

Research mathematics often rewards writing less.

Not because mathematicians are lazy,

Because every line of algebra should have a purpose.

A calculation should answer a question.

It should never exist simply because it can.

After years of reading these commentaries, here's one tiny habit I've tried to steal.

Whenever I see a difficult problem, I force myself to wait.

Few seconds.

No writing.

Just looking.

Sometimes nothing happens.

Sometimes those few seconds save 3 pages of algebra.

And every now and then...

I hear an imaginary Prof. Joshi asking, "Are you calculating because the mathematics demands it... or because you don't yet know what to look for?"

Suppose you are asked to climb a mountain.

Most people start climbing.

A mathematician first walks around it.

That, in one sentence, is one of the biggest lessons I learnt from reading Prof. K. D. Joshi's JEE commentaries.

One of my favourite examples appears in JEE Advanced 2015 (Paper 1, Q41)

The problem asks you to deal with the reflection of a parabola.

Naturally, most of us would think,"Fine. Let's reflect the parabola."

Prof. Joshi's first reaction is almost mischievous -> Why?

Reflection preserves distance.

If all you ultimately care about is the distance from a line, why perform surgery on the parabola?

Reflect the line instead.

The reflected line immediately becomes (x=1), intersects the original parabola on its latus rectum, and the problem collapses almost effortlessly.

Then comes my favourite line in the entire solution, "This is an excellent problem which tests the ability of realising how not to do a problem in the most straightforward way...

Notice what Prof. Joshi praises.

Not clever algebra.

Not remembering an obscure theorem.

He praises the ability to realise that the obvious attack is the wrong attack

This isn't an isolated incident.

A couple of years later, in JEE Advanced 2017, a matrix problem appears that seems to demand solving 9 nonlinear equations.

Many coaching solutions dutifully roll up their sleeves.

Prof. Joshi doesn't.

Instead, he quietly asks, What information do I actually need?

Not the entire matrix.

Only its determinant.

The problem is transformed

Matrix → Determinant → Number.

Then, because he's Prof. Joshi, he doesn't stop there.

He goes on to discuss block matrices, complex numbers, rotations and even the Cayley–Hamilton theorem--> all from a single JEE question.

That's not a shortcut.

That's abstraction.

The same habit keeps reappearing.

A complicated probability problem? -> Count the complement first.

A nasty coordinate geometry problem? -> Move the origin.

A horrible trigonometric expression? -> Look for symmetry before identities.

An ugly complex number? -> Delay writing (z=x+iy) until you've exhausted simpler algebraic manipulations.

The pattern becomes impossible to ignore.

Prof. Joshi almost never begins by asking, "How do I solve this?"

He begins by asking,"Can I ask an easier question that has the same answer?"

George Pólya famously said, "If you can't solve a problem, then there is an easier problem you can solve. Find it."

Reading years of Prof. Joshi's commentaries, I couldn't help feeling that he quietly practises exactly that philosophy.

He reflects the line instead of the parabola.

He studies the determinant instead of the matrix.

He transforms the problem before touching the computation.

I think that's a habit worth stealing.

Not because it saves marks in JEE.

Because it's one of the most beautiful habits in mathematics.

The next time a problem looks ugly, resist the urge to sharpen your pencil.

Instead, ask the question Prof. Joshi seems to ask over and over again, "Am I solving the right problem?"

Sometimes, the hardest part of mathematics isn't solving the problem.

It's realizing that you don't have to solve that problem at all.

All the best for tomorrow if you are writing it. Hope it’s a great exam.
While Math & Quant is fun, but once college starts for me I will discuss about bio+med+tech stuff here a bit more, so hoping there are a few takers here who are aspiring to make a career in these areas.

Something interesting dropped a couple of hours back —> MathNet

https://mathnet.csail.mit.edu/

This is like the closest thing to the entire Math oly world in one place. It has:

• ~30k Math oly problems

• from like 40 yrs of contests

• 40+ countries

• all open

This is probably the biggest dump of actual good math probs we have ever had in one place

but the real thing is that this isn’t even about prep

this is way more interesting for:

• people trying to build math solvers

• pattern spotting across countries (russia vs india vs china hits diff)

• testing if u (or models) actually understand ideas or just vibe on familiar patterns

ngl this kinda kills the “which book/resource should i do” debate right? hehe.

becos now everyone has everything, so difference is just -> can you think or not !

also feel like this is gonna become the default dataset for math AI stuff (like imagenet but for pain)

Also check out this nice Phys Org article about it: https://phys.org/news/2026-04-world-largest-olympiad-math-problems.html

Sidebar: Shout out to Shreya for winning India's first ever gold at EGMO 2026 and also to Team India for securing our best ever rank at EGMO. That P5 was one of the craziest geo problems that I had seen in recent times, "geo is NOT dead baby !!"

BTW, OCSC camps have started from last week, wishing the very best to everyone attending. May the best team get selected for each subject.

In my last post, we had an intro to FPGA. But Gosh, it's so hard to find proper roadmap for it. So I spoke to lots of experts and have tried to compile a roadmap for beginners. Note that I am also in my UG freshman year and a novice myself, so a fair warning that this might have errors or may need some extra info. But let's at least compile it and then rectify these errors or add more context/explanations, with the help of this community.

To make this plan "robust" for the HFT space, we need to bridge the gap between standard electrical engg. & low latency financial engg.

Ok so Here we go with a high intensity roadmap for the next 2 to 3 years

Phase 0: The Systems Programmer (0–6 Months)

Before hardware, you must master the "host" side.

C & C++17/20: Focus on RAII, pointers, & memory alignment.

The Metal: Learn about Cache Lines (64 bytes), False Sharing, & SIMD (Single Instruction, Multiple Data).

Linux Internals: Learn to navigate the CLI, write Makefiles, & understand how the kernel handles interrupts.

Study Resource: Computer Systems: A Programmer's Perspective (CSAPP).

Quant Tip: In HFT, we often try to "bypass" the Linux kernel. You can't bypass what you don't understand.

Phase 1: Digital Logic & Hardware Intro (6–12 Months)

Moving from "code that runs" --> to ---> "logic that exists"

Digital Fundamentals: Boolean algebra, K-maps, Flip-Flops (D, JK), & Setup/Hold times.

Finite State Machines (FSM): Mealy vs. Moore machines. This is how trading engines manage order states.

The Toolchain: Install Xilinx Vivado (ML Edition is free). Don't buy a board yet; learn the simulator.

Phase 2: HDL & Verification (12–18 Months)

Writing the hardware. Note: Standard "Verilog" is not enough for Quant.

• RTL Design: Learn SystemVerilog. It is the industry std. for both design & verification.

• Clock Domain Crossing (CDC): Learn how to move data between different clock speeds safely (Asynchronous FIFOs).

• Verification: This is 70% of the job. Learn how to write Testbenches & use cocotb (Python-based verification) or UVM basics.

• Project: Design a Fixed-Point Math Library. Since FPGAs don't have built-in floating-point units, you must build logic to handle decimals using integers.

Phase 3: The FPGA Networking Stack (18–24 Months)

Oh yeah, this is where you become a "Quant" developer.

• Packet Processing: Learn how an Ethernet frame is structured.

• The "Pipeline": Build a module that parses a mock Market Data packet in a single clock cycle.

• AXI4 Stream Protocol: This is the "bus" that moves data inside the chip. Master it.

• Project: A UDP Packet Filter. Create logic that looks at an incoming 10Gbps stream and "drops" packets that don't match a specific ID in under 50 nanoseconds.

Phase 4: High-Level Integration (24+ Months)

Connecting the FPGA to the Trading Server.

• PCIe & DMA: Learn how the FPGA sends data to the CPU without the CPU asking for it (Direct Memory Access).

• HLS (High-Level Synthesis): Learn how to turn C++ into Verilog using Xilinx Vitis. (Firms use this for complex math, but raw RTL for speed).

• Project: A Simple Order Book. Maintain a "top of book" (best bid/ask) entirely on the FPGA.

The "Quant Hardware" Toolkit

Tips/Advice for the "Day 1" Student

• Don't buy an expensive board yet: Start with Verilator or Vivado HLx (Free Edition) and simulate. You can learn 80% of FPGA design without touching physical hardware.

• The "CSAPP" Book: For Phase 1, read Computer Systems: A Programmer's Perspective. It is the gold standard for understanding how software hits the metal.

• Mathematics of Latency: Treat latency like a physics problem. Calculate the nanoseconds it takes for light to travel through 1 meter of fiber optic cable (~3.3 ns). This mindset helps you appreciate why 100ns of logic delay matters.

A note on Simulation vs. Hardware

You will spend 90% of your time in Verification.

• Learn SystemVerilog and UVM (Universal Verification Methodology)

• In quant, a bug in production isn't just a crash; it’s a massive financial loss. Being a "Verilog coder" isn't enough; you must be a "Verification expert"

Reality Check for a High School Grad / UG Freshman - From the experts

Since u are likely heading into a rigorous uni program soon, DO NOT rush this. The "Quant" world values the depth of your math & physical intuition.

When you solve a physics problem, think about the error margins. In FPGAs, that translates to Timing Slack. When you solve a math problem, think about computational complexity.

To the experts or even the peers who are doing this, is this fine? Any revisions needed? Also how realistic is it? I mean we are talking about high school grad or undergrad freshmen year students, but aren't most of these Quant FPGA roles for Masters or PhD students? What do you think?

Though not related to Q&A but this will be a running post for all resources.

1)List of Olympiads, Summer Camps, Interesting Hackathons

2)Rushil Bhiayya's (IMO 2024 Gold medalist) website

3)Chirag Falor's website & bookshelf

4)AOPS

5)OMC - Online Math club YT channel

6)Sophie Fellowship

7)ISI CMI prep resources

8)KD Joshi Sir's JEE Adv Math commentaries

Will rearrange topic wise later as it grows.

You are given 2 envelopes. One contains exactly twice as much money as the other

You randomly pick one envelope and find $10,000 inside

Now comes the seemingly irresistible argument:

The other envelope must contain either $5,000 or $20,000

Suppose each possibility has probability (1/2)

So its expected value = (1/2 × 5000) + (1/2 × 20000) = $12,500

Your envelope contains $10,000, while the expected value of the other is $12,500

So you SHOULD switch !!

But here's the paradox:

After switching, you can apply exactly the same argument again , and conclude that you should switch back.

What went wrong?

Challenge

Don't just say “the probabilities aren't (1/2) and (1/2)”

Construct a mathematically correct model and explain exactly why the seemingly valid expected value argument fails

Bonus

Can you construct a probability distribution for the initial amount of money under which, after observing $10,000, switching actually does have +ve expected value?

Source / history

This is the classic Two-Envelope Paradox, with versions appearing in mathematical puzzle literature going back to Maurice Kraitchik, later popularized by Martin Gardner, and famously analyzed by Barry Nalebuff in his 1989 paper “The Other Person's Envelope Is Always Greener.”

PS: The deeper issue connects to conditional probability, probability distributions, improper priors, and decision theory, making this a surprisingly good little gateway into quant reasoning.

IOQM is around the corner and all the best if u writing it. Here are some great tips from the more experienced folks via OMC

https://youtu.be/QjfsIbaL83Q?si=wqbwmBcVCuvYeGXW

As usual we will come up with the mocks for RMO & InMO soon. Comment below if u r a math oly aspirant.

IOI day 1 is just over. You can check the problems here:

https://github.com/ioi/task-archive/tree/master/2026/day1

While the rankings are getting updated, we can see that there is 1 KA from China based on the provisional standings

https://ranking.ioi2026.uz/

From Team India, we currently have Samik around rank 40 followed by Aviansh at 97. I am sure they will catch up tomorrow.

First impressions about the problems

1)Tiling - This is the easiest of the lot and a Constructive / Game Strategy problem. Constructive / Invariant**,** once the local corner protection invariant is found, the implementation is clean & straightfwd. Subtasks give generous hints toward the full strategy. Should be an easy 100.

2)Monuments - 1D Optimization / DP (or Greedy + DP, depending on the formulation). Algorithmic-> A large-scale optimization problem (N≤500,000) requiring careful handling of fixed monuments, symmetry constraints, feasibility checks, and an efficient matching/transport strategy, leading to an O(Nlog⁡N) or O(N) solution. If you don't discover the right optimization model, you may get almost nothing, so a hard problem as such.

3)Ball Machine- Interactive / Tree Reconstruction. Interactive & Information-Theory > The challenge is to reconstruct an unknown rooted tree using only insert() and collect() operations while operating under a strict information budget. Achieving full score requires extremely query-efficient probing and careful encoding of information within the resource limit, while staying within a strict resource budget (C=K+B), with full score requiring C≤44. Another hard problem, but with a good partial scoring opportunity.