The unexpectedly hard windmill question (2011 IMO, Q2)
The famous (infamous?) "windmill" problem on the 2011 IMO
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Special thanks to these supporters: 3b1b.co/windmillthanks
Home page: www.3blue1brown.com
The author of this problem was Geoff Smith. You can find the full list of problems considered for the IMO that year, together with their solutions, here:
www.imo-official.org/problems...
You can find data for past IMO results here:
www.imo-official.org/
Viewer-created interactive about this problem:
/ interactive_windmill_v...
And another:
aalluri7.github.io/windmill/
I made a quick reference to "proper time" as an example of an invariant. Take a look at this minutephysics video if you want to learn more.
• Spacetime Intervals: N...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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These animations are largely made using manim, a scrappy open-source python library: github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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“I know what you’re thinking, those do happen to be all prime numbers” nah I wasn’t thinking that
I don't even know what a prime number is.
@@Paddy656 Numberphile (a YT channel I've been following for years) has plenty of videos regarding prime numbers. You should check them out! Long story short a prime number can only be divided by itself and by one. So 2,3,5,7,11,13 are some examples.
@@OguriMichiyo its a joke . Everyone knows what a prime no is
@@OguriMichiyo it's joke but I agree about what you said about Numberphile.
@@OguriMichiyo Don't patronise me, nerd. Everyone laugh at the nerd
Came for 3Blue1Brown, stayed for 5Blue5Brown
WhatYouDidThere is a subset of ThingsISee
I'm more of a 4Blue4Brown1Pivot guy myself
LoL
That is actually really really good!
Can someone explain?
"If a made up windmill prepares you for a real problem, who cares that it's a fiction?" Wise words man
A constant.... People need to remember this....
It tells why you do maths
@Logan Post +1 and the image was so exquisite, it is my wallpaper now :)
F
that gave me chills. beautiful words
I paused this video months ago trying to solve it. I ran into all the traps you listed out after giving about the solution, and ultimately gave in and watched the video. This is one of the most unique failures I've had when trying to solve a problem and an incredible lesson. Thank you.
As Grant said in the chessboard video, puzzles are a rare gift and it takes ages to forget the solution. Cheers mate! You've inspired me to not give up on hard problems.
I thought they meant any point which makes it so that it is possible to never touch a certain point
The clicking sound when a point gets hit is really satisfying.
no it is very annoying a much softer bump sound would be better
you are so right i love it!
Reminds me of the sound of a Geiger counter
Go watch 3blue1brown's collision video
@@timr3682 That's why the radiation sign looks like a windmill
Came for the video, stayed for the dope clicking noises
I only hope you haven't yet watched the bouncing blocks videos so that you can go away and discover them anew!
ASMR? I feel ya
Came for the videos, came for the noises
oh, yea man, spin that line faster, I need more clicks
*click click*
I can't help but think the reason so many participants struggled with this problem is because it's so difficult to visualize. The graphics in this video make discovering the solution seem almost obvious, but with a pencil and paper? And a question written in that kind of language? It's very far removed from what you typically see in the study of mathematics.
it's not really that hard to visualize, just draw a few points and use your pencil as the spinning line. Such language is nothing unusual for competitive math and while it might be true that contest math and school math are 2 entirely different things, problems you usually solve at school usually involve using some formula or definition you've learned recently which is far too easy and you can't really call that a challenge, which a competition, especially one like IMO, is supposed to be.
It is actually very hard to visualize. Stating it's easy won't make it easier. Doing it on paper is cool but it take a lot of time if you want to try different starting line And starting points like in the video.
i think they can visualize it, but using pen and paper to turn imaginary visual into practical visual would take a lot of time so they prioritize other questions, and because Q3/6 is harder it probably took the biggest chunk of time.
Actually I think people confused the wording ANY point vs a point in which I would’ve gotten it wrong since ANYONE point is not true Also a windmill process is not obvious at all
Actually apparently the #2 scorer thought the problem was too trivial but he didn’t write more than a 2 line solution and 2 lines can never grant full 7 points shrug
The click every time the pivot passes off is god tier
See Y’all In 5 Years When This Is In Everyone’s Reccomended
yep
I hope i get a girlfriend by then
It's how I found this video lmao
Bluezzz same
Already happening and I'm a piano channel
After watching the video: 1. I still don’t know the answer 2. I still don’t know the question
The whole crux of the maths problem was...Thin line go round and round. Little dots go clickey clickey. That's my reading of it anyway.
@@GirGir183 BIG BRAINNN
It's not explicitly laid out in the video, but the video gives us all the needed information: - the video has proven that if you take a point and line that "cut" the set in half, then when you rotate 180 deg both halves are inverted (all the blue points are now brown, and vice-versa). It only works if you cut the set in half "perfectly" (with a little cheat when you have an even number of points) though. - following the same logic, after you've gone 360 deg you're back to your original position. - now, an important point made in the video is that the *only* way for a point to switch sides (become brown when it was blue and vice-versa), is to first become a pivot. - so that means that when you've travelled 180 deg, since every single point is now a different color, they *must* have become a pivot at a point or another (otherwise they would still be the same color). So here we've proven that there is a combination of "starting point +line" that hits every single point. - and, when we've travelled 360 deg, we're back to square one - which simply means that as we continue turning, we're going to do the same pattern indefinitely, hitting all the points over and over again. So, not only are we hitting every single point; we're hitting every single point an infinite number of times. There's not really an equation to describe this, or at least not an easily palatable one. You're better off writing a logical statement. A condensed version of what I've said above, which I believe would be accepted as an answer, could be: "for a set S of 2n+1 points (odd number), there is a point P in S through which we can trace a line that has exactly n points on each side (for a set of 2n points (even number), one side has n-1 points). Since the change of pivot results in a "side switch" for the previous pivot point, and that when the line has rotated 180 degrees, every point that was previously on the left of the line is now on the right, and vice-versa, then it must be that every point has been the pivot at least once. After a 360 degree rotation, the system will reset, and we can conclude that each point of S will undergo pivot and switch an infinite number of times."
Well the question was displayed, so perhaps you meant you didn't comprehend the question. Of course, I'm just teasing because when I first saw the question I was rather confused as to what it wanted myself.
@@quentinrizzardi1763 bravo, that was amazing.
On the "social" point, I've noticed this phenomenon in all sorts of things I've taught to people: programming, dance, woodworking, video games, etc. For human beings, it seems like an essential part of learning something is forgetting what it was like not to know that thing.
True. I have experienced this many times when learning. I don't know whether you're religious or not, but this reminds me of Isaiah which says "Forget the things of old; behold, I will do a new thing...".
I remember attending a math camp and my instructor was one of the contestants of the 2011 IMO and he got this question on the test. This has one of the best solutions that I’ve seen
“You want to believe a result before you try too hard to prove it” Excellent!
Mathematicians are anti- science confirmed
Best take-away from this video for me!
@@Kahitar can you give me timestamp I missed it
this goes well with "if you can't prove something is true - prove it is false and the find your mistake" those two are probably the only way I'm able to cope with college level maths
@@sambishara9300 Why is that anti-science? Scientists have to make hypothesizes before trying to prove (or disprove) them.
3Blue1Brown: "I guess I will try to exite people with this maths problem" Internet: "thAt CLickIng nOISe Is SatiSFyIng"
Honestly I get scared of gama-rays when hearing this😆
There is no 's' in math. Why do people keep spelling it with an 's'?
@@shawncaton9489 I think it was numberphile that covered it in a vid. If I remember correctly, it comes from most (English speaking) Europeans considering mathematics as a plural and translating it to the shortened version "maths".
@@shawncaton9489 There are a lot of "S" in math, because math it's plural, just because a new born country thinks it's singular, the rest of the world disagrees, and no, Europeans don't just "Consider it Plural", the word IS plural, even without the "s".
@@shawncaton9489 mathematic (n.) "mathematical science," late 14c. as singular noun, mathematik (replaced since early 17c. by mathematics, q.v.), from Old French mathematique and directly from Latin mathematica (plural), from Greek mathēmatike tekhnē "mathematical science," feminine singular of mathēmatikos (adj.) "relating to mathematics, scientific, astronomical; pertaining to learning, disposed to learn," from mathēma (genitive mathēmatos) "science, knowledge, mathematical knowledge; a lesson," literally "that which is learnt;" from manthanein "to learn," from PIE root *mendh- "to learn." As an adjective, "pertaining to mathematics," from c. 1400, from French mathématique or directly from Latin mathematicus.
Now I know why my university adjusts the grading scale according to the number of points students were able to get. I always thought it is to limit the number of people who pass since the first year is a sort of entrance exam. Now I think it is reasonable to think that sometimes a problem is just way harder than they anticipated so they are more merciful on grading that one. I am often amazed by the fact that we are expected to be able to prove theorems that mathematicians a few decades ago were unable to proof and I am just a student. Of course, it is easier for us because we know that it is provable and we know that we probably need to use the tools that were presented to us in the course, but a few mathematicians died without ever seeing the solution to this problem.
I had an astronomy 101 professor whose tests were 10 questions each worth 10 points. He told us that the only way to get a perfect score on a question would be to answer it as thoroughly as he would, which is why 50% was an A.
after watching your videos for a year, i’m baffled at how you consistently explain not only the solution to a math problem, but the process of thinking for an entire topic within math. Props
For your 2M subscriber special, can we see you tackle an IMO problem you’ve never seen before, to see how your brain works without having a ton of time to animate and think? You are incredibly insightful!
Hmm...intriguing idea...
Love this idea.
3Blue1Brown agreed!
Understanding the way you handle a problem from the beginning would be extremely interesting, please do it !
Like 'Mathematics Train'
Let's just take a minute to appreciate how much effort was put into creating such a well animated video.
I know; I always wonder how educational channels make animations like this
this conversation sounds like an intro to an ad for an animation program
@@thomashorne2607but there's no good tutorial videos I guess...
@@Uluc994 you can check out TheoremOfBeethoven channel for manim tutorial
Szymon Duniecki expect what he calls brown looks more gray but that doesn’t really matter
3Blue1Brown As a non-mathematician I greatly appreciate the Don Quixote reference you made at the end of a very satisfying mathematical puzzle. I love your videos. You provide just enough information in one moment that I can make the stab at your next point before you quite get there in your video. Very satisfying, even when I get it wrong, but especially that one or two times I've actually gotten it somewhat right.
Fun fact (I think this is correct) but apparently Lim Jeck (the 2nd place scorer) saw this problem as trivial and gave a 2 liner for it and hence got 5 marks which is insane. He apparently missed 2 marks because there was no way a 2 liner answer could get full 7
if he actually managed to fully explain that in 2 lines he should’ve gotten those 7 points length should ideally never be a factor i think the reason is because you can’t fully explain it in 2 lines
what did he say? can we find it anywhere?
3Blue1Brown: "A question that anyone could understand." Narrator: "The viewer could not understand."
freeze frame, black and white, and background voice. "It was not, in fact, understood."
That made me chuckle
I read that in Morgan Freeman's voice
@@wisdomwielder perfect lol
@@wisdomwielder I read it in Ron Howard's.
my boy georgios getting a perfect score in the single hardest question that only 0.01% got a perfect score on, but then getting a 0 on the one 60% did. rip
It probably took him to much time, so he had no time left for the other tasks
it's 1% not 0.01% btw
greek kid
@@spotifyhd890 Question 1-3 and 4-6 are given at 2 separate days. The irony being pointed out is that Georgios Kalantzis (4:58) got a 0 on question 2 but a perfect 7 on question 6 (the hardest question) the next day.
A true Geometry lover, my prof at school was one of the people that "trained" the greek math team, he was very proud of kalantzis.
I found your channel yesterday. The animations, explanations, and the problems are amazing. I'm really enjoying it.
Beautiful video. Beautiful channel. You have no idea how inspiring your videos are to me and so many others. I think your content goes far beyond mere tutelage, far beyond helping people trying to pass mathematics classes. I'm no mathematician, far from it, but your videos give me no other choice than to conclude how beautiful this complex world is, even when the outside world can be chaotic. Thank you, Grant.
As a physics student, I really appreciate the take-away message of "look for an invariant". Really good advice!
MiSta BlackJack Me too! I’m about to start a physics PhD and throughout my undergrad nothing has been more important in solving problems than invariants. No surprise that my favourite mathematician is Emmy Noether - whose theorem laid out the basis for finding generalised invariants in physics! Right now I’m working on my masters project (I graduated in July but the project didn’t quite get to where I wanted it too, so I’m putting in some more work this summer) and the one thing that’s really stumping me is calculating a particular sort of invariant called a Chern number (you might have heard of them, but if you haven’t they’re the invariant in the quantum Hall effect for example).
but you can never stick to only one concept, no matter how attractive it was the last time you met it.
As someone who will start studying physics this fall... Any good last minute tips?
@Darklorddestroyer14 you sit on a throne of lies darklord, olympiad my ass.
Yep
Yes I was definitely thinking "101, 563 and 2011? Those are all prime numbers"
That is the joke
Hey! We almost have the same last name. Sorry, but it's rare to find another xD
Yep haha definitely (cough)
Yeah..."Lumpkin and Lumpkins" could definitely be a cop show.
I liked how you added a philosophical aspect to this problem because what you said at the end is so true. Your solution seemed easy to me, but then again how would I know to go with that solution, and how would I know to look for a constant.
An appreciation would be much less for such type of videos. It just blew... Mathematical passion brings you here and what vibe this channel gives is always a real pleasure to have! Thanks for such a beautiful video.
The inclusion of the animated graphics makes this much easier to understand. It’s good to remember that the kids taking the test didn’t have this advantage. Anyway fantastic videos they let my feel smarter than I am.
... I'm the only that tough on using a pen to simulate the line?
@@blasecube No
I´m of the opposite opinion, nothing will me make feel dumber than comparing myself to the very best you can find on this planet.
Welcome Homo VIdens. You're the living proof of how we "evolved" into a new kind of stimulae analysis.
Did they have rulers? Enough paper to make one out of it?
I love the videos where you explain math questions that where given at real math competitions. This video inspired me to start taking Math Olympiad much more seriously, I would love to see another video like this!
I really enjoyed this one. I find your videos on general problem solving helpful as well as pleasurable. Well done.
*4.5 hours pass by* Lisa Sauermann: Oh nice perfect score Me: *still frantically drawing a connect the dots picture of a bunny*
You sir, are a genius
A B : 1, Math Guys : 0!
It just goes to show us, once in a blue moon a girl will do better at maths than a boy. But that's about how often, no more than that.
@@GirGir183 incel detected lol
No. It's just what I've observed around me in the decades I've been alive.
*All truths are easy to understand once they are discovered; the point is to discover them.* Galileo Galilee
*quantum physics enters the chat
@@hacherskanon3082 The truth of quantum physics isn't difficult to understand, just difficult to accept. The truth is that quantum mechanics doesn't simply model reality, it IS reality. Copenhagen, Everett, and other "interpretations" are all attempts to deny this reality by trying to explain it in terms of the "reality" our eyes show us, and to invent explanations for the things our eyes can't see but which we know are there, and it is this process which introduces all of the confusing and strange elements of modern quantum physics which cause it to seem difficult to understand. People have a tough time accepting this, because mathematics is a tool invented by humans with arbitrary rules we chose and it feels really weird to think of it as something which exists on such a fundamental level. It'd be interesting to see someone attempt a proof of this principle: that any system of mathematics which can be proven to be consistent and can be shown to accurately model reality is functionally identical to any other system of mathematics which meets the same criteria. This would show that mathematics has more in common with language, in that you can always convey the same ideas even if the words are different, which would show that it exists on a much more fundamental level than merely being a human creation.
Did he know organic chemistry tho?
Galilee must be the Chinese copy of Galilei.
@@x.x-JoJo-x.x lmao
Grant, you're my Idol. I love your work and your ease to explain things, and to carry deeper messages than just computations in your videos. You made me cry for the beauty of what you teach, thank you so much
Math always was difficult to grasp. Visualizing and correlating what the function means visually makes it much more stable and grounded to show what the function defined is trying to outlay. It’s like seeing the alphabet for the first time associated with each word - rather than a word trying to focus on what the word is trying to say rather than starting with the letters. Thank you - this method of conveying math makes the difference between trying to keep up versus having it visualized so it’s easier to grasp. I was never shown this way of learning in any schools growing up.
Is this problem hard? Well yes, but actually no, but actually yes
Can you solve it? Well no, but actually yes, but actually no
@@Fins-T Can you solve it? Only after 3Blue1Brown gives an excellent proof including awesome animations.
With some problems like this, how hard you find them depends a lot on how quickly you get the right idea. If you get it, it seems trivial. If not, it can take hours. I didn't find this problem particularly difficult, and that was similar to 2018 Problem 3, where I got a key idea very quickly.
@@mathymathymathy9091 k
Mathymathymathy exactly, I didn’t really find it extremely difficult either but it all comes down to finding a setup in which the solution is obvious!
“You want to believe a result before you try too hard to prove it” I finally understand flat earthers.
I mean that's how every theory works. You see an observation and you decide what you think it means. From there you decide you believe your own theory. You then need to announce or publish your theory. From there its just a matter of defending your theory from any and all logical and rational attacks. If someone presents you with an relevant observation your theory can't account for then you have one of two options. You can either accept your theory is wrong or you can accept your theory is currently incomplete. Flat Earthers have failed to defend their theory, thus it is not widely accepted.
@John Smith Or you could look for obvious flaws in your theory. Or try and look if there is another already made theory that is mutual exclusive with your own and try to disproof that one first. If you can´t, there is a good chance your own is wrong.
@@hamsterfromabove8905 did you watch the flat earth documentary on netflix?
but i dont understand trangenders though lol
No, flat earthers actually "research" their stuff. Most aren't uneducated, so they aren't so easily convinced to change their minds from starting off believing the Earth is a globe, to thinking it's flat. The problem is, that since they think they are educated, they think they are capable of doing their own research, and they think that they have covered all their bases with whatever messed up "science" they have, they think they have tested this rigorously. This of course, would have to be coupled with a strong ability to conjure conspiracy to even consider that what everyone else knows as fact isn't actually true.
7:58 I was trying to solve it myself, but I could not, and after seeing just this single frame I understood everything. Of course you still need to see that every dot has changed color, but the first idea is the hardest and the most important. Beautiful.
I love it when there are the dots, and then the line goes through them. thank you!
Ok so Lisa didn't have beautiful youtube graphics so how did she write the proof ?
Drawing dots on a scratch paper, putting her pencil on the dots and spinning it around, recognizing the pattern that repeats
@@thelostdonkey1585 That's how she figured it out, but that doesn't suffice as a rigorous, pointworthy proof.
@@beerus553 And they say that girls are dumb at math.
This is still a descriptive solution, not a mathematical one.
@@user-eg6xu7cr8e yes, but it is ok if it is correct and logical. would mr. Spock say :)
Nobody: 3blue1brown: I know what u r thinking... Those do happen to be prime numbers
I DEFINITELY thought that, yeah.
@@terner1234 I didn't get that far. I was trying to figure out why the number of participants wasn't 606. 101 countries, 6 participants per country. Then I realised, oh right, people might have fallen sick.
@@chaosredefined3834 I didn't even think about the numbers, I just made a joke, it's good you even got that far
@@chaosredefined3834 more like, not all countries had the money to send their competitors.
unfunny format
This is one of my favourite of your videos, touching upon so many interesting ideas! Fabulous!
Extraordinary exposition. This is how love for mathematics is born. It would be helpful to pose such questions to very young minds even in junior schools without demanding formal proofs; just watch how they tackle the conceptual part. Love your videos.
Interestingly, Lisa Sauermann just finished her Ph.D at Stanford this spring, according to her academic website :)
lol what a nerd
@@yourlordandsaviouryeesusbe2998 Why is anything funny
@@Mylada such stoic
Very interesting remark!
@@Rahul-cb4jb I'd rather have a sense of humor than be a humorless jerk like you.
The clicking sound is so satisfying..... And also the proof
I think you mean "satisfying"
@@zpotatoes7923 "satisfactory" a factory of countinuous satisfaction
sat·is·fac·to·ry /ˌsadəsˈfaktərē/ adjective fulfilling expectations or needs; acceptable, though not outstanding or perfect.
sat·is·fy·ing /ˈsadəsˌfīiNG/ adjective giving fulfillment or the pleasure associated with this.
1)Now I want too see the Q6 problem, lol 2)Tao's analogy actually heavy advocates for not necessarily trying to solve those problems, but learning how they should be solved
Just based off of the animations and playing around with it myself, I think the proof has something to do with the points in relation to the points they would be "connected" to (A more rigorous definition would be "The points in which the line would hit first in either rotational direction if it could pass through the points") having an angle
(Edited) A few people ask "but what is the formal solution?". Take a look at this writeup, which you'll find is essentially the same as the video. It would get you full marks on the test: artofproblemsolving.com/wiki/index.php?title=2011_IMO_Problems/Problem_2 The fact that an argument doesn't reference equations and sets doesn't necessarily make it incomplete. Sure, you could explicitly define what words like "left" and "right" refer to by defining such and such cross product with such and such parameterization of the line, but for most readers, it wouldn't actually remove any ambiguity. The purpose of formality is to make all the terms used unambiguous to all readers, not to dress up the language to involve sufficiently many symbols. If you understood this video, you understood the full solution.
Please do a series on Groups, rings and fields 🙏🙏🙏
💞💞💞.
Is there a part 2 to this one where you give the actual answer?
Cool
@Thomas Wilkinson Err, because no solution is given?
It's obvious....in hindsight. Give that shit to me without this explanation and I'd just stare for 10 years.
That's the beauty of this problem. The solution is rather simple, but coming up with the solution on one's own, as Miss Sauermann did, takes considerable insight.
😂😂😂😂😂
The social element of this puzzle, of not being able to empathize how challenging a puzzle is, has a lot of applications outside math. I can see this principle aiding in video game design, in mental health research, in leadership/management development, and so on
Even politics. How many times have you scratched your head thinking "this should be obvious to everyone!" when you're in a disagreement
This video is the embodiment of "obvious in retrospect". And I love it.
You are the embodiment of a twat.
@@julioservantes8242 comes on man
@@julioservantes8242 Yeah man comes on
I always like to say, "All Math is Easy .......once you understand it". This is video is a very nice illustration of that principle. I started saying that back when I got my math degree and I noticed that all my 'advanced' math textbooks were still being titled "Introduction to..." or "Elementary ..." ---- never once did I have the satisfication of carrying around a book titled "Really damn hard advanced ...." The people who wrote the books, thought the stuff was 'easy'..... and eventually it was.....
What a deep thought btw, every book is named as "basic, eleventary.." xD
You should write world's first "fundamentals of really damn hard advanced calculus (w/applications)" book.
“This textbook will first melt your brain then marry your mom”
i beg to differ
EVERYTHING is easy when you know the solution, but this is the approach of the dumbest people.
This is by far my most favorite video...possibly ever. Mathematics is merely the highest class of abstraction that we use to understand some instances of reality and from which we use to create new instances of reality. Once you abstract what a “cup” is, you can create infinitely many instances of a “cup.” Like the “cup” of points in the problem that creates failure and which contains the point that leads to success. Teaching is the same way. By understanding all the different perspectives (angles of the windmill) that a student addresses a problem, a teacher can interact productively with every student and encourage productive interactions between each student. Those interactions mean that at some point, each individual is a student (brown spots; the ground) or a teacher (blue spots; the sky). The cyclic (spinning windmill) manner of teacher student interactions can also be viewed one-dimensionally as a reciprocal function over time. One person provides information, the other processes that information and returns new information. This is represented in The Socratic Method wherein there is no “teacher” or “student,” but rather two partners journeying on the path of discovery together.
Great animation and analysis of the problem.. A huge congrats to Lisa for solving all the problems perfectly.. i see no comment here praising her for this stunning feat..
I appreciate the Don Quijote cameo at the end :)
Didn't expect that
i legit almost cried
Don Quixote is the reason that if I ever get a donkey I'll name it Oatey...
3B1B is expert in both math and literature, he's definitely a renaissance man
Very nice touch indeed
This channel does everything: Sociological analysis, philosophical ponderings and now he's even getting poetic
Sometimes I just return to this video to enjoy how beautiful math is. How good example this problem is that math is not only about numbers and computing, but math is in fact the art of modeling problems, stuff that "you don't even know how to start with". Beautiful problem, and sincerest congratulations to @3Blue1Brown for this video, animation, and effort.
Hi 3b1b, really nice video! I have recently started to use manim to make maths olympiad videos and the package is really awesome. I would like to thank you for providing such amazing tool! Cheers!
"If a made up windmill helps you prepare for a real problem, who cares if it's a fiction" This sounds like something that should be an ancient chinese proverb
I think it's a Don Quixote reference.
Reminds me of Hamlet. Windmills prepare you to fight giants...Thats a real problem.
@@vaevictus4637 I mean... he did animate Don Quixote into the video...
if the balance it off the line will fall off from center to outside..
Holy wow, these kids are truly uber-geniuses to figure out how to do this all by themselves.
On the shoulders of giants. But, yes.
yeah well i didn't understand even after he showed us the solution so lmao
Wow that comment just came to life with cringe
@@nathanaelarnquist That specific problem seems to be a good testament of true ingenuity rather than knowledge passed on by "giants".
@@RamHomier knowledge molds the way of thinking.
3b1b's videos are always a fun space to come to
Omg 3blue1brown I love u so much! Every video on your channel is an inspirational masterpiece. I wish you happy 2024!
This video is literally “Well yes, but actually no” personified
well yes, but actually no
Well yes, but actually no, it's videographied
It took me 11 minutes to realise that you choose the colors blue and brown lol
But they weren't 3:1, simply 1:1!
now i realise LOL
@@shizotypical Look at 8:13 there it is kind of 3:1
10:59 lmao
haha same here!! 10 mins in and it finally clicked why brown but not just red OMG
Outer points and convex hulls is where my mind started going. This is a fantastic video about pure maths.
This problem is so beautiful. A few months ago I felt overwhelmed by the example, having just gotten into competitive math, however actually watching and fully thinking along I can see how truly elegant this solution is
“You want to believe a result before you try too hard to prove it” This once happened to me. I believed a result, and my friend believed a different result, so I said "Fine, I'll prove it". In the middle of the proof, I figure out that I am wrong.
1AmGroot that's always fun
1AmGroot very vague example like thats just rewriting the question and giving it to your teacher as the answer
What were the 2 results, and what was the problem?
@@emperorjimmu9941 That's a good story. Thanks!
@@emperorjimmu9941 That "problem" is very misleading though. 1 / 3 = 0.333... when you're dividing (working with real numbers). But a fraction 1/3 or 3/3 is "1" because you're abstracting away the underlying value. 0.333 * 3 can never equal 1. The catch is really that you're working with two different units, or rather an estimation of the value of 1/3.
I would love to see her acutal answer to the question. Like, how to you but that in mathematical writing
yeah, the statements and reasons for the stuff
probably looked similar to this artofproblemsolving.com/wiki/index.php/2011_IMO_Problems/Problem_2
After watching and taking notes I believe the correct answer is "yes"
@@someoneontheinternet3090 dude the correct answer is "spin thru middle-ish, change the color-ish"
Ah after reading it twice what eluded me was that the "middle" point is in the relative center of the X axis, but the line drawn starts at initial vertical position, and it's the combination of the two that ensures it runs trough the middle.
Rewatching, only just noticed that the very beginning seamlessly transitions from the thumbnail. Love that detail.
I don't know why but that ending had me BALLING, crying my eyes out. Thank you for your channel and everything haha, math is so fun and amazing .
Cool
Infinity is a concept created by mathematicians to represent how much I love this channel
“Problems 1 and 4 are doable” PFFFFFFFFFFF
I press x on that one
@@Anonymations I press F on that one, cuz I'm PC gamer not a console player.
Nd yeah F
F
It’s because the people that go to IMO are literally geniuses
peaceful voice and music, wise insights, great visuals, and interesting maths. i love this channel so much ❤
So for the past half year, from time to time I was trying to approach this problem from different angles - mostly related to convex hulls. Even though I did not manage to solve it on my own, the experience was really nice and I've come up with some cool reasoning.
That's the beauty of problem solving.
This was a VERY pleasant experience. I did not expect this video to have such a deep and beautiful message, and I bet even people who whole-heartedly hate math and exact sciences can appreciate it, assuming they can actually sit through the video and resist the urge to click off. Edit: Wow, 200 likes! I never had that many before, tysm!
My focus of study is in the social sciences, and though "whole-heartedly hate" might be too strong a term, I have always been terrible at math and exact sciences. I struggled mightily with getting into the right mode of thinking for discrete math and it is the main reason I did not pursue computer science. Getting a well-articulated glimpse into the mathematical/exact sciences mode of thinking and reasoning into a problem has been very valuable for me. "Find an invariant" was a wonderful insight for me personally, and hindsight truly is 20/20... an important lesson in humility.
@@ddogg14 it's very interesting how much reasoning and logic can be different, but still just as useful and fitting when you look somewhere far from your field. Thank you for the comment :)
I watched and at the end I said out loud: "Wow, that video was f***ing fun." The sounds and the smooth animations and the little light flash on each point as it becomes the new pivot are all great. I can tell you really pay attention to the sensory experience of your videos.
like mines? :D
as a high school student aspiring to teach math this provided quite a bit of insight for just one problem, thank you
Wow! Really appreciate the effort you put into these videos!
Me: Haven't done math in 14 years. KZhead: You know what i think you'd really enjoy? ... Strangely enough, youtube was 100% right.
I find your comment weirdly motivational. I'm studying physics and maths is stressing me out TO NO END at the moment, and it's really frustrating... but then I read this and try to imagine what it's like to not do maths for 14 years and I can't even imagine that. Hell, I miss maths over a long summer break, no maths for THAT long, I just couldn't do it. It reminds me of how much I love maths, even if it drives me crazy occasionally.
@@baguettegott3409 Yeah same for me, coming close to going to university and I need to get an A* in Maths and Physics. These videos help me.
People: Is this question hard? 3Blue1Brown: yes, but actually no, but actually yes.
2/3 yes?
Vsauce2: So it really is a simple problem, right? WRONG!
Oh no it's in a superposition of being both hard and not hard!
@@valeriobertoncello1809 Schrodinger nods in approval
Oviously not but typically yes
I have never passed exams. My math is basic. At the end I have asked a question. Would love some answers, thank you. What I have been doing over time is watching simple explanations on youtube. These are pretty complicated but as I watch each one, I add to my knowledge and understanding. Math is so interesting, thank you for this. Can't wait to learn more. I get the gist of it but am not able to explain it ❤😷 I am loving this at 06.33am. It's so calming too, his voice/ watching the windmill go around. Wow wish everyone could understand how amazing this video is. The pivot point with 4 on left 4 on right, so easy to see and understand. A constant☺ What's amazing is that I screen shot the Question, then watched the video, then read the Question and I thought that's simple enough to understand. Why did nearly everyone fail the question and answer ?
I appreciate this showing up in my recommendations. What a great video.
"Say, half the points blue, and the other half brown." But that won't be 3 blue to 1 brown. I know this doesn't help with the current problem, but I can't help but be upset at how close this came to that without quite going there.
I went there too 😀
Although, did you notice how clever he was in the animations? Even with half the points blue and the other half brown, he still managed to have the screen 3 times brown and once white :')
He could use 4 points and as it is even, take the point passing through the line to be blue and.... Yeaah 3b1b
ARGH! All he needed to perform was to shift line l’s starting point proportionately to the side and still have been able to solve the problem!
"The solution is something that anyone watching this video could understand" Press X To Doubt
yea, i still dont understand what the solution is
X
@@kefinnigan2 In order to make all points be hit an infinite amount of times, on whatever starting point chosen, the line you start with must be angled in such a way that it divides the dots in half.
@@kefinnigan2 same amount of colored dots on both sides after a full rotation = it never moved from its original starting point and therefore *cannot* ever move because theres no other variables
X
오늘 도파 방송보고 왔는데 자기가 진짜 바쁘다 or 현직 문과다 하는 사람들을 위한 요약 앞에 7:54~9:00부분은 도선생님이랑 봤으니까 따로 1번 더봐 그래야 감 좀 올듯 9:21~10:10 회전중심으로 선을 2등분했을 때, 방향은 시계방향으로 고정이니까 위쪽부분의 선은 무조건 오른쪽 점에 닿고 아래쪽 부분의 선은 무조건 왼쪽 점에 닿음 그리고 닿았을 때는 무조건 색이 바뀌고 닿아야 바뀜(이제 전문용어로 필요충분조건이라고 하지요?) 10:11~11:28 180도 돌렸을 때 직선을 180도 돌리면 똑같은데 이제 앞에 조건을 다 만족하려면 처음회전중심=180도 돌렸을 때 회전중심 이여야함 이 때 180도 돌았기 때문에 처음이랑 비교했을 때 왼쪽 오른쪽 색만 뒤바뀜. 근데 색이 뒤바뀌었다는 뜻은 곧 만났다는 뜻이니까 따라서 이 앞부분 7:54부터 나오는 중심은 무조건 문제의 조건을 만족함 11:29~12:40 점 개수가 짝수일 때는 회전중심에도 색이 있다고 가정하고 똑같은 과정을 하면 됨 이 때 180도 돌면 회전중심만 색이 바뀌고 위에 점 개수 홀수일 때랑 상황이 똑같음 12:40~14:09 이 문제가 주는 교훈중 첫 번째: 대충 머리속에서 선이 돈다고 생각했을 때 왠지모르게 ㅈ밥같아보이는 무언가가 생김-> 문제가 쉽다고 생각함 but 영상 앞에서 보여줬다시피 이 문제는 괭장히 어렵고 이 문제 낸 사람들도 난도조절 실패함. 실제로 수학업계 종사자들은 자기가 하고있는게 얼마나 어려운지 감을 못 잡는다고 함(이 영상 제목의 이유) 14:10~14:55 무언가 고정돼있는 고정값이 중요하다 14:55~끝 수학 문제는 실생활을 푸는데 중요하다 이런 개 무쓸모같은것들도 언젠가 필요해질 수도 있다. 졸려서 글이 잘 안써져서 ㅈㅅㅈㅅ.. 이해안가면 좌표찾아가서 보셈 +이더리움 화이팅
그래서 답을 적을라면 어떻게 적어야하나여~?
@@user-gi1sj5ms3z 저 위에 베댓 중에 있어요 영어가 딸려서 직접 찾아보심이
What an insightful comment.
Sir, the way this problem was worked out is such a good way to teach. I wish this was how public education worked.
Damn. The problem is easy to understand, the solution is easy to understand, but it's extremely difficult to come up with. It's all so beautiful. This is real math! :-)
Pretty much that was my experience when I participated in IMO 1991 hosted by Sweden. The unusual problem of that year was a problem combining graph theory with number theory. Question was easy to understand, and proof was easy to explain (could possibly fit into one tweet), but man was it difficult to come up with. Needless to say I didn't score very high (no medal) but I did manage to score a full 7 points on that problem. The feeling was worth it.
@@bornach really appreciate that .. i always wonder what these good student do when they grow up ? do they continue into education or research field or something else ??
Have you looked much into the coffin problems?
@@bornach Beautiful problem. The key observation I assume being that consecutive numbers will always have a greatest common divisor of one, and then you have to figure out how to "break up" the graph into sequences of consecutive numbers.
Honestly the most satisfying feeling in the world is solving a ridiculously difficult math problem. I live for it. Guess that's why I'm an Engineer 😂
Every contestant after solving the first question '' ah, this paper is so easy'' 2nd question ''I'm about to end this man's whole career''
Lol what about the people who dint even solve the very first problem. Poor kids.
@@SantoshKumar.budithi no shame in not solving an imo problem. I don't know about imo contestants, but in similar Olympiads the median score is usually zero
@@SantoshKumar.budithi They aren't poor kids ! They're among top 6 mathematical geniuses of their country. I bet you weren't even in top 6 of your highschool. -_-
@@austinconner2479 But it can still drive you insane, though. I watched this video 3 days ago and since then struggle to find the solution for problem 3 ...
@@SantoshKumar.budithi Another comment saying "kids".You know that most of the contestants are 17 or 18 years old right?
사람들은 자신의 이해를 확실시하기 위해서 혹은 증명하기 이해서 다양한 문제들을 만들어 냅니다. 그 중에서도 이 영상에서처럼 국제적인 수학 협회에서 만든 문제를 풀어볼 기회가 생겼다는 사실이 영광입니다. 이런 문제를 소개해주셔서 감사합니다.
Solving a (math) problem is way harder than understanding its solution. Brilliant video, as always!
*How about a series on **_Essence of Probability?_*
Yeess
Yesss
No !!!
Actually I'm pretty sure that will happen since there is a folder inside the "active_projects" folder of Manim (his animation software) under the name of "eop" (essence of probability) with a file called "what_does_probability_mean.py". github.com/3b1b/manim/tree/master/active_projects/eop
Essence of Statistics*
"And anyone watching this video can understand" Me: "Oh ok nice, usually I'm not too good at math but lets give this a go" *First sentence of problem is said* Me: "Mission failed, we'll get 'em next time"
Hahahaha
It basically means that you have a bunch of points on a flat area
"It's not over yet, we'll get 'em next time" -CODM domination Lol
The funny part is that math problems are often easier to understand once you've read ALL sentences, because then the various givens and everything become a whole.
haha
Find the constants in the chaos. It's such a simple yet powerful concept. I love it.
Very interesting and I will remember to look for the invariant when solving problems. Like first looking for all the edge pieces of your puzzle then grouping the remaining by color… Thanks for the video!!
The last thing you said about empathizing with not understanding something, this is something every teacher should think about daily when teaching. People who teach the same stuff year after year forget how it was when they didn't understand it and can get impatient when someone doesn't understand it right away.
that Don Quixote reference was a perfect touch. i want to cry.
Omg I thought I was wrong abt that
actually, I _did_ cry.
Where? Can you please tell the time stamp?
@@tanshufy23 it is on 15:35 the windmill, figures on the left and of course the sentence. that sentence was somewhat close to the whole point of the book.
Wow, I came here to comment exactly this. I love all of you here!!!
This windmill question is taking me back to the Borsuk- Ulam theorem and Broken Necklace Problem
the combination of calming voice and calming music is the closest thing to ASMR without being explicitly ASMR
My long term takeaway from this video will be "look for the invariant".
I actually used this to solve a problem I was struggling with on Mind Your Decisions.
There is a book called problem solving strategies by Arthur Engel. It's a really great book that sorts Olympiad problems by solving principles rather than the sub field of mathematics. One of the chapters is on invariants which has 100s of problems like this one. There are other chapters on things such as the extremal principle, induction, colouring proofs, etc.
Invariant is one of the most powerful solution finding methods. Meaning you're right :)
Totally agree!
“This problem seems hard, then it doesn't, but it really is” Riemann’s Hypothesis and Fermat’s Last Theorem in a nutshell
The riemann hypothesis does not seem easy. Nor has it ever
No. The proof of Fermat's Last Theorem has over a hundred pages and you need heavy mathematical background to understand it. It's very much unlike the proof in the video, which requires almost no mathematical background to understand, it's just really hard to come up with on your own.
Collatz's Conjecture too
@@pendragon7600 Anyway, it seems easier than it is. There actually is an extremely simple explanation: www.quora.com/What-is-the-simplest-way-to-explain-the-Riemann-hypothesis-to-a-young-child, but you know hoow actually hard it is.
@@user-nb6zu3rk4f That is one of the most amazing way's I've ever seen to approach the Implication of the Reimann hypothesis!