(Early view) An unusual problem from the world's toughest high school math test

Really nice explanation paired with minimal animation to capture the required insight. It reminds me of other 'parity' problems like the mutilated chessboard (domino tiling) problem where once you have the insight, most of the apparent geometric details become irrelevant. Is it possible to generalize this argument to higher dimensions? For 3D with S containing no co-planar 4-point subsets and a plane pivoting on a line between any two points, for instance? Not sure how to define the next pivot line once the plane hits a new point.

I'm very much for even hugely awkwardly shoehorning something #3 human into this very lovely abstract problem, although you may not welcome the edits involved, and my idea remains ill-formed. Can you make the observation on empathy (imagining yourself on the other side) connect to the way the line/problem divides the points/students into two side/understanding halves? And when you change your perspective by rotating the line 180 degrees, you can move from one half to the other? The visuals underneath might move some score breakdown analysis from the prelude to the end. (I did find the current flow actually buried the lede - I wanted to hear the problem statement earlier!)

Anthony Bailey

I like the idea of talking more about invariants, but feel kinda indifferent about "real math." I know some people want everything to relate back to physics and engineering, but I like math for its own sake. P2 is beautiful in its own right. Maybe it has applications, but I'm not sure I would emphasize that too much.

Kevin

Oh, interesting, so you're suggesting that the desire for an "infinite" result is suggestive of seeking invariance?

3blue1brown

I'd say #1 seems to follow most cleanly from the video's current stopping point.

RedAgent14

1988P6 is the Vieta jumping problem, right?

RedAgent14

I think option 1 would be best but diving into option 2 would be insightful too if there's time! also just wanted to say how amazing satisfying the clicks were, was like mathematical ASMR?

Wonderfully elegant explanation! I wonder if you could motivate finding the invariant more clearly. Atm you talk about “putting numbers on vague ideas”, but the idea of counting the number of points on both sides still felt sudden. One potential option could be to remind viewers of the “continues infinitely” requirement, and lead from that to finding an invariant. Either way, I enjoyed the proof a lot!

that was delightful. you do a fair number of proofs on this channel but you don’t often talk explicitly about the process of finding one, or of convincing oneself that it is correct. perhaps that could be something new to comment on? i think any of your suggestions would be great too.

issa

I would be interested to see a competitor's proof. I think that could be mixed with #3.

I would appreciate a brief restatement of the solution as a way to conclude the video. right now it kinda feels like it leaves me hanging a little bit.

Majed Samad

Hah, tried this before viewing the solution. I had a more convoluted way of finding the middle section the line should pass through in order to get the desired cycle. Once I realized the inside-outside dichotomy I hit upon the idea that recursively removing convex hulls from the set of all points would eventually condense it down to the essential middle points. As long as your line starts by running through this set, you're good. I think, anyway. I got stuck here, as while I could have coded this up and see it working, proving correctness by building on this convex hull property was tough to get going. The counting argument is much more elegant. +1 for the importance of invariants! (And many thanks for bringing such a fun problem to our attention)

Now do 1988P6 :−) 1. Invariants are great and make lots of seemingly complex problems become easy. However, those problems tend to be of “toy” olympiad variety. What solutions to real-life problems feature invariants? Cryptanalysis comes to mind, e. g. Enigma, impossible differentials, even though those things are kind of the opposite to invariants. 2. That isn’t a topic that would resonate with most viewers I’d guess. (I did it at school, it was fun, but most your videos are way too complex for schoolchildren, even the bright ones.) Also there always are suspicions of foul play at the IMO (I heard the country’s coaches translate the solutions for the judges? How does someone solve P6 but hardly anything else?). 3. This foray into psychology sounds off-topic. Tangentially related to things like Moscow State University anti-Jew entrance exams.

Roman Odaisky

A very nice problem.I think there is a small typo. You say that AMC stands for the American Mathematical Contest but the legend in the box says American Mathematical Content.

Great problem! I think #2 is interesting, but from a different angle. Could it be that the IMO participants' extensive experience actually made this problem harder for them? I think this problem could be an excellent hands-on classroom activity, using some circles on the floor for the points and a long stick or taut rope for the line. I don't know for sure -- would like to try -- but I wouldn't be surprised if in this context, students who don't consider themselves good at math might come up with the solution, whereas it's possible that IMO students were trying to connect to other problems. For myself, I was also looking at the convex hull, and then taking the convex hull of remaining points, etc, until there was a point, segment, or polygon in the middle. I got too tempted to watch the rest of the video... maybe this approach can work. But maybe having the concept of a convex hull interfered with me looking at the problem as in the video.

I'll throw in a vote for #3. During this whole video, I was thinking that I had no idea how to solve this problem, but that once I'd seen it, it would seem so simple, which is exactly what happened. The difference between being able to understand a concept vs generate that concept is huge, and has huge consequences on how impactful a person can be in contributing to a theoretical field. It would be interesting to hear your thoughts on what makes some people better than others at the generation side, and whether you think there are good ways to get better at it.

> Also would be interesting to see if this problem is intrinsically 2-dim or if you can generalise it some how. The screen at 8:30 reminds me of cosmic webs ( in a purely visual way). You could try to reformulate the windmill in terms of pivoting about two points and switching out when you get to a 3rd, but it's not obvious that the plane will return to the starting orientation.

Nate

I suppose I would most rather have option 1, but if it didn't make things too long I would love to see option 2 thrown in as well. Perhaps focus primarily on option 1 but then touch a little on option 2, if it doesn't make the video too long. It just feels to me like a meaningful connection can be made between the two options. After all, one way a person might justify math competition would be to yield to its applicability to "real" math. Still, if there is only time for one option I would love to see how these odd, but challenging, problems can be viewed in light of much larger math topics.

It really is so easy to fool oneself into thinking this is an easy puzzle once you see the answer. But that's the delight about having the IMO score data, in some very real sense we can *quantitatively* say that it is hard (or a certain kind of student). I guess the question is what _would_ make you think of seeking an invariant here, when your mind might be more naturally drawn towards thoughts of angles, convex hulls, etc.

3blue1brown

Good to know. To be clear, my own conclusion on #2 is not that it shouldn't be competitive. In general, competition is a great way to bring out greatness, in whatever form. But certainly during a developmental phase, it's worth asking if aspects of the competition run the risk of doing more harm than good. For example, if a student is the deep contemplative sort, the pace of the AMC might make them feel "bad" at math, if if they likely aren't.

3blue1brown

(whoops hit the return button) More generally though regarding the end, I wonder if there's a way to really double down on how following your nose leads to new insights, and even if those insights don't always lead to the right path, it's fruitful to double back and look for alternative forks in the trail you might have missed earlier. That's kind of what mathematical exploration is all about, right, and it might help to ameliorate the sense that divining the solution is the essence of math. You kind of have it in there anyway and many of your other videos too.

Delightful content as always. I got sidetracked wondering if there wasn't a nice little inductive argument based on your "inside outside" comment. Something like: start with any three in your scattering of points, now add a point, if it's on the "outside" of your first three just toss it in and all will work; if it's on the inside the start on that point instead. Now add in another point, repeat the procedure just described. I got a little lost tracking the details, and indeed the more I thought about it the more your solution started rearing it's head :)

I like Option 1. I would love a quick nod to another (less tricky?) example of a problem that relies on the same fundamental insight re: invariants, or utilizes the same chain of reasoning. In general I always find resemblances the most useful kind of "conclusion." :) Great work by the way, loved this!!

Mark Mulvey

By your "year" do you mean that you were present at this competition?

Jacob Mirra

All three conclusions are good opportunities. (2) is the weakest to me (in other words, I vote for (1) and (3)) because math, for better for worse, often *is* competitive. You're passing exams at university. You're using your math skills to compete for a job in the real world. You're using it in industry to seek a competitive advantage for your company. And it's no better in the academic setting, where you're working for tenure, publishing results, applying for grants. So, while I can get on board with the whole, "Math is beautiful, let's all be innocently curious all day and not, like, compete, dude," it's maybe a little... dishonest?

Jacob Mirra

Good point. Actually, I'd say it's the opposite. It's intuitive clear that you can "rotate the picture a little bit", but to make it rigorous, you need to say something to the effect of, "since there is a finite number of points, there exists an orientation theta for the line such that for any pair of points (P,Q), the line L is not parallel to PQ. Select this orientation."

Jacob Mirra

Beautiful. I agree with what many said, relating it to the importance of invariants sounds like a good idea.

I thought about the convex hull too. Interestingly, starting with the line outside the convex hull is equivalent to coloring all the points one color. In other words, "All the (nonpivot) points are the same color if and only if the line starts outside the convex hull". I am guessing that Grant chose not to introduce the concept of "convex hull" because it gives the mistaken impression that you need some "esoteric math knowledge" to solve this problem---which you definitely don't.

Jacob Mirra

I would love to see option 2 and necessarily how working with problems like these affect your problem solving skills

I'd like to see option 2. Although Matthew Gruen might be right that this would belong in a different video. Besides from what Gilles Englebert already mentioned I'd want to add some additional points. I'm currently working on my bachelor thesis so I'm studying mathematics at a German university. And I often feel like math has a big problem with being too intimidating (even for me often enough) which I think prohibits people from entering math who might bring an interesting point of view to the table. Of course it is useful to be extremely smart and a good problem solver and those contestants surely will make great contributions but you don’t have to be. I feel like to opposite is true. Math is such a large field and there are so many problems and interesting questions to work on that one can make a valuable contribution just by taking the time to think about some strange problem that no one else seems interested in. Of course maybe a genius would be faster but there are not many geniuses… Additionally looking at the topic I work on in my thesis currently math just doesn’t feel competitive. There might only be half a dozen people who work on something and they essentially work together and help each other. Math is first and foremost a team sport. On the one hand because one can accomplish more by working with others and on the other hand because if no one reads what you did and no one understands and therefore believes it whatever awesome thing you might have done is essentially worthless. And to come back to these competitions I don’t think they promote the idea that contributing to mathematics doesn’t take geniuses nor how important it is to work with others. To summarize this: I think these competitions are great to get students interested who are competitive in nature and recruit the 0.001% of the best problem solvers but it promotes a totally wrong impression of how math is or should be done and is intimidating for less competitive people.

i vote for 1

It's a pleasant surprise to see my year chosen in a 3Blue1Brown video. The windmill is quite fascinating. :)

I personally would be interested in the second Option: How good/representative is it to promote interest in maths through competitions, and are there better ways to do so? The people participating in the olympiads sure have fun doing so, but there is a big difference between problems in olympiads and questions mathematicians tackle in the real world: scope. Competition problems are elegant, have already been solved and are contained to a couple pages. Real life problems tend to be messy (at first), unsolved and open-ended, maybe not even well-posed.

Also, one uncertainty about the proof I was left with at the end of the video is if, given some chosen set of points and a fixed orientation, if there are two points along the middle line rather than one. This makes the parallel lines argument a bit more gnarly, since it's not the line itself, but also whether it's before or after pivot transfer (the "if it was anywhere else" phrase in particular). One way out is just to choose a slightly different orientation of course. It also seems like it wouldn't be a problem in a more rigorous formulation, just for the intuition part.

thefifthmatt

A suggestion which is related to 1 and 3. My first guess for formalizing the "middle" was to use the convex hull of the points. I think you could show that by starting on a point within the convex hull, you always stay "inside", but that might not get you to the goal of hitting all the points. Introducing an alternative tactic which doesn't work as well, might help point out what is so good about the left/right invariant you used in the video. This relates to mathematicians coming up with better and better bounds for certain problems by changing they way they think about it.

Gabe

The importance of invariants seems real nice ! Also, there is a typo in contest at 42 seconds :)

This seems to be a popular choice, but I think option 1 is nice - because it can answer a question of motivation, which tends to important to people when I'm talking about math with them, but also because it shows that intuition *is* learnable, when backed by proofy things (abstractions), each giving you a new way to approach a problem. Option 3 may be good to touch on given how much setup there is for it. Option 2 is interesting but may fit better in a different video, as the human element doesn't feature directly in the video itself.

thefifthmatt

I’d like to see option 1 and 3. Together they make an excellent point about maths.

I also vote for 1: "How does a puzzle like this relate to "real" math? (Talk about the importance of invariants)". Another insight. I would have liked to see a bit more of the proof, how to demonstrate that this holds in all cases. For example, I was thinking: Is there any way to position the points so this method will skip one?

In the cases I'm imagining, the windmill does indeed still return to its median-point starting position after 180 or 360 degrees.. but the invariant of (number of blue points == number of brown points) doesn't hold, during the cycle. Are the accepted answers published somewhere? I think there may be more to the problem than just choosing a point P and line L which partitions the set evenly.

Shawn Van Ness

This seems either wrong (or incomplete) to me. I can easily imagine constructing a set of points such that the windmill will hit consecutive points above (or below) the pivot.

Shawn Van Ness

After seeing such an "easy" answer, it really made me want to know why something like this might be so hard (especially since my sense is that "invariant" arguments aren't that uncommon, though perhaps as a physicist I am biased in that regard?)

Henry Reich

Option 3 for sure!!!

Henry Reich

I’d vote for 1 also. It’s always great to see how apparently different problems and domains connect. Also in the video, it might be good for the narrative to remind the listener that the problem was to show that a solution line existed, not to test for all possible lines.

Awesome video! I would like to hear more about invariants and problem solving strategies in general moving forward. I really enjoyed this video and well as your video on the Putnam probability problem. I think these contest math question videos are a great way to discuss the process of problem solving which is very much applicable to “real” math.

Let it be about the relation to real Maths!

I kept thinking about computational geometry concepts like convex hulls, rotating calipers, and the ham sandwich theorem, which all seem like they'd be relevant but they don't turn out to be. It's intriguing how much you can accomplish with so little heavy machinery.

Max Goldstein

My vote is for the "role of invariants". Clearly the fulcrum in solving these problem (and many others).

Daniel Armesto

It´s great how you illustrate the process from intuition to formalizing to proof.

Daniel Armesto

Searching and using invariants in phisical or mathematical problems is certanly a good method to know and teach. It's this kind of "trick" or strategy that studends should be aware of. To be able to transfer it and apply it to this kind of problem, proofs at least for the commission the talent of the competitors and for me the talent of the commission to find a beautiful and unusual problem as you mentioned in the video. Competitions are often poor on empathy, because its easy to forget the love for all their participants :-)

Great video! I think talking about finding invariants would be the most interesting.

Hi, feel free to DM me, or send me an email, and I'm happy to talk about other things that you might be more comfortable with, and what's bugging you about Patreon.

3blue1brown

Thanks, Grant! Awesome problem, and awesome video as always. Of the options you gave, I’d vote for (3). Once I went through the video and internalized the solution, it seems an obvious answer, but it’s clearly a hard problem from the numbers. For teachers (and tutors, TAs, parents, etc.) it’s important to remind yourself of this point, and try to think about a problem from the perspective of someone new to it. I think (1), i.e. the importance of invariance, is a good way to end as well, especially if you could share some insights into the kinds of problems/patterns that make you think of using invariants in order to solve the problem. Finally (2), i.e. whether competition like this is good for math (or in learning in general) is a very large topic that I feel strongly about. I don’t feel it’s a good way to end your video because I feel this topic requires a lot of time to delve into. You can also consider ending the video on the topic of starting off the solution to a problem with a vague, hand-wavy, idea and slowly formalizing it step by step.

I vote for 1. I think invariants are really cool, but not receive enough attention so it eould be great to talk about them a little.

I second that as I was going to ask the same thing, and expected the video would concluded on seeing the formal proof.

What happens if the pivot does not have the same number on each side? I feel like we went down the path of "in the middle" and didn't come back to the edge. The set up of equal points on each side, and lingering around certain regions (possibly more accurately some notion of density) feels reminiscent of velocity vector field for the evolution of a pendulum S4 E1? It seems that starting in the "middle" is stable, and other initial conditions may not be? Just wondering if there was a connection there. Does the formal proof contain trig? The 360 and 180 really TRIGgered that thought. (I am not even sorry for that one) Also would be interesting to see if this problem is intrinsically 2-dim or if you can generalise it some how. The screen at 8:30 reminds me of cosmic webs ( in a purely visual way). Personally, I think the best way of ending it would be to include as many examples of duality, or at least strong links to other fields (gut instinct tells me there are more than first meets the eye). I think its the problems that have crazy elegant solutions in seemingly unrelated fields that really capture the beauty of maths. Sort of a more detailed version of point (1). A more selfish reason is I don't have the maths knowledge to know the difference between cool connections and coincidences, but you do ;) Point (2) might be a good discussion for Ben, Ben and Blue if you're still doing that?

Your presentation of the problem is great as always. Perhaps not the feedback you wanted but the visuals in the video kept tricking my eyes. Points that were blue but switched to brown would stay blue in my periferal vision until I directly looked at the point. Also, as the line rotated and points toggled colors, the points in my periferal vision would tend to disappear if I kept my eyes centered. It felt a bit like a combination of http://www.rpdms.com/satillusion/index.html And https://www.inverse.com/article/41595-hermann-grid-illusion-black-dot I'm not sure you should change anything but it may have been related to the white ring around every point.

I really, really like this video. The solution is elegant and simple to explain (no background knowledge needed!), but the insight is hard to discover. As for the ending, I think your first one is better. Talking about empathy is nice, but may reach fewer people, because it targets teachers more than students (unless more 3blue1brown listeners are teachers?).

Vincent Zalzal

So beautiful! Is there somewhere a collection of problems with a similarly drastic change from really hard to almost trivial over the course of one "simple" idea?

Georg Wille

Invariants!!! Wooooo!

Really nice video! I vote for "Is it good to make math competitive like this?" Completely unrelated to what you asked, but just a random thought: I was wondering if there a logic behind selecting ~250K, 12K, 500 and 60 students for the AMC, AIME, USAMO and MOP?

Point (1) would be very interesting to the math enthusiasts (including young ones) Point (2) probably needs a bit more background on the test in an international sense. I remember it was a big deal back in Iran when I went to school, but in the US, I never heard of it up until now! :D Point (3) would be a pure 3b1b ending ;)

Alipasha Sadri

Is there any chance you can share the formal proof to this question? I'd be particularly interested to see one from the competition, if available.

Toby Archer

I agree with others that teaching about invariants is most mathematical and most useful.The empathy point is important for teachers: the first time I TA’’ed an introductory programming course I was startled by how hard some things were for beginners that seemed elementary to me (recursion, for example, or accumulating a result by appending to an array or concatenating strings.) I had quickly to re-learn beginner’s mind in order to be useful to my students. As far as competition, I get the impression that many mathematicians are competitive, math is an art, art is competitive. I think it is good to have the outlet for those who are. I think those who are not can & ought to be encouraged to enjoy participating without the pressure of competition.

One of my favorite videos! Another vote for both points 1 (next-to-last) and 3 (last). I wish I had spent more time with it - I got part-way, but was too afraid that I'd never get it. Perhaps even give a hint about looking for an invariant, and recommend that people spend a while with it. Or provide software to let us play with it ourselves - that would also help.

Neal McBurnett

Thanks for the catch!

3blue1brown

I dunno, I'm not dumb but I've never liked proofs or been any good at them. I'm sure it would help my understanding of mathematics but not being able to do them hasn't stopped me being a top-performing engineering student.

Toby Archer

You have the wrong meaning for AMC, it says content instead of contest. But still a very nice explanation.

Steve

I dunno if I can keep my Patreon status. I want to do it, because these are the best videos I've ever seen, and they're important for my sanity. But Patreon is an awful company with terrible principles that makes stupid technical decisions. Can I keep my status but, you know, just send it to you via Paypal or something?

I think the invariant point would prove the most useful technically. I can personally say that invariants is a technique that's incredibly useful yet underrated - I messed up my Oxford interview by not knowing how to use them! You could also maybe link this to CS in a way - loop invariants are essential to prove any property of an algorithm. However, I also like the empathy point - so many people (me sometimes included!) fail to accept that other people just **can't** understand something as explained - on both sides it feels like talking to a wall.

_ericBG

The joke about the odds of all those numbers being prime was hilarious. That is an interesting problem, and it has a wonderful ah ha moment at the end. Re: The ending I think all three of these points are worth discussing in some fashion. However, whether or not it is good to make math competitive doesn't seem to fit in tone with the rest of the video and how the proof ends. That discussion would be better served at the outset of the video, if you are going to have it at all. I like the idea of talking about the applications of invariants as an ending, but I would be tempted to mention that within the video itself (if you are willing/have time to modify the script, recording, and animation) as well. As you mentioned in the group theory video, people who watch this channel aren't afraid to get into the math itself, so knowing the terms to match with the intuition, for me at least, makes it easier to apply what I learned here to problems I'm solving. It also (for me at least) gives some insight into what the right type of questions to ask are when approaching problems, as in what questions give you a foot hold to work from.

I vote for the first option. I love when a proof relies on showing that some problem is equivalent to a seemingly unrelated one, for which a known property like an invariant easily solves both. Also at 0:42, the text says "American Mathematics Content", but should be "Contest", right?

Kevin Davis

I think the first point (invariants) is the most mathematical, but the third point (empathy) is more profound. It's oddly difficult to remember the effort one put into learning something the first time, and it's really important for teachers (official or otherwise) to remember that.

The jump at 1:30 is hilarious - I'm so used to 3b1b videos flowing smoothly throughout, I thought that was fun. Frankly I don't mind the current ending that much, but I'd most like to hear your thoughts on the competition aspect. Great work, can't wait for the continuation of the differential equations series!

Minor typo at 0:42 --> "Content" instead of "Contest". This is an excellent video! I especially like the way it's paced so the viewer can have the key insights before they're given. As far as conclusions go, one takeaway is that it's easy to underestimate the difficulty of finding the right frame of mind, or having a conceptual breakthrough. Students often wonder for example why it took so long to invent the number zero, given that it seems so obvious to us now. The point is that from the other side of that conceptual leap it isn't nearly so obvious.

Point 3 is nice, maybe as a sign-off to the video, but I think point 1 is very interesting. Expanding a bit on how invariants show the order in potentially chaotic dynamic systems (like conservation of energy for pendula, or this windmill constantly changing pivots, etc) is really cool. Whatever you do, it'll be great! Awesome video, awesome problem.

I like the second one -- I'd love to hear your thoughts on if competitions make you "smarter" when it comes to maths. If it actually means anything to be able to solve these problems.

PseudsPie

I'd really love to hear your thoughts on the merits of making math competitive, i.e. the second one!

Of the extra points, the third resonates most for me.

Primer