HN Theater @HNTheaterMonth

I have a master's degree in math and a PhD in STEM so I know how linear algebra works. So when I say it's excellent writing, I don't mean perfectly calibrated for me or you. I mean it's excellent writing for a wide audience to get the gist of it. I think that's very difficult to do for math. An average 12 year old would not understand matrices without a very clear explanation.

Having taught math to both children and undergrads, I think it's very easy to underestimate how easy it is once you already understand it. A sentence like:

> for any ε > 0 and 0 < δ < 1, using poly(dimension, #nonzeros, condition, 1/ε, 1/δ) floating-point operations, our algorithm finds a solution having less than ε error with probability 1 - δ.

Would confuse the shit out of most people.

This video explains really beautifully what I mean: https://www.youtube.com/watch?v=M64HUIJFTZM

Speaking of the IMO, I saw this video recently and thought it was pretty interesting: https://www.youtube.com/watch?v=M64HUIJFTZM

⬐ chongli

This phenomenon, whereby a person who knows the "trick" to solve a puzzle cannot accurate gauge its difficulty, seems to extend beyond mathematics. Adventure games (including text-based, parser-driven, and point-and-click) suffer badly from this problem. They are chock-full of puzzles that only make sense in hindsight (if at all). They can be really fun though!

⬐ baddox

Because of this fact, I can only imagine how difficult game design must be (particularly puzzle games). The "perfect" experience is for the player to have just enough difficult figuring out a solution that they feel clever when they succeed. I remember feeling this way several times in my first play-through of Portal, but then I remember Portal 2 feeling way too "guided" (sometimes even easy), as if they had play-tested it to death.

⬐ OscarCunningham

Tanya Khovanova has a paper https://arxiv.org/abs/1110.1556 with a list of some more of these problems with easy solutions that are hard to find. She calls them "Jewish Problems" because they were used by USSR universities to discriminate against Jewish applicants. The applicants would fail to solve them but the university would justifiably be able to claim they were easy.

⬐ dxbydt

Hey, thank you so much for posting this! I picked a random problem ( the one on logs) in that paper and solved it under 1 minute. I feel like a million bucks now!

⬐ eruci

This can be formally solved by constructing the arrangement of lines passing through each set of two points, then computing the dual of the arrangement.

The cell with the maximum depth on the arrangement contains the points in the "middle", meaning they have as many points on one side, as they have on the other.

Then you can prove that a line starting on any such point will visit every other point an infinite number of times.

⬐ boyobo

Just pick a point and rotate a line around that point continuously. Keep track of the number of points on the left. Since this count is essentially continuous (the jumps are of size 1), at some point during this rotation you will have an almost-balanced configuration (same number of points on left and right, up to parity error).

⬐ eruci

Yes, but it must be a point with certain properties, such that the same number of points are on either side of the line initially, otherwise, it would not work.

⬐ boyobo

I am giving an alternate proof of your implicit statement "There exists a line which has the same number of points on each side". You did this by computing duals of cell arrangements. I am arguing that you don't need to do that.

The proof I outlined will work for any point. Initially it might have the wrong number of points on both sides but for some rotation it will have the correct number of points on both sides.

⬐ x3n0ph3n3

This guy makes excellent math visualization videos -- some of the best I've ever seen.

⬐ quickthrower2

I recommend watching the "But what is the Fourier Transform? A visual introduction." https://www.youtube.com/watch?v=spUNpyF58BY. Beautiful.

⬐ ranie93

Yes! I really enjoy them as well-- and he has his animation software on github: https://github.com/3b1b/manim

⬐ Chris2048

Would be interested in how this stacks against mathbox2*

* https://acko.net/blog/mathbox2/

⬐ carapace

This is hella cool.

"Knowing when the math is hard is way harder than the math itself"

But then maybe the math is hard only because it's not being explained well? (I hope it's uncontroversial to suggest that our current methods of teaching math are not the best of all possible worlds.)

I get that this problem came up in the context of of a math puzzle contest, and that some people enjoy solving puzzles. I am questioning their utility as an educational device.

I kinda think that we should teach math as fast as we can so that we can concentrate on the stuff that's really hard, not just apparently hard because someone is being coy with the easy routes.

⬐ boyobo

Are you trying to say that math is hard because the teachers are withholding all the tricks?

> I am questioning their utility as an educational device.

Puzzles like this aren't found in mainstream math education contexts. As you acknowledged in your post, they are only found in math competitions. What do you mean?

⬐ carapace

> Are you trying to say that math is hard because the teachers are withholding all the tricks?

Kind of, although I don't think they do it deliberately.

Things like teaching logarithms without a slide rule.

> Puzzles like this aren't found in mainstream math education contexts. As you acknowledged in your post, they are only found in math competitions. What do you mean?

You're right. Let me try again.

Check out William Bricken's "Iconic Math" http://iconicmath.com/ or "Proofs without Words" https://en.wikipedia.org/wiki/Proof_without_words or the other 3Blue1Brown videos for that matter.

I think that most math seems hard to most people only because we are not creative in the ways that it is presented. We should use science to figure out how to present math so that people get it as fast as they can, in part so that we can find and concentrate on the actually hard math problems.

E.g. Alan Kay using Smalltalk to teach calculus to little kids in the context of modelling falling objects, to me kinda proves that it shouldn't take a whole semester to teach calculus to teenagers.

⬐ boyobo

> Check out William Bricken's "Iconic Math" http://iconicmath.com/ or "Proofs without Words" https://en.wikipedia.org/wiki/Proof_without_words or the other 3Blue1Brown videos for that matter.

Yes, those references demonstrate that some mathematical facts can be demonstrated easily, given the right presentation.

I agree that there is large gap between current mathematical presentations and the optimal presentation.

I am curious how effective the optimal presentation is. How easy can we make math? We have to be careful of the trap that the 3b1b video warns us against - when we understand something it is very difficult to put yourself in the shoes of a beginner. Something that looks like an elegant and clear presentation may seem like gibberish to the beginner (look at the YouTube comments on the 3b1b video).

I tried to google "Alan kay teaching calculus smalltalk" and didn't find anything. I am curious to see how much he actually taught the kids. It's clear that the typical college student, after taking a typical calculus class, doesn't really get the point of calculus. They may be able to follow some algorithms for differentiating and integrating but I don't think they understand when they can apply calculus.

⬐ carapace

I went and looked the Alan Kay thing up. It's not calculus, just acceleration. Sorry! It's described in "The Real Computer Revolution Hasn’t Happened Yet" http://www.vpri.org/pdf/m2007007a_revolution.pdf in the section titled "Children Discover, Measure, and Mathematically Model Galilean Gravity"

I think an important piece of the puzzle (no pun intended) is customized presentations with feedback for individually-tailored education. But then it occurs to me that group activity is also crucial for learning, eh? I'm not an expert.

⬐ _Microft

Since this is a puzzle that will certainly nerd-snipe a number of us, could someone who already watched the video tell us if there is a "spoiler" moment in it or if we could watch it bit by bit in case that we get stuck?

⬐ lelf

The problem:

Let S be a finite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill we mean a process as follows. Start with a line l going through a point P ∈ S. Rotate l clockwise around the pivot P until the line contains another point Q of S. The point Q now takes over as the new pivot. This process continues indefinitely, with the pivot always being a point from S.

Show that for a suitable P ∈ S and a suitable starting line l containing P, the resulting windmill will visit each point of S as a pivot infinitely often.

⬐ mlevental

if not all points in P are coplanar then what does it mean to rotate clockwise? perpendicular to which axis?

⬐ bcyn

They are all coplanar, but no three points are colinear. The first sentence just means that there are at least 2 points in the set.

⬐ ufo

The start of the video is talks about the problem and illustrates some concrete examples. At 6:50 it starts the solution, by presenting a key invariant.

⬐ jerf

There's maybe a step or two's worth of that, then the answer. The video is not explicitly structured as a set of clues, it only accidentally half-does that because it builds up to the answer mathematically, but I would not say the steps are "even" in size.

⬐ Tallasatree

is this in essence a convex-hull problem, except instead of closing, we continue?

⬐ AnimalMuppet

I don't think so. In a convex hull, you only visit points on the hull of the set, not in the interior. In this problem, you visit points in the interior too (if I understand correctly).

⬐ saagarjha

You get to choose the line so that this circuit is possible. If you didn’t, you’d essentially be able to form a convex hull and thus not visit all the points.

⬐ utopcell

No. In fact, in the end he shows a list of approaches one might think are related but turn out not to be, and computing convex hulls is one of them.

⬐ bcyn

I think he presents a line of reasoning that also helps to formalize why finding a convex hull works.

⬐ jhncls

Grant Sanderson manages to combine his compelling narrative style with elegant animations. The solution to the chosen puzzle only depends on very straightforward mathematical knowledge. Grant shows a possible route towards discovering the invariants that rule this puzzle.

It is a kind of puzzle to try out on your own, knowing that a rather simple mathematical reasoning solves it. It doesn't rely on some obscure mathematical knowledge as might be suspected from an Olympiad problem.

If you want to see Grant Sanderson, instead of only hearing his voice, he uploaded a Q&A where he explains his motivations. Mainly, creating educational math videos that other people would not be able to create.

⬐ jhncls

I love it how Grant Anderson manages to combine his compelling narrative style with elegant animations. And how the solution only depends on very simple mathematical knowledge. Grant shows a possible route towards discovering the invariants that rule this puzzle.

I think it is a kind of puzzle to try out on your own, knowing that a rather smooth reasoning solves it. And that is doesn't rely on some obscure mathematical fact as might be suspected from an Olympiad problem.

⬐ prvc

You can obtain the problem statement on this page: https://www.imo-official.org/problems.aspx

⬐ throwaway_bad

The video is not so much about the problem solving process so it's not good for hints. The visualizations alone makes the problem a lot easier off the bat.

It's mainly about showcasing a problem that is objectively hard (based on IMO competition results) then revealing the simple thought process that will lead you to an "obvious" answer in a few short minutes. The main point is that there are problems where once you know the "trick", you can't accurately judge how hard the problem is anymore.

I think people would appreciate that point a lot more if they struggle with the problem for a bit first!

⬐ gorgoiler

This is great, the visualization is so helpful. Even better I think would be if the point field was counter rotating and scrolling such that the windmill was constantly falling forwards and backwards either side of being vertical, keeping the two sets of points bisected and on either side of the screen.

⬐ d--b

Mmh intuitively, I would have thought that the proof would involve the enveloppe of points. If the line starts with a section that crosses inside the enveloppe of the set of points then it remains so, and hits all points, while a line that starts outside remains outside and so can avoid some points.

Any formal proof along those lines?

⬐ ghusbands

By envelope, you likely mean the convex hull. By "crosses inside the envelope", you probably mean it intersects the hull. That does not work. If you have a triangle inside a triangle and a central point inside that, that gives you seven points. If you start on the inner triangle, intersecting the outer triangle (convex hull), you can repeatedly visit the points of the triangles without touching the central point.

⬐ tromp

I enjoyed watching this similarly insightful video [1] on "The hardest problem on the hardest test" of the Putnam Competition.

[1] https://www.youtube.com/watch?v=OkmNXy7er84

⬐ fspeech

The video gives excellent intuitions. But do try writing down a rigorous argument after watching it!

⬐ sAbakumoff

From the same channel : https://www.youtube.com/watch?v=jsYwFizhncE overview of very elegant connection between blocks collision and PI.

⬐ ikeboy

I looked up the problem without the solution, sat down and thought about it for about 15 minutes and it seems trivial. Spoilers ahead:

Lemma 1: if a line rotates indefinitely, then the condition is satisfied. Proof: it can never skip a point (because no 3 points are collinear) in each 360 degree rotation: at some point in the rotation it will hit that point and that point will be a pivot. Indefinite rotation yields the required claim.

Lemma 2: each step after the first has a minimum rotation. Proof: after the first step, we have two ordered points A and B, and a deterministic way of knowing how much to rotate. As the set of ordered pairs is finite, the lemma follows.

It immediately follows that the line rotates indefinitely and therefore the conclusion.

I suspect there's some error here I'm missing, because this proof feels too simple.

⬐ prvc

Great presentation. I wonder whether all correct solutions submitted on the contest day had the same solution.

⬐ gambiting

Now I really want to know what question 6 was and an equally informative explanation what made it so hard!

⬐ mrosett

It looks like a pure geometry problem: https://artofproblemsolving.com/community/c6h418983p2365045

⬐ authoritarian

Numberphile has a couple videos about it

https://www.youtube.com/watch?v=Y30VF3cSIYQ

https://www.youtube.com/watch?v=L0Vj_7Y2-xY

⬐ AlexCoventry

The actual question is at https://www.youtube.com/watch?v=Y30VF3cSIYQ#t=5m31s , if you don't want to hear a 5 minute rant about how hard it is.

⬐ AlexCoventry

I'm going to have to watch this later, because the cat I've got on my lap appears to be deeply alarmed by the blinking eyes of the "pi" avatar.

⬐ rvz

The approach to solving this problem looks very elegant to viewers with/without a mathematical background and the author's use of visual explanations towards solving it step-by-step helps untangle the ambiguities in this puzzle.

Correctly proving this without assistance is one thing, but explaining it to non-mathematicians via a YouTube video sounds so difficult that some I.M.O candidates may struggle with this. Even so, I think the author is perhaps a professional/skilled mathematician or both which greatly helps explain this proof in a concise fashion.

On the other hand, I find that problems like this may be (ab)used in the future for technical interviews at financial/asset/investment management institutions for software engineering roles. Over the top indeed, but I think it would very difficult to justify using mathematical proof questions in interviews.

⬐ Analemma_

> On the other hand, I find that problems like this may be (ab)used in the future for technical interviews at financial/asset/investment management institutions for software engineering roles

I'm not really worried. In general, the trend for hiring software engineers has been away from silly puzzles, not towards. Microsoft and Google both used to use them and now they don't, and other companies have been following along. In general, hiring fads and follow-the-leader are not great, but in this case it's for the best that other companies have taken their lead.

To be sure, there are still lots of other problems with how developers are hired, but the stupid puzzles at least have mostly faded away.

⬐ codingslave

Really? If anything I thought that the use of puzzle/algorithms questions is only accelerating. Microsoft asks leetcode, as does google.

⬐ Analemma_

I don't like leetcode, but at least leetcode problems are actual algorithmic/programming questions. What I was talking about in my comment is the "Why are manhole covers round"-type questions that used to be endemic but have fortunately mostly vanished. A pure mathematics question like in the linked video, with no connection to algorithms/CS, would fall under that definition to me.

⬐ th0ma5

Does having an algorithm memorized help anyone in their daily jobs at either Microsoft or Google?

⬐ majormajor

This assumes that everyone just has a huge set of canned answers memorized, versus putting together some on-the-fly combinations of some building blocks like maps and different things to do with lists.

If you can memorize the whole world, that probably would be useful day to day too - I'm sure you'd see a lot of stuff you could use the shit you memorized for! - but it doesn't seem to be happening much.

⬐ th0ma5

I think this is the most reasonable answer, and perhaps the memorization stuff is more of a measurement of who paid the most attention in the most recent of class taking... I just feel as I progress in my career and do ever more complicated things, I'm shedding more and more ready knowledge of any specifics, but perhaps making and ever more complex map of how to find what I need.

⬐ majormajor

For specialist roles and experience, I'd definitely interview differently. Hard to do live coding there, but if you can tell me exactly about how you've solved problems in the space, that's awesome.

Most of where I've see the "how would you manipulate this array" type of stuff is generalist stuff or new-to-the-particular-subdomain candidates, where if I asked for exactly what I'm looking for them to know, they'd fail. Gotta just look for people who can learn it on the fly fast instead, and I haven't found any better proxies yet. :|

⬐ ianmobbs

Yes! A former engineer at Google put together this blog post[0] about an interview question he asks and its pertinence to what's done day to day at Google. It sheds some light on the usefulness of having these algorithms memorized.

[0] https://medium.com/@alexgolec/google-interview-problems-rati...

⬐ th0ma5

This is a post more about something this person finds interesting rather than something I can tell they seriously mean to evaluate a coworker or employee. They even say they don't know why similar things haven't done well as ways to evaluate people to hire??

⬐ Sean1708

Maybe I'm In a slightly unique position because the role I hire for is "Algorithm Developer", but if you've memorised an algorithm I will keep asking you questions until I find an algorithm that you haven't memorised. I don't care whether you know the "correct" algorithm, I care that you can find a working solution to an unknown problem (since that's literally your job) and I care that you know how to assess the good and bad parts of your solution.

⬐ th0ma5

I like this response quite a bit. Sounds like I could forget all the names, be fuzzy on the specifics, but if I've been doing the work it couldn't help but show eventually.

⬐ DougBTX

Yes

⬐ th0ma5

The other reply disagrees... Why would memorization of something so easily referenced matter?

⬐ mrosett

Having an algorithm memorized? Probably not. Being able to select the correct algorithm for a particular problem? Absolutely.

⬐ th0ma5

From memory?

⬐ trungdq88

Yes

⬐ th0ma5

See some other other replies. I have yet to come across anything compelling that supports this.

⬐ zorked

My very unofficial interpretation of hiring processes like that isn't that what's demanded is that people know algorithms from memory, it's that they are so familiar with algorithms that knowing from memory isn't hard to them.

⬐ th0ma5

Maybe, but what are these really applicable to? Embedded systems, high performance or maybe graphics loops... I'd think anyone working in a higher level interpreted language would not need to memorize much of anything. Probably not Java, R, Python, etc. And these tests all seem to be about implementing them rather than applying them which seems like something most people shouldn't do if there is a version out there already that is known to be working.

⬐ andrewflnr

> Over the top indeed, but I think it would very difficult to justify using mathematical proof questions in interviews.

Programming is literally isomorphic to finding proofs (Curry-Howard FTW!). From the other perspective on proofs, they're about communicating technical concepts in a clear way, which is a vital skill for a developer in an organization. So no, I don't think it would be that hard to justify. I was kind of joking at first, but that's actually pretty compelling...

⬐ Dylan16807

And a human is roughly isomorphic to a donut. The differences overwhelm the similarities.

⬐ vecter

While you make a theoretically true statement, it is practically useless and irrelevant. I have a background in computer science and can cobble together some proofs. Writing production software at a tech company is quite different from writing proofs for most software roles. I work with talented engineers and I doubt most of them could write a simple mathematical proof (mostly because they don't have the background or experience, not for lack of ability).

⬐ fooker

I got asked a mathematics question during a Google interview in 2016. Got the answer right but could not completely prove it was correct.

⬐ cc81

What type of position was it? I'm thinking it is perfectly reasonably if the position is within AI R&D or something like that but less reasonable if you are web designer.

⬐ fooker

It was non AI R&D, fairly reasonable.

⬐ gbjw

The author is Grant Sanderson (https://en.wikipedia.org/wiki/3Blue1Brown) who has an undergrad degree in math from Stanford and worked at Khan Academy before starting his YouTube channel 3Blue1Brown. Also, the student who is mentioned in the video (Lisa Sauermann) as having solved this problem at the 2011 IMO (and attaining the only perfect score) just recently started as a Prof. at Stanford (http://web.stanford.edu/~lsauerma/) as a 27 year old.

⬐ jedberg

Wow. She went straight from getting her PhD at Stanford to teaching there. That's almost unheard of. She must not only be a brilliant mathematician but an amazing teacher too.

⬐ vecter

Academics are given tenure-track positions at top universities for the quality of their research, not because they're amazing teachers.

Lisa is undoubtedly brilliant. She may also very well be an amazing teacher also, I don't know. My point is just that one should not assume that.

⬐ rsj_hn

Not only do teaching skills play no role in getting tenure at a place like Stanford, teaching is actually a threat to research productivity and research universities will hire professors who do as little teaching as possible, leaving most of it to older who professors who are no longer productive researchers or to post docs, teaching assistants or other staff.

To understand how this works, the researcher will apply for some grant, say $300,000 to study some question in geometry. Now, why does a mathematician need grant money when their only tools are a paper and pencil (maybe a laptop with Tex installed)? First, the university gets 1/3 of that money as "overhead", so the researcher is left with $200,000. Then, the researcher will pay to "buy out" his teaching load which is more money paid to the university, say $150,000 to not teach 2 classes for a year. With the remaining $50,000, he may spend money to fund a post doc to come and assist him for a semester. Again, that money goes to the university. So the researcher may get $300,000 but it all ends up in the pocket of the University, which in turn pays him a good salary with the assumption that he keeps the grants coming. A place like Stanford gets about 1/3 of its funding from these research grants, 1/3 from its endowment, and 1/3 from tuition. It hires researches to get the grants, grad students and adjuncts to teach, and the sports teams and other events help with endowment.

Thus research professors are hired on the basis of their ability to avoid teaching loads, not on their teaching skills.

⬐ jimmyvalmer

Yes, this. Neither students nor teachers are well served by this grossly inefficient enterprise that wears young faculty so thin that it effectively violates several OSHA guidelines. I've always believed the college lecture format is an utter waste of time for everyone. I dream of a day when universities all become pure research institutions, and undergrads teach themselves with the help of AI feedback systems.

⬐ toxik

That sounds dystopian and unlikely to ever work. Learning is a social endeavor.

⬐ jedberg

> say $150,000 to not teach 2 classes for a year

Is that why she is teaching two classes in her first semester, including one which is lower division? Usually you don't put the crappy teachers in the lower division classes, you give them graduate seminars.

I appreciate your cynicism, but based on her teaching load, I'm going to guess that she is also a good teacher.

⬐ rsj_hn

Not sure why you are accusing me of cynicism or arguing that a teaching load of two classes (typically 6 hours per week of instruction) in one semester is something to be proud of for a full time teacher. Go to your local teaching college to see people teach 4-5 classes each semester, or to your high school where they teach 5-6 one hour classes each day -- and do it without a bevy of grad students to grade papers for them, hold office hours so that the professor doesn't need to, answer student questions, hold seminars, write and correct exams.

I am merely describing to you how this stuff works. My descriptions are accurate, from the overhead that universities take to the shifting of teaching loads onto adjuncts and grad students to the relative weight of teaching on research hires. You can verify by discussing these issues with someone else who went through the grad school experience and saw it all first hand -- I did it at Stanford.

As to why such an anodyne and factual description of reality strikes you as cynical is something you have to come to grips with. There are reasons for this system. Lots of grant money is available -- should it not be available? Should we not be funding this stuff? Given that grant money is available for research, it makes sense that specialists who are good at getting grants would be allowed to do that -- get grants -- whereas others who are good at teaching be allowed to do that. Obviously universities are going to compete to find these specialists and will pay them well. The only problem here is that when people think of Stanford as a great research institution (which it is), they just assume that is must be a great teaching college, which it isn't. It's pretty mediocre on that front, yet that's what people assume, because they think a good researcher must be a good teacher. Listen, many good researchers can't even speak english at anything approaching a college level. At Stanford. They aren't there to teach. Researchers do research, and teachers teach. That is probably the thing that is upsetting you, but really a moment's reflection should tell you that these are all simple consequences of the multiple hats a research university like Stanford is expected to wear. If you want a good education, go to a teaching college -- there are many out there.

⬐ boyobo

> > say $150,000 to not teach 2 classes for a year

Is that why she is teaching two classes in her first semester, including one which is lower division?

Not really sure what you're saying. The fact that she's teaching just means that she isn't using (or doesn't have) a research grant to `buy out' of teaching.

> Usually you don't put the crappy teachers in the lower division classes, you give them graduate seminars.

Again, I don't know where you got this idea from. Usually (in a math research department such as the one at Stanford) whoever's arranging the teaching assignments doesn't look at an instructors teaching credentials at all, unless they are egregiously bad. So all we can conclude is that she isn't absolutely awful at teaching.

The fact of the matter is that many researchers (due to their incentives) view teaching, especially lower division courses, as a chore, so really anyone in the department who wants to teach such a course is not going to get much opposition.

⬐ jimmyvalmer

> Usually you don't put the crappy teachers in the lower division classes

Since when? The hard-and-fast rule is junior faculty are assigned intro classes. We often give youthful teachers higher marks than crusty, doddering emeriti, perhaps for good reason, perhaps not.

> I appreciate your cynicism, but based on her teaching load, I'm going to > guess that she is also a good teacher.

GP did not question her teaching ability, but your inference of said ability from her impressive ascent at Stanford. It's a bit like inferring LeBron James must really be mature since he entered the NBA straight from high school.

⬐ jpxw

Assistant Professor*

Quite a big difference

⬐ dannykwells

Lisa is great but the role she holds is not tenure track. Its closer to a postdoc.

https://professorpositions.com/szego-assistant-professor-at-...

I'm sure she will get a tt job when she wants however.