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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.

4 comments

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.

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.

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.

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.

> 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.

> 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.

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.

> > 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.

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.

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

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.

Assistant Professor*

Quite a big difference

> 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.

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

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.

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

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

From memory?

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.

Yes

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.

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.

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. :|

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...

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??

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.

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.

Yes

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

> 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...

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).

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

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

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.

It was non AI R&D, fairly reasonable.