3Blue1Brown - 2019-08-04
The famous (infamous?) "windmill" problem on the 2011 IMO Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/windmillthanks 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: https://www.imo-official.org/problems/IMO2011SL.pdf You can find data for past IMO results here: https://www.imo-official.org/ Viewer-created interactive about this problem: https://www.reddit.com/r/3Blue1Brown/comments/d0b0qw/interactive_windmill_visual_program_download_link/ And another: https://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. https://youtu.be/WFAEHKAR5hU ------------------ These animations are largely made using manim, a scrappy open-source python library: https://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: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u 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. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown
Let's just take a minute to appreciate how much effort was put into creating such a well animated video.
I prefer to take it for granted and move on.
@Ninety 9 you might be red-green color blind
@CaptainLandr0ver yeah I don't understand ether
@Thomas Horne Thanks!
Szymon Duniecki - You are wise. It's important to be both grateful and kind.
Ok so Lisa didn't have beautiful youtube graphics so how did she write the proof ?
@ထ Anthony ထ if you're sure that your initial comment was a hilarious joke, with the deleted comment it was replying to as context, can you explain the joke?
@um thank you... fellow stranger
@MinusPi just have faith, she had described well the reasons too... You know IMO ain't a piece of Joke bro...
@M I think you are confused between numericals and reasoning, both are mathematical.
There is no real line between "description" of the solution and "mathematical proof". What is a proof, anyway? It's the description of a series of mental steps, each step simple enough to be understood and verified by the reader, such that the last step leads to the statement that is proven.
How large these steps can be for the reader to be convinced, depends on the reader, and requires some finesse. If some steps are intuitively convincing but appear to demand more rigorous proofs, the proof might actually appear to be less formal to an outsider than an expert; the outsider will see some intuitively correct but lacking arguments, while the expert will know that every step is correct, because he/she has dealt with similar problems before and could fill these "gaps" easily.
Consider the "most" formal way of proving things: Using nothing but the axioms of the logic you're working with. There aren't many. If you wanted to show that the statement at hand follows from just these axioms, you'd be applying these same, trivial, syntactic axioms a hundred, thousand and million times over, making the proof immensely tedious to read. What you'd have to do is introduce simplifications; some steps are clear and don't require maximally rigorous explanation. Which level, between this maximal rigorousity and saying "the statement is intuitively clear", is enough, depends on the reader.
the german lady received herPhD in mathematics from Stanford in Spring 2019 :O
Leyrann That’s really incredible!
@Leyrann She not only shaved off some years but also found the time to give birth to her daugther autum 2018. But it seems she found some shotcuts as she olny studied from 2011 to 2014 in Bonn (germany) and then right of started as a grad student at stanford. No matter what, she has an impressive agenda.
@Ganimede And the guy from Hong Kong is now a doctorate student and lectures tutorials in the mathematics department at CUHK.
@Insaf Idris Whatt ?! I am an Engineering student at CUHK may I know which tutorial he lectures on ??
I think she's at MIT now. impressive maths skills and a mother too. girls tend to multitask lol
“Problems 1 and 4 are doable” PFFFFFFFFFFF
@Aaron Kou I wouldn't say genius but yes they work and study really toughly.
I can do some p1/p4 it is doable one year of training is enough for some p1/p4s
@BatMAN Bin SuparMAN It's a combination of talent and hard work. One without the other is like striking a nail without a hammer. Without the hammer, sure you can try really hard and push a nail into wood. That's what doing hard work without talent feels like. And a hammer without a nail doesn't get to show it's true utility. That's what happens when you have the talent but not the hard work to utilize it.
@David S. I was a nominee for one of my country's IMO positions (which I considered too much of a hassle so I didn't even try to do it lmao.) And yes those are beyond doable, all you need is to just have studied for some time (a couple of months IIRC.) and not be unlucky with the questions and you'd be pretty much set.
not even hard bruh
The clicking sound when a point gets hit is really satisfying.
that's what I intended to say!
You should play Peggle.
sounds like a geiger counter
My ears feel relived after my eyes saw such a complex windmill problem cuz of that clicking sound
Dohh, I found another who is much like me.
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
@SpotifyHD 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.
"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
The don Quixote reference was SO GOOD
@Logan Post +1 and the image was so exquisite, it is my wallpaper now :)
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!
@Valerio Bertoncello Schrodinger nods in approval
Can we get a 10 hour video of the windmill just spinning around?
It would be interesting to implement this in the real world by having rotating shafts on a board that can be electrically magnetized, and rotate a steel bar as the line.
As the steel bar goes over the next point, the change in magnetism or light triggers that electro magnet and the former pivot point is turned off.
Would be a cool visual effect.
Idk why but I have a feeling it wouldn't turn out to be that eye pleasing. First of all how will you make the line go through the past pivot?
And also with so many parts, entropy will tirelessly work against you, so even if you manage, it will be a "sometime" pleasure.
"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.
Her: The hint im giving is so obvious!
The hint:
The Don Quijote de la Mancha reference at the end really got me. I love it.
...and it gets me again every time I rewatch it. Something about it is strangely touching.
3Blue1Brown: "I guess I will try to exite people with this maths problem"
Internet: "thAt CLickIng nOISe Is SatiSFyIng"
The pi guy: smart math things
My monkey brain: hehe click clack
it is
@Leo179 "Mathematics", as in: the various fields of mathematical science, is plural ('maths' being a shortening of this). When you say "math is fun", you are referring to "mathematical science", the process by which maths are used, which is singular.
@Shawn Caton the study of mathematics. It’s meant to be plural
@Allen Rubin Math without the 's' can also be plural. Examples would be: Sheep instead of sheeps, or fish instead of fishs. They just don't look right but we know what is being said. Sentence example: Math class was difficult today. To a 'proper' English speaking person, this may sound off a bit when said aloud, however to the common English speaker it is the other way around. Hearing 'maths' to us sounds like someone might not be fluent in English just yet and made a pronunciation mistake.
There at the end, this video went from "whoa, that's really neat" to profound :o
I'm noticing a lot of people asking about the actual "mathematical" proof. In fact, it is not too different from this, and there is minimal mathematical notation:
https://artofproblemsolving.com/wiki/index.php/2011_IMO_Problems/Problem_2
To be a proof, a solution need not be bogged down by equations and greek letters with complicated subscripts. It only has to be logically sound, and the above solution is logically sound. Often, the more precise notation exists to make solving problems easier or more efficient, but it should not distract from the heart of mathematics, which is pure problem solving. I believe this solution embodies that brilliantly, because it is little more than an exercise in logic. It is, in some ways, the purest form of mathematics.
"The solution is something that anyone watching this video can understand."
You're giving me way too much credit, pal.
2:21 Omg english subtitles were activated and I thought the dot at the enf of the sentence was the pivot
When you have three it’s like spinning a pencil around three fingers.
I love the humbling conclusion of this video.
I've been told that i could do amazing things with my intellect and should apply for mensa and shit.
I dont understand any of this.
I knew i was retarded but not THAT retarded x(
도선생님 생일 축하드립니다 ㅎㅎㅎ
Came for 3Blue1Brown, stayed for 5Blue5Brown
@Red Pillaging no plz
Harvard wants to know your location
HajahajhHF
Wtf all we are geniuses🤣
😂😂😂
This is something that can be understood easily, by visualising scenarios in the mind, but is very hard know how to explain & prove mathematically!
자 드가자
"knowing when math is hard is way harder than the math itself" .. me smells the lurkings of PvNP.
I’m just here for the ASMR clicking sound and his therapeutic voice
I gained 3 IQ just from touching this video
@Pashimanu 007 There was an 11-year old Belgian boy on the news the other day who was already doing Master's classes in Applied Physics, helping the professor with his research. Normally people are like 21+ before they do stuff like that :) Some people are just born with some crazy genes that give you hyper intelligence.
@ikinoktace from 1 to 2
@Vaylinne You're ignoring simple facts. The eleven year old boy did study advanced math. You can't magically know it. You first need to have the willingness to study something beyond your school. Assume you're right. Assume that the guy has very high intelligence as it's defined. If he didn't have any interest in maths whatsoever, he wouldn't have even been known. By thinking that intelligence is everything , people brutally underestimate the importance of passion and hardwork. Nikola Tesla is supposedly said to have high IQ. Yet he studied 17 hours in college everyday which spoilt his health and didn't get a degree when he stopped studying completely. :) Even geniuses need to study a lot to understand stuff. No one gets it magically. Yes some people do have better abilities but that can only give you a headstart. Beyond that everyone has to work. Headstart isn certainly not enough to complete a race and a person without a headstart can also complete the race
@Pashimanu 007 There's no possible way to judge someone's IQ from a youtube comment
@Vaylinne btw if IQ is indeed fixed and reflects someone's ability in maths then I disagree because if you observe people around you, the ones who perform better on a maths test are the ones who practice those hard problems over and over. You can probably try it on yourself. It's my experience too. Mathematical ability shouldn't be compared to IQ as it's not static as implied by IQ
"wE cAn AlL dO iT"
This has to be one of my favourite videos I've ever watched. Really made me think. My mathematical curiosity has been reignited.
슈바 ㅋㅋㅋㅋ 쥰내 어렵네
I've got an evening to myself and I'm gonna try to do this and see what I get, ill be taking notes as I go along here. I probably won't come up with what's in the video or even a viable solution at all(I'm not very formally trained in math and am pretty young) but we'll see where it goes. You can continue if you feel like reading
edit: just finished the video, my proof works but its a far cry from the elegance of the one shown.
The first thing I notice is if the points make a convex polygon then the problem is already solved, the line will just rotate around the given points. This immediately means that any combination of 1, 2, or 3 points will follow this rule. The first time we see an instance of a set of dots that may not follow this rule is when we have four points. If three form a convex triangle and the remaining point is inside that triangle and you start at a point on the triangle you will not follow the rule. This of course then means that we must start from the point within the triangle. When we do this we find that it is possible to satisfy this condition. This means all 4 point systems satisfy the condition. So far I have shown that any set of points:
*That form a convex polygon
or/and
*is of length 1,2,3,4
must satisfy the condition.
Here I think the problem can be restated slightly: At all points, there must be a loop through which you can return to your original point, this loop may not hit every point but there must be at least one for every point. So the proof becomes showing that for any set of points there cannot be a point that has no loop back to itself. Another way I may be able to prove this is by showing that if an N dot system has a solution then an N+1 dot system must also have a solution. I've got some options for where to go with this proof as of now.
I've been thinking and I've realized that any system with N-1 dots making a convex polygon and 1 dot in the interior will also satisfy the condition. To prove this let's go back to my first statement that if all dots have some path back to themselves the condition is satisfied, I can show that this is true in the setup I just mentioned, firstly all N-1 can reach back to themselves by simply traversing along the convex polygon (remember the path doesn't have to touch every point there just has to be one such loop for every point) and the center point will have a path back to itself in 3 steps [inner point -> outer point 1 -> outer point 2 -> inner point], therefore the entire system will repeat.
I think I've found something more general, I need to think a bit but this might be it. Ok, so the only way that a set of points does not satisfy the condition is if there is some convex polygon that the line rotates around and every other point is inside that polygon, there is no other way for the condition not to be satisfied. I can prove this,
Imagine a set of points P_0, ..., P_n that form a convex polygon, if a point P_j is added on the outside of the polygon then it will be hit no matter what because in order to hit points P_0 through P_n it must loop, meaning the line will turn 180 degrees and in doing so must hit P_j no matter where it is outside the polygon.
So let's try to prove that every point must have a loop through which it returns to itself. Let's assume a set of points P_0 through P_n and let's start at an arbitrary point P_j, from here one of two things can happen either:
*The line returns to P_j in which case the proof is done
*The line rotates around every point other than P_j in a convex polygon with P_j inside.
But as I have already shown the setup in which we start at an inner point surrounded by a convex polygon must hit all points.
QED
I don't think this is 100% solid but it's alright
After watching the video:
1. I still don’t know the answer
2. I still don’t know the question
did you have sound on?
I think I found the solution. Please consider my bad english and also math words. Solution : Define 'outer point' as if we line some two points and if there are zero point on one side of that line we can call the two points as 'outer point'. If we delete all the outer points of the given points, then there would be another outer points. If we keep delete all the outer points then there will be last points that contains no other points inside the area. If we start our line from that area, that will be the answer. Is this right?
same
@Traptinreality this is contest math
Question, super complex, simplified: random dots, good, make straight line, good. How you know you touch all dot if you rotate that line like a windmill.
Answer, start in the middle.
"The solution is something which anyone watching this video can understand"
Literally most of our dumbass who came here just because it was recommended: lmao
This literally added a great tool effort me. When I problem seems really hard, start by looking for invariants
Imagine the amount of work that must have gone into this video, the script writing, social and mathematical analysis, the animations, and everything else. Then imagine the kind of mind it must take to, after all that, be able to make this into a Don Quixote meme. This channel is too good.
완벽히 이해했어!!(하나도 모름)
Came for the video, stayed for the dope clicking noises
@ Spicy Jew - So relieved you've opened dialogue and to know I'm not alone; I thought I needed intervention for giving in to the hearty goodness of those satisfying clicks.
@Vojta Hála It's not 3.6 blue points, it's 15 000 red points!!!
@Rick White
Memes everywhere !
same
That's some ASMR shit right here: soft voice, clicking sounds, calm ambient music you name it!
This reminds me of kindergarten. Figuring out how to solve problems on my own and, unfortunately, ignoring my teacher. Pretty epic!
정말 재밌고 유익한 영상이에요. 저희집 강아지 마루쉐가 참 좋아해요
올려주셔서 감사하고 번역해주셔서 감사합니다
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
All truths are easy to understand once they are discovered; the point is to discover them.
Galileo Galilee
Galilei*
@hachersk Anon Quantum physics isn't hard to understand, it's hard to visualize. We know the rules, we can prove the rules are right by doing observations, we can use the rules in our calculations, we just can't connect them to the world as we see it in the way you can with, for example, gravity. The most important rule when doing quantum physics is: don't think about it too much, just utilize the rules you have learned. The more you think about it, the harder it becomes.
Galilei
"It is only the clearest of minds which sees that which once seen is clear to everybody." - Dr. R.L. Moore, mathematician and educator
Wiles' proof of Fermat's last theorem has entered the chat
I love how on a few parts of the video his voice sounds sooooo close to Phil Schiller (from Apple)
I was watching Cody ko before this and uhm I'm woundering how this got in my recrmdstions
中文字幕是用Google翻的吧
很不順
댓글 보니까 홍보 대성공이네
I appreciate this showing up in my recommendations. What a great video.
When I saw the video recommended, I first thought nothing of it but then I recognised the problem and that was a funny moment. (I'm one of those who solved it back in 2011)
gdhr - 2019-08-05
“I know what you’re thinking, those do happen to be all prime numbers”
nah I wasn’t thinking that
Shotgunboy07 - 2021-01-10
In fact, i never had a thought watching this whole video.
PFunk _ - 2021-02-13
@Nevan hibinibbbi big m he bbiibbbibibib
Soatnod - 2021-02-18
Ngl there are actually people who don't instinctively know what a prime number is, like me, a non native speaker of the English language.
Loofus Doofus - 2021-02-19
@Paddy the Daddy Wow U should probably not watch dis video Noob and look it up
Adam Doughty - 2021-02-27
Prime numbers have only one divisor