The hardest problem on the hardest test

"But Grant, the Putnam's not really the hardest test, and if you really want to find the hardest problem..."

Yeah, yeah, of course.  But if having an imprecise-but-not-totally-off-base title means this lesson reaches more people, great! 

For one thing, it's just a very elegant solution well worth sharing.  But I'm always fascinated by the question of how one would find elegant solutions that just look like they were plucked out of thin air.  That is, rather than perpetuating the illusion of innate genius by just showing the solution, try shedding some light on what background/tactics might lead someone to the relevant insight.

More often than not, the answer has more to do with someone's past experiences than with the problem-solving techniques they used in the moment of the puzzle.  But every now and then there actually is a lesson to be taken away in terms of how you might systematically expose yourself to interesting insights, and the goal of this video is to share one such lesson.

Collaboration with MinutePhysics

Henry Reich, who runs the channel MinutePhysics, has been incredibly helpful to me as I've started up the channel. He's offered mentorship and advice when I needed it, he gave some meaningful encouragement early on, and he's just that rare combination of very talented and very kind.

So when my friend Evan Miyazono, a newly minted physics Ph.D. from CalTech, mentioned that he'd be willing to help me out if ever I wanted to cover something in quantum mechanics, I thought it could be a good opportunity to actually work with Henry in creating something together. Fast forward a few months (which involved one particularly fun weekend in Montana when Evan, Henry and I all got together with a few other physics folks), and we have two videos for you.

The one on my channel is meant to lay down some of the fundamental intuitions for the math behind quantum, without giving too much of a "look at how crazy quantum is!" vibe. At its heart, a large portion of quantum mechanics is about wave mechanics. While the field certainly has some strange components, some of the things that people often think of as weird (superposition, uncertainty principle, etc.) are ideas that come up very naturally in the context of waves. It's just that the meaning of these waves in the context of quantum gives these ideas some unusual physical manifestations.

The approach I decided on was to spend most of the video just talking about classical electromagnetic waves, giving a feel for the meaning of superposition and polarity in that context. Once that's laid down, it's more meaningful to talk about what actually becomes different in quantum mechanics. What I love is that the formulas for describing polarization barely change, it's just that the constraint that the energy of light comes in discrete little chunks gives a slight twist to the physical manifestation of this math.

The other video, which we put on MinutePhysics, is about Bell's theorem. This is a really clever experimental result, with very deep implications. The entire collaboration centered around this project, really, since it's a topic that Henry had wanted to do for a while, and one which I found thoroughly captivating when Evan first told me about it. I hope you enjoy!

A trick to visualizing higher dimensions


Perhaps it's worth saying a few words on why those interested in math care about higher dimensions.  Why care about 20-dimensional space when the world we live in is not 20-dimensional?  Broadly speaking, spatial reasoning is not just useful for describing the physical world, it's a sort of problem-solving tactic for any scenario where you want to think about collections of things (like lists of numbers) as individual points of some object.

For example, viewers of the "Who cares about topology" video will remember how nice it was to associate pairs of points on a loop with individual points on a Mobius strip.  But this is only associating collections of two things with individual points.  As the relevant collection gets larger, the corresponding object is harder to hold in one's mind, since it lives in a higher dimensional space.  For example, there is something analogous to a Mobius strip that represents unordered quintuplets of points on a loop, but seeing it would require being able to visualize 6 dimensions.  That doesn't stop mathematicians from working with such objects, though, it's just that the work becomes a bit more abstruse.

The tactic I provide in this video is not some magic make-you-feel-like-neo-from-the-matrix visual that lets you hold a full high-dimensional object in your head.  Instead, its main function is to emphasize that what we really mean by "a point in n dimensions" is "a list of n real numbers", and moreover that it can still be fruitful to think visually about that idea even when you can't conceptual the list of numbers as an individual point in space as we can for 1, 2 and 3 dimensions.

Ever wonder how Bitcoin (and other cryptocurrencies) actually work?

I don't know about you, but I used to think of the prefix "crypto" as applying mainly to secret messages.   One of the neatest takeaways from the study of cryptography as a whole is just how much more widely applicable the mathematical tools at its core are.  Really this prefix serves more as an antonym for "trust" than it does as a synonym for "secrecy".  One of the first times this really sank in for me was upon reading the Bitcoin white paper.

I decided to cover this topic now partially in light of the recent rise in attention towards other cryptocurrencies; a level of attention which is not always accompanied with a real understanding of what these currencies actually are.  While there are a handful of really good explanations of the blockchain I've seen out there, it seems that the more prominent search results one will get when trying to learn about them lean too heavily on vague analogies with gold-mining, and the more technical ones don't always motivate why the system works the way that it does. (An excellent exception, by the way, is this post by Michael Nielsen).  

Here I try to frame things in the context of you stumbling into inventing your own cryptocurrency, which hopefully helps clarify why the various technical constructs involved come into play the way that they do.

The supplemented video here was something I originally had as part of the script of the main video to illustrate what "infeasible" means in the context of digital signatures and cryptographic hash functions.  It's too peripheral to justify the amount of time it would add to the main video, but I had fun thinking through the example, so it was pulled out into an isolated footnote.

All possible pythagorean triples, visualized

This is kind of a fun one.  The first time I ever learned about the formula for generating pythagorean triples, it was in the context of stereographic projections onto the unit circle with rational slopes.  The visual intuition of that is nice, but the actual algebra to work out a formula is quite a mess, and (at least for me) the final result is easy to forget.

However, you can land on that same general formula just by squaring complex numbers with integer coordinates, which lends itself to a different visualization entirely.  I still talk about the projections onto the unit circle, as it makes for a very nice little proof at the end, but I hope this offers a different perspective even to those familiar with the question of finding pythagorean triples.

Pi hiding in prime regularities

There are only so many times that one can use the phrase "This is my favorite piece of math" before it starts to lose meaning.  So instead I'll restrain myself and say this is my favorite piece of number theory.

A surprising number of formulas relate things the Riemann zeta function, as well as some of its cousins, to pi.  Most notably the values of zeta on positive even integers.  Here I explain one example from this family, where it's most clear how circles are involved.  What I think is fascinating is that you could set out to find a formula for pi by counting lattice points in the way described here, with no intention of talking about prime number or multiplicative functions or any of that, and moreover the final formula itself seems to be clean of primes, but nevertheless the natural course of exploration seems to have this gravitational pull towards patterns in primes.

The fact that similar formulas arise from the zeta function itself, which is known to encode the information regarding the distribution of primes, is one more piece of evidence that this connection is not just a coincidence. 

Euler's formula with introductory group theory

Two years after uploading the first video, I wanted to revisit the same topic.  For one thing, I just talk way too fast in the first one, and I think many of the animations are unclear.  But more than that, as I've come to see that people seem to prefer long-form videos, this was a good opportunity to take the time needed to talk about what group theory is, better setting the stage for how viewing numbers as actions can give a little insight into this formula.  

It also helps set the stage for how this formula generalizes in the context of Lie groups, which if I'm ambitious I might want to cover down the road.


Who (else) cares about topology

After I put out the first "Who cares about topology" video, someone pointed me to the paper that this video is based on as another great example of a topological fact being used to solve a non-topological problem.  There's an interesting philosophical point to be had in the fact both this and the proof in the first topology video are non-constructive.  That is, they show that some desired solution must exist, but not how to find it.  This is true of just about every case I've seen of topology touching other fields.  

This makes some sense, given that topology is all about understanding what properties are invariant under a wide variety of alterations you can make to some object, while specific solutions to specific instances of a problem tend to depend on the peculiarities of that instance.  Nevertheless, I'd be curious to a fully constructive application of a pure topological fact, if that is not a contradiction.

Fractals are typically not self-similar

I remember being blown away when I first read Mandelbrot's "The Fractal Geometry of Nature" about how utterly reasonable the definition of fractal dimension is.  Moreover, as the title of the video suggests, I think that was the first time I saw that the original motivation for defining fractals did not have self-similarity as the central focus, but instead centered around capturing the idea of roughness that persists at all levels.

If I could revisit the video, one thing I might change is how I discuss the definition of a fractal.  I wanted something concrete that a viewer could come away with, a way to fill in the blank as they say "fractals are not self-similar, instead they are _______".  But I fear that focusing on non-integer dimension might not do appropriate justice to the nuances in trying to define roughness.  I opted not to say that fractals are shapes whose fractal dimension exceeds their topological dimension because I didn't want to get into topological dimension.  But even that definition misses a few rough objects.  A rather nice analogy that one commenter referenced is that defining fractals for mathematicians is a bit like defining life for biologists:  We all know what it is that we're trying to encapsulate, but every time you try to nail it down with a concrete definition it seems to neglect some of what you want or to admit some of what you don't want.

Tattoos on math

This one is definitely a very different video for me.  When a friend of mine was getting a math tattoo, I decided to record it with my phone just in case it might make a fun video.  After pondering it, and deciding that there could be a nice narrative about the man-made permanence of conventions along with a mini-lesson on how to interpret trig quantities, I decided to just go for it.

Visualizing the Riemann zeta function and analytic continuation

While working on (well, procrastinating from) an "Essence of Calculus" video, I was playing around with visualizing various complex functions.  When I put in the zeta function, it struck me as so pretty that I just had to make it the next project.  This was especially true when I realized it could give a nice way to motivate the idea of analytic continuation, which is one of the more opaque aspects of this function that stalls people trying to read more into it.

Hopefully, this can help motivate people to learn about complex analysis more generally.

Binary, Hanoi and Sierpinski

This pair of videos includes several really surprising relationships, between binary/ternary counting, the Towers of Hanoi puzzle, and Sierpinski's triangle.  I was also pretty excited to be able to do a project with Keith Schwarz, a phenomenal CS lecturer at Stanford.   Anyone curious should look at some of his projects at

This is also surprisingly relevant to the space filling curves video I did some time ago.  In those, I included a brief animation of a curve that fills Sierpinski's triangle, but I'll admit that it was rather unintuitive to me.  This here was the first time that I felt a deep intuitive reason for why such a curve should exist, and can naturally be defined. 

Who cares about topology?

This is, quite possibly, my favorite pieces of math of all time.  As I say in the video, I think it was the first time I realized why mathematicians care about the properties that certain shapes have which don't change as you squish and morph around that shape.  Sure, the mobius strip, torus and Klein bottle are mildly interesting to look at, but until you see one used to solve a problem, contemplating these shapes feels more like an intellectual indulgence than math.  

I never actually define topology in the video, doing so only really makes sense if you have some history with more abstract math, but I do think the viewer who understands the cleverness here has an appreciation for the spirit of topology itself.


Triangle of power

Who doesn't love a notation rant?  I came across this idea on math stack exchange (credit to Alex Jordan), and thought the symmetry here made it well worth sharing.  Also, this has a bit more of the "popular appeal" factor than some of my other videos.

The Hilbert curve

This video actually started when a student who had gotten interested in doodling Hilbert curves asked if I could animate him one.  Fractals are a fun thing to program, and I actually went a little crazy trying out what different space-filling curves look like, hence the supplemental "Fractal charm".  

As I was creating it, I thought about how it's very rare for people presenting space-filling curves to mention that the hard part is ensuring these things are well-defined.  That is to say, to define what it means for a family of curves to actually have a limit.  It also reminded me of a thought I had a while back about how translating visual data to audio data would benefit from using space-filling curve approximations.  In short, I realized there was some good fodder for a story there, so the animation for my student turned into a full-blown video.

The Brachistochrone

I have huge respect for Steven Strogatz, and the work he's put into math outreach.  So when I saw that he had tweeted some nice things about my early videos, I reached out to see if he wanted to work on something together.  Doing the Brachistochrone, and communicating Mark Levi's clever insight, was all his idea, and I'm really glad he suggested it.

Music and measure theory

There are many facts in math in which I've been able to follow the proof, but have nevertheless been stumped to find an intuitive understanding of what's happening.  Such facts have a funny way of becoming topics I want to create a video about.  In this case, the fact that you can cover a dense set in the real numbers with open intervals whose lengths sum to an arbitrarily small constant absolutely blew my mind.  

After mentally toiling for a while, I finally felt satisfied when I just sat down and thought through what it would feel like to zoom in on an irrational value which was not covered.  Thinking about the conditions for being such an irrational reminded me, surprisingly, of when I learned about why 12 is such a natural number of chromatic notes to have in a scale.

Binary counting

After "What does it feel like to invent math", I wanted to do a shorter video, and to experiment with something that lacked narration.  Having failed to do due diligence, it was only after I had made heavy progress into the video that I discovered just how unoriginal this method of counting is.  Whoops!  Nevertheless, I'm glad I made it.

For the first time in one of my videos, I actually filmed something.  After using my phone to film my right hand counting to 31, I played around with some image processing techniques.  This involved finding the appropriate combination of a Gaussian blur, Canny edge detection, and some more custom operations to thicken and prune the footage.