The hardest problem on the hardest test

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A difficult Putnam question with an elegant solution.
This video was sponsored by Brilliant: brilliant.org/3b1b
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/putnam-thanks
Home page: www.3blue1brown.com/
Solution to the puzzle mentioned at the end: brilliant.org/3b1bindicator/
These videos exist thanks to Patreon:
/ 3blue1brown
A different write-up of this solution:
lsusmath.rickmabry.org/psisson...
1992 Putnam with this problem:
kskedlaya.org/putnam-archive/1...
A problem with a similar flavor came up on the 2005 Putnam A6. Give it a try! The solution for that problem, by the way, was written by Calvin Lin, a friend of mine who works at Brilliant. Small world!
kskedlaya.org/putnam-archive/2...
kskedlaya.org/putnam-archive/2...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Korean: tebaioioo
------------------
Animations 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 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.
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with RU-vid, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown
Reddit: / 3blue1brown

Опубликовано:

7 дек 2017

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Комментарии : 20 тыс.

@knownbynone558 3 года назад

The video: "The hardest problem on the hardest test" Me who barely passed math classes: “Hmm, lets see.”

@kaydasilva595 3 года назад

hahahaha same here. Barely passed math but tought: well maybe i can solve it

@lukeskywalker691 3 года назад

Even if you ace your Maths quizzes and exams, I don't think that you can completely understand what he's saying

@pianoingels7128 3 года назад

@@lukeskywalker691 i had descriptive geometry in my architecture studies and was later a tutor for it, and as architects we are generally pretty good in spatial thinking and i understood the problem and even his solution almost fully, but i would never thought of it myself ofc haha

@netaropati7738 3 года назад

Same

@pianoingels7128 3 года назад

@@krewpi it is just about finding things you love to do. i was very average to under average in math, but when i started studying architecture it really began to make fun and i ended up being tutor.

@Kali-bs7oj 3 года назад

You know it’s getting lit when the smart kid brags about getting 3/120 points

@Umar-yi2ot 2 года назад

Hello 👋

@Kali-bs7oj 2 года назад

@@Umar-yi2ot excuse me, mr vertical version of my logo

@melviness4769 2 года назад

@@Kali-bs7oj you mean horizontal? You're the vertical one lol

@Kali-bs7oj 2 года назад

@@melviness4769 I saw it as vertical since the colors are stacked vertically but yeah I see now how I kinda am the vertical one haha

@RoxelanaLisowska 2 года назад

Because hes only smart not genius xd

@isaacmalik3714 Год назад

what i find even more amazing is that someone had to come up with all of these problems and solve all of them

@drawwithme4964 Год назад

ikr

@quantumgaming9180 Год назад

EXACTLY, I hold respect for professors that are able to make great problems like these , not too easy to spoil the fun, yet not impossible for the students to solve by themselves.

@kinghosty Месяц назад

@@quantumgaming9180maybe they just put something rly random that looks hard and then someone solved it and after the solution looks right they jusf accspged it

@matthewgeary1811 2 года назад

I remember watching this back in 2017, my freshman year of high school, and then falling in love with this channel and math itself. Here I am in 2022, a freshman math undergraduate, still knowing damn well that I could never solve this by myself lmao

@mgara514 9 месяцев назад

Lmfao

@gazingstar-zz9gf 2 месяца назад

hahahahhaha@@mgara514

@matthewganun4444 3 года назад

Imagine taking this test only to see some kid start flipping a coin

@TheBrickagon 3 года назад

😂😂😂😂

@muhammadumar8903 3 года назад

I'd defo start laughing at him and myself

@nataliegrayhandle 3 года назад

Gotta be creative to figure it out

@onlygrumpycat8380 3 года назад

Whenever you realize it’s not multiple choice😰

@luketaylor1305 3 года назад

@@kellycong8388 i think he understands the joke mate.

@enjoipls. 3 года назад

"If you choose 4 random points on a sphere...." I think I changed my mind can I go home now?

@magicmulder 2 года назад

My math prof, first lecture in first course: “Let’s take a simple example. Let E be a Banach space with norm M…”

@azysgaming8410 2 года назад

Exactly. That line itself made me lose my hope.

@lordartoriuscollbrande7434 2 года назад

Is it okay if anyone here didn’t mind if I only just asked if anyone ever wanted to ever watch my videos that I made? I hope that’s okay if I just asked. I don’t mean to bother anyone.

@RoxelanaLisowska 2 года назад

Why u are asking this answer your personal questions by yourself

@man-vd4zx 2 года назад

Exactly😂 a rational mind would give up in that point😂

@acurefordeath 2 года назад

You can further extrapolate the simplification to a one dimensional problem: given the space between two random points on a bounded line, how many solutions will contain the center? Given a first random point, there is always a 1/2 chance that the other random point is on the other side of the midpoint. You can then extrapolate from one to three dimensions by cubing the solution. (1/2)^3 = 1/8

@AbdulIsik 2 года назад

Works in the 0th dimension as well. Given a random point on a point, what's the probability that they'll be the same point? (1/2)^0=1 😅 crazy how math works.

@shubhamtokas5995 2 года назад

you're just taking the assumption that it can be cubed. there needs to be serious proof behind it.

@minecrafting_il 2 года назад

@@shubhamtokas5995 but that might be easier than the original question

@kennethluo4934 2 года назад

that's really actually cool, idk if it's just a coincidence or if that's actually how it works cause you'd need to continue testing that in 4+ dimensions unless you could come up with a rigorous proof like in this video

@jordanliu9747 2 года назад

@@kennethluo4934 1/2^n It's correct in every dimension. It's the proof that's the hard part though

@clone2255 6 месяцев назад

About 5 years ago, I watched this video and got inspired to study maths in college and take this test. This Saturday, I took the Putnam and solved 1 question!!!!

@kuolich287 6 месяцев назад

that is so cool! great job :D I'm thinking of taking maths in college too but I'm worried it'll put a burden on me as I'll be taking the maximum of 5 subjects. But this is so interesting :(

@sinan.7328 4 года назад

Teacher: "The test won't be that hard." *The test:*

@bluegamer07 4 года назад

Bratans Bratuchas 1.7k likes but no comments?

@sinan.7328 4 года назад

@@bluegamer07 yes, whats up with that?

@bluegamer07 4 года назад

Bratans Bratuchas strange.... well whatever I am glad to be the first one I guess lmao.

@sinan.7328 4 года назад

@@bluegamer07 it isnt strange at all I bought 2500 likes first then the other 200 came by themselves

@MK-sc9le 4 года назад

Bratans Bratuchas why would u buy likes though?

@nujak3271 3 года назад

This guy: explain all the math Me: still trying to figure out what a tetrahedron is

@Artist_of_Imagination 3 года назад

@akdarklight156 3 года назад

oh me too

@somerandomdudewhodoesstuff3107 3 года назад

I’m gonna okay myself but a tetrahedron is a 4 sided 3D shape if I’m not wring

@asaniwasabi 3 года назад

@@somerandomdudewhodoesstuff3107 like the Egyptian pyramids?

@wesburrows3733 3 года назад

@@asaniwasabi no i think the pyramids are square based 3d triangles (pyramids)

@caitlinphillips8243 2 года назад

Your teaching abilities are phenominal, you have explained this in a way that makes perfect sense to me without simply removing relevant details to avoid simplifying them, which is what most 'teachers' would do and consider it a job done.

@cheese5065 2 года назад

I’m in high school and I’m about average in math, I was so close to seeing this video as a write off and leaving. I’m so glad I decided to stick around, you explained this in a way that even the average can understand, you are an amazing teacher and this video was extremely entertaining!

@joshyoung1440 10 месяцев назад

What do you mean seeing this video as a write-off?

@shanedsouza189 3 месяца назад

It's a way of saying "math is too difficult and I'm not interested in learning this because I'm not going to use it in my real life anyway".

@momgay2118 19 дней назад

This is not for average university students

@elijahcole6786 2 года назад

High school teachers be pulling this out then saying “This is a college level class, it only gets harder from here”

@azdirtnaper 2 года назад

TRUEEEEE

@cheesefries7174 2 года назад

💀

@thegrim418 2 года назад

That's actually a great idea. April fools will be brutal next year.

@thedarksaber 2 года назад

@@thegrim418 oh no

@mosio.2504 2 года назад

My math teacher did a huge ass equation and said "Its primary school knowledge" I don't even know how to deal with fractions

@RingtoneCentral 4 года назад

*In class:* The solar system has 8 planets *In tests:* exactly how many stars are there in the universe

@manuel-no5zb 4 года назад

you have spoken the truth

@pao2725 4 года назад

Probably 24 stars

@masteryooda9087 4 года назад

I actually don’t know lol. Something like 2,000,000,000,000 x 100,000,000,000

@slattypatty9982 4 года назад

Nova nah maybe like 17 24 seems like too much

@pao2725 4 года назад

But why are we here just to suffer? Hm i guess 17 is more scientifically possible than 24... Thanks

@QuantumDivergence 7 месяцев назад

Wow, the animation, and music, the way you explained all of this. This is a work of art. Well done.

@TheSurprises711 Год назад

I really love how you break it down ...it makes so much sense and you are super understandable! Thanks for what you're doing to teach these concepts. And breaking it down beyond the answer to this question to what you can apply to all other situations is GOLD! 🥰

@antihero8214 4 года назад

teacher: cmon guys i taught u this

@dharmang 4 года назад

Haha yes fucking teachers be like "Common peeps i just literally taught you THIS exact problem yesterday"

@Quarkee 4 года назад

us: Noo, you taught the nerds that. The rest of us were busy raising our hands, waiting for our turn to ask for help..

@luukabrassart7635 4 года назад

You had this 3 years ago...

@awesomekb3065 2 года назад

Me at school: "ugh maths is boring" What I watch when I go home:

@beangobernador 2 года назад

it boring cuz u too slow git gud noob B) my PB: 7 minutes 38 seconds 12 milliseconds git gud

@itsatrap3703 2 года назад

Awesome 🤣🤣

@itsatrap3703 2 года назад

@@deleted-user-82u3ht92 Sad but true..

@butcherpete9252 2 года назад

@@beangobernador tf is you talking bout

@Hamsters_areCute 2 года назад

@@butcherpete9252 LMAOO

@Ajbolt89 5 месяцев назад

My brother recommended this channel, and this is a great introductory video to your content. The math is above my head, but the discussion about the process of problem solving is relevant to anyone. Cool!

@lordyt3689 2 года назад

I love this way of visualization 3Blue1Brown as well as other Math related RU-vidrs use. It really helps to understand these kinds of problems on a deeper level. For example, I doubt I would‘ve gained the same key insights from just reading through the listed write-up in comparison to your great video. I wish this method would be used in actual classrooms, however, with the dawning age of smart boards and such we might get tho this at some point ^^.

@nicholastorres603 4 года назад

teacher: im just gonna skip this question, u guys should have learned this last year

@atikshsingh1598 4 года назад

teacher last year: im just gonna skip this question, u guys should learn this next year

@unknownvulture6189 4 года назад

@@atikshsingh1598 😂

@classymuffin4589 4 года назад

Why is this situation so damn relatable...

@rithanipriyanka2393 3 года назад

teacher who taught us the previous year : I'm just gonna skip this question, u guys should have learnt this last year

@mattjohnson4013 3 года назад

@@rithanipriyanka2393 teacher: this ones self explanatory

@ApenstaartjeYT 4 года назад

Girl: I don’t get why he doesn’t get my signals Her signals:

@lunaa9512 4 года назад

lolllllllll

@centralprocessingunit2564 4 года назад

AHAHAHAHAHHAHAA

@momov4060 4 года назад

damn I didnt think id find something this accurate out here

@manifestationhappiness705 4 года назад

Lololol

@omniyambot9876 4 года назад

G: he is cheating His boyfriend:

@goblinsharky Год назад

I’ve watched this video every year since I was a freshman in high-school and now I just finished my freshman year of college and I finally understand every part of this proof and wow is it elegant

@Ashborne002 Год назад

Watching it again 4 years later and I can see the Alice-Bob problem solving applied here too. Subtle shift of perspective from numbers to concept is all it takes from solving one particular question to solving an array of similarly framed questions

@calebsherrock8749 3 года назад

This is hard enough to understand with diagrams and a helpful teacher, but imagine how smart you have to be to solve this with just pen and paper...

@bombfire8280 3 года назад

And limited time

@slatined4576 3 года назад

Some people even solve these things mentally 💀

@ytdude_g2042 3 года назад

@@slatined4576 i did bro. When he said it was 0,25 chance of the point cirkle to be in the area of the triangle i immidiantly knew it was 1/16 for the sphere

@ytdude_g2042 3 года назад

@@slatined4576 naw im kidding i dont know the answer lmao

@renedan7247 3 года назад

Its not about being smart, math revolves about knowing concepts, and recognising where you can use them. You might as well make an analogy to your native language, you know what words to use and when - thats math, but its a foreign language.

@mynameisjeff5397 2 года назад

You know shit is real when the kid with photographic memory starts re-reading his notes

@tShuka_ 2 года назад

*god help us*

@arandomcowjustchilling4237 2 года назад

@Lil Quack Satan has left the chat

@jasongrenbox6997 2 года назад

Wait.. then whos left?

@mynameisjeff5397 2 года назад

@@jasongrenbox6997 Zeus?

@keanux5906 2 года назад

@@mynameisjeff5397 Zeus has left the chat

@derekfordyce9 2 года назад

I took this exact test in college. I remember this question and it's great to finally see the solution.

@KejriwalBhakt Год назад

And what was your score? Has anyone scored 100 percentage?

@derekfordyce9 Год назад

@@KejriwalBhakt haha, 0

@jorgerivera5003 8 месяцев назад

Every now and then I come back here and always end up amazed. Love this video, I'm in med school, but I participated in math olympics through out all my highschool, love maths

@Jj-gq3sl 3 года назад

Him: Explains hardcore math What I learned: ah yes moving lines inside circle very nice

@salwaabdullah1682 3 года назад

LOL😭

@erikhalvorseth3950 3 года назад

Youre not alone Sir. Still I wish I could have had a math teacher like him

@JoseRojas-hl7sn 3 года назад

Same

@NeelTigers 3 года назад

It’s not hardcore math lol..there’s just too much of it and you gotta have great intuition of the different patterns in your mind..so basically you need to be very creative and imaginative to solve these..

@ahnrho 3 года назад

Damn. I remember when 3blue had fewer than 200k. Wasn't too long ago, either. How it's all grown.

@ankit_singh2704 4 года назад

Me: Watching cute cat videos RU-vid: So I have heard you like tetrahedrons

@andreasfDE 4 года назад

Your cat watched this video about tetrahedrons.

@momgo6533 4 года назад

Andreas F they’re only one step closer to world domination

@Steven-nq7fx 4 года назад

CreamHeroes? I know it's cringe, but I'm just curious

@justhingalit9780 4 года назад

Same HHAAHAH

@jonathanshapiro6593 4 года назад

I am c a c k l i n g

@totes_not_chad 4 месяца назад

This video was very insightful, always love your videos. Thank you!

@tundraxd4896 11 месяцев назад

This is really good. Too many people tend to think the math is just memorizing a bunch of formulas, when in reality, they are like logic puzzles, almost like sudokus. So for all the people who say that math isn't your thing, maybe memorizing isn't your thing. This is way beyond my league, and yet you explained it so that I don't need any prior knowledge and equations.

@TheSteinbitt 10 месяцев назад

But you need to know the rules before you can play.

@day99988 3 года назад

teacher: “no calculators for question 5!” the question:

@Ryan-wp6vr 3 года назад

He did all of this without a calculator...

@parallelogrum7967 3 года назад

@@Ryan-wp6vr Yeah he did, if you exclude the graphing software

@oscDaBeast 3 года назад

@@parallelogrum7967 that was for the video not for the answer

@joeyhardin5903 3 года назад

i dont think a calculator would help you with this question

@hendric6149 3 года назад

even with calculator i wouldn't even know how to use against this question

@Flow86767 4 года назад

“Take a moment to digest the question” Me : *What is a tetrahedron?*

@Silk0926 4 года назад

So true

@adeenala 4 года назад

And now Im dead thanks hahahaha

@ariwang8613 4 года назад

If you're actually wondering it's a triangular pyramid with 4 faces lol

@Flow86767 4 года назад

Ari Wang Thank you kind stranger!

@_____......_____ 4 года назад

U shouldn't be here .... 😀😀

@IkBenNoLife Год назад

Thank you for explaining this in a way i can understand my man. Keep doing your amazing work as you deffo got my sub

@wren5291 2 года назад

I have just realised that math is fragmented. It doesn't matter how much of it a person may get wrong, what's important is that one experiences each of the questions within their own categories, gaining wider experience of each category respectively, and then having all the tools combined to connect all the pieces of the puzzle. That's what I just realised anyway thanks to this video👍🏼

@DBJ99_ 2 года назад

Me only a few minutes into the video: “What’s a tetrahedron?”

@casperthegamer7176 2 года назад

Not related but go pack go

@DBJ99_ 2 года назад

@@casperthegamer7176 hell yeah GPG!

@frds_skce 2 года назад

Alright, real answer here *Tetrahedron is like pyramid but all the edges are parallel.* Think of it like cube and cuboid. All 12 edges of cube is the same length , meanwhile there's 2 different pair of different edges of cuboid with different length. In this case, *tetrahedron is a prism with 4 triangular faces, 6 straight edges, and 4 vertices (aka corners).* Hope this helps you

@OsinakiKun 2 года назад

I am so not understand any of this... But I still watched

@bobcostas6272 2 года назад

Build more pylons

@husseinmohammed1431 4 года назад

Teacher: there will only be 3 questions Me: thank god Test: 1,1A,1B,1C, 2, 2A, 2B, 2C, 3, 3A, 3B, 3C

@Anikinoro 4 года назад

@Angel Buceta this isn't Instagram

@Anthony-tp1md 4 года назад

More like 1, 1A, 1Ai, 1Aii, 1Aiii, 1Aiv, 1B, 1C, 2A, 2B, 2Bi, 2Bii, 2C, 2D, 3, 3A, 3Ai, 3Aii, 3Aiii, 3Aiv, 3B, 3Bi, 3C, 3D, 3E

@grada.r5049 4 года назад

@@Anthony-tp1md more like 1 a b c d e f etc. 2 a b c etc. 3 a b etc

@somewhat_human 4 года назад

Test B-Sides bottom text

@kd204 4 года назад

Basically physics tho... My teacher: this worksheet is due tomorrow. Don’t worry, it has only 3 problems Worksheet: 1a, 1b, 1c, 1d, 1e, 1f, 1g, 1h, 1i, 1j, 2.....

@bentuu816 4 месяца назад

Every single video without fail manages to blow my mind

@kaygur._. 7 месяцев назад

I'm still in highschool and understood the way you approached this question,my mind hurts but at the same time feels relieved and proud.

@tobe4336 4 года назад

Me after 3 mins: I understand Me after 6 mins: I understandn't

@Hahny 3 года назад

Me: after 2 seconds i understand Me: after 1 min. Im'nt understand

@SPlompie 3 года назад

Damn this is underrated

@nickbilly7329 3 года назад

It’s actually simple. You just have to break it down into simpler forms and use your understanding of formulas and equations to prove and solve it

@romilgoel4191 3 года назад

Did you understand ?? *Yesn't*

@Rookie1706 3 года назад

@Hamentsios10 2 года назад

After watching this I had a great discovery. Big enlightment. I can safely say, that i've come to the genius conclusion that I understood nothing.

@daveasence8948 2 года назад

Ahh i too was enlightened to the fact that my noggin containing cerebrum was able to decifer nullity to what the pictorial visual was about.

@Hammy_tch 2 года назад

😂

@Hammy_tch 2 года назад

@@daveasence8948 😂

@guesswho9541 2 года назад

this is 100% me 😂, i never felt so relate in my life

@Hansulf 2 года назад

What? That was super well explained, came on

@meingutername2158 2 года назад

In 1D it would be 2 points "randomly" selected out of 2 points on a line, probability of 1/2 that it is the different point and the connecting line goes through the center. For 4D and above I think it's quite helpful to move away from the visual intuition and describe it formally, but it is still the visual intuition that was the key to the solution.

@BurgoYT 29 дней назад

Yes good point and the 1D case would be very easy too, the probability that two random points on a line when connected contain the centre. Doesn’t matter which side of the centre the first point is on, the 2nd point just needs to be on the other, which is of course a 1/2 chance as you said!

@PunmasterSTP Год назад

Before coming across 3Blue1Brown, I never knew just how much simple intuition could be transmitted by one man to an incredibly large audience.

@TheRealGuywithoutaMustache 3 года назад

Imagine coming home tired from learning math at school, only to find this in your recommended.

@kingdinodragonite3470 3 года назад

Why can’t i see your sub count?

@EAOO 3 года назад

@@kingdinodragonite3470 cause ur bad

@mrgoose3453 3 года назад

Everywhere I go I see your face 😂

@gamercatt1235 3 года назад

Heyy

@user-jd4so3hr3t 3 года назад

This is the first time I see you without 100 likes

@h4brid992 4 года назад

Professor's: Put's effort making a hard question Me: **Leaves it blank**

@cozmos1553 4 года назад

He'll be sad

@user-bz1xk3pm2v 4 года назад

Oh I see what you did... your pfp

@mabo_lombo1533 4 года назад

Profesor: (ur profile Pic)

@mateusribeiro9384 4 года назад

My monography be like

@Crazylom 4 года назад

Put is You made a _greeeeat_ mistake

@sarimshaikh5224 2 года назад

A good teacher is very important in every student's life,you are absolute good teacher 👍

@niloufarashayeri704 2 года назад

I lovveee these sort of math questions,,, and I love them even more when someone just explains this well how they can be solved so that I don't have to think much :D ( because usually I cant solve them alone, but no doubt if by chance i do solve something this hard its the best feeling in the world)

@radian2.012 3 года назад

Me watching this: *"Hmm yes, tetrahedron."*

@justaman7124 3 года назад

Hahaha

@eddiemartinez8277 3 года назад

Bro that was the moment when i realized "yea i shouldnt be here"

@Royal-sp9pb 3 года назад

exactly my reaction lmfao, this comment made my day

@asianalliancebs6474 3 года назад

hmm yes, 10 times 12 is 120

@toneasyy 3 года назад

Bruh exactly lmao

@enricoiraldo9057 3 года назад

During a math test I get so anxious and scared of making arithmetic mistakes that when I have to do 6 divided by 2 I use the calculator

@anasmaher4783 3 года назад

I swear i do 1 plus 1 and things like that on calculator

@suk6556 3 года назад

me.

@olegdobinda8354 3 года назад

That’s just sad 😦

@Mike_Toreno 3 года назад

6 divided by 2? It's so easy lol, It's 12 *Checks in calculator* Oh...

@TheBrickagon 3 года назад

I once got 9,90 points out of 10 on a test just because I wrote 6/3=3 😂😂

@EvilNinja182 2 года назад

I did the problem with a square and it made it way simpler to do it with the circle! It’s really interesting because framing the question in a certain way makes it so that the shape of the object is almost arbitrary! I inscribed the circle in a square. I then divided it into 4 parts by the diagonals. Take an edge. That edge is 2 of the 3 points of your triangle. Now move the last point and note the area under which the middle is covered. That area should be only one of the 4 parts of the square that you cut using the diagonals. Using the same diagonals, you get 4 of these situations because there are 4 edges and each of them gives you 1/4. If you take a fresh square now and make 2 cuts that serve the same purpose as the diagonals in the other case, you’ll find that you have another set of 4 configurations that give you an average area of 1/4 because each set of 2 is mirrored. Since the circle is inscribed and we used areas we can use the same cuts for the circle and it still holds. If you do the same thing with a cube and make 4 cuts. You can do the same thing but you get 1/8. This shit is so cool

@1imag337 2 года назад

i may be years behind on anything like this but you explained it in a way that makes sense

@TudorLaurini 3 года назад

what a beautiful mindfuck

@paxera2913 3 года назад

@Malik-im8zg 3 года назад

I changed this so you will never now why those people are crying

@minglei779 3 года назад

@@Malik-im8zgchill brub

@thenumberfour6455 3 года назад

@@Malik-im8zg you do realize that "useless comments" make up 99% of youtube right?

@abstrakt8709 3 года назад

The Number Four too true

@remy629 3 года назад

As a wise turtle once said..... “I don’t know”

@aulalivre5481 3 года назад

Nem eu.

@DP-ot6zf 2 года назад

If a turtle said that to you, it's time to check your meds.

@QAYWSXEDCCXYDSAEWQ 2 года назад

I love these videos, they very interesting and really well done

@jimmybfmv9023 Год назад

Amazing explanation man totally loved it

@sahazmarkk5624 2 года назад

When I'm studying maths for exams, it's a nightmare but when I'm casually trying to comprehend it through videos like this, it's so fascinating.

@dangomangothe1941 2 года назад

I feel this so much, just having a maths exam a few days ago..

@sahazmarkk5624 2 года назад

@@dangomangothe1941 I hope that it went well xD

@dangomangothe1941 2 года назад

@@sahazmarkk5624 oh god as an atheist I hope as well

@sahazmarkk5624 2 года назад

@@dangomangothe1941 lmao, good luck for results

@Arcessitor 2 года назад

@@dangomangothe1941 cringe

@7Dok 4 года назад

Me: Watching Minecraft RU-vid: So i herd u leik polygons

@eggrolltheillegal6201 4 года назад

This is at the same level of some minecraft redstone.

@Ja-sm7sz 4 года назад

I was just watching minecraft then saw this

@joleeen 4 года назад

I've just watched a minecraft vid, wtf.

@holyfknsmokesigotmymoneyup 4 года назад

Timothy Moore same

@juliourigunera9748 4 года назад

Same dude.

@ArashArfaee 2 года назад

I love how it's emphasis on the key being in how we think not the solution itself.

@momentoyal930 2 года назад

very interesting

@stevecoffman2559 Год назад

Being brilliant at maths is one thing. Being brilliant at perception/framing is a completely different skill. Good video!

@slackfazed5788 3 года назад

It's all cool in the exam until the atheist kid starts praying

@ten1162 3 года назад

asian atheist kid*

@kittenmimi5326 3 года назад

@@ten1162 uhmm that might mean the apocalypse is coming

@nicknawd7757 3 года назад

That’s me lmao

@zzztriplezzz5264 3 года назад

@@user-dk3et3hw9t what’s wrong with believing or not believing in god? Can people just get over it?

@richardbernadino6449 3 года назад

@@user-dk3et3hw9t whether god is real or not no one knows! Doesn’t matter to much, just live a live after your own morals

@AnupSingh-kw3ww 3 года назад

Imagine solving questions like this with just pen and paper.

@z.e....3175 3 года назад

You write and scribble.

@pommes1287 3 года назад

@@z.e....3175 and flip a coin

@lukeskywalker691 3 года назад

This is Mathematical Olympiad

@pin2398 3 года назад

For eg.. JEE

@AnupSingh-kw3ww 3 года назад

@@pin2398 I'm preparing for the same ; )

@angrybhalo1671 4 месяца назад

It's the 7th time I'm watching this and still amazed how great this problem and the solution is

@wisechimp6159 5 лет назад

In class: 2+3=5 Homework: 2x3=6 Test: 4 points are randomly chosen on a sphere, what is the probability that the tetrahedron formed by connecting the points also contains the center of the sphere?

@user-vn1zi9mx3z 5 лет назад

Illuminati final test “If I have 8 oranges in my right hand and 7 apples in my left hand how many bananas I ate in lunch “

@justliketheoldone4767 5 лет назад

There are 69 likes should I like it?

@justliketheoldone4767 5 лет назад

@@user-vn1zi9mx3z Me: Can I go home?

@user-vn1zi9mx3z 5 лет назад

Just Like The Old One no until you get the correct answer

@xOxAdnanxOx 5 лет назад

EricYyt Zero bananas because you only have oranges and apples in your hands.. I am thinking to take this Putnam test

@doubleyell_ 4 года назад

Me at 10PM: ok let’s get some sleep tonight Me at 3AM:

@enriqueortiz1421 4 года назад

Double Yell me rn

@omaralmadani4580 4 года назад

Double Yell me rn

@DiggoryJiggory 4 года назад

3:43am rn

@eitanav111 4 года назад

@Ares_TZ 4 года назад

What should I say? I live in Germany and its 4:56 am

@tiemen9095 Год назад

I think it often really helps with 3D problems to simplify to 2D, and 1D and see if you can find a pattern. That gives you an idea about what the answer might be; trying to proof or disproof your idea often gives you new ideas about how to approach the problem. This is especially powerful when you try to solve problems in 4+-dimensional systems.

@ishanvianand31 5 месяцев назад

this video made me hooked to your account and made me love mathematics, thankyou

@Josealock 2 года назад

It is quite interesting how it can be simplified by just visual representation, doing it by formulas on the other hand is probably pretty confusing though.

@attempt58 2 года назад

Having eyes is a blessing.

@Josealock 2 года назад

@@attempt58 True

@beybrain7896 2 года назад

@@attempt58 And a brain to visualize with

@evilhutdug4665 2 года назад

@@beybrain7896 nothing really matters without the brain

@coconuutz1892 2 года назад

@@evilhutdug4665 well yes, as without a brain you have no conscious

@outsidergameing921 3 года назад

I swear every elegant solution of a math problem is basically a big bruh moment

@clipkos3009 3 года назад

so true doe

@membranekeyboard781 3 года назад

*when you get your test back*

@hiro_6015 3 года назад

yeah i always think for so long only to find out the solution could be found out by a third grader

@debroopghosh7845 3 года назад

Maths is really beautiful tough problems have simple solutions but complex understandings

@rayankasam4784 3 года назад

Under what circumstance would anyone be able to find this.

@PedroHenrique-ic9vr Год назад

I was never good in math, but is so cool when someone presents something that you would never think about it specially when you get a visual representation, great vid thank you youtube algorithm.

@entertainment1257 2 года назад

Wow, nicely visualised!!!

@jackhall1150 2 года назад

All the jokes in the comments aside, you've just explained something to me, a person who is terrible at math, in a way that actually makes sense. I think that says something about your teaching ability

@kukokeko 2 года назад

truly. where was this guy in 6th grade?

@seal2560 2 года назад

@@kukokeko Where was this guy during my entire schooling career (so far)

@valediction9174 2 года назад

facts bro why can’t teachers be like this

@Forward.4 2 года назад

Exactly. Teachers should be taught how to teach first

@smirfbroil6934 2 года назад

@@valediction9174 ;( we getting there, but teachers still payed like shit and not actually taught how to teach, just to show

@SkullKingize 4 года назад

Class: Solid, liquid, and gas Homework: Plasma Test: Fabricate anti-matter then harness the energy and relay it to energy required to the construction of several dyson spheres spanning the M31 group[5 marks] Me: The mitochondria is the powerhouse of the cell

@saaramb3452 4 года назад

This comment is underrated

@Aakash_Goswami1 4 года назад

Answer (D): None of the above

@ganeshprasad9851 4 года назад

@@saaramb3452 you can prepare anti matter using particle accelerator.. but you need at least a gram of that antimatter to harness the energy for building the Dyson sphere around a star...let us a assuming that you want to build one of those several Dyson spheres around our very own sun... Then you need start recruiting at least half of our current world population as the labor force required for this project. Assuming that you are a very good public relations company and have recruited half of the world population for this horundus task, still you need atleast a gram of anti matter to use it as start off to provide you energy for this start up job, but our current particle accelerators at CERN can produce about 100,000 anti protons per day and each of these last an average of 10 seconds.. So a gram means ahh... Let's assume it to be hydrogen's anti protons for calculation purposes 1 mole of anti-Hydrogen is 2 grams that (because hydrogen exists as H2) which means 1 moles of anti-Hydrogen contains (6.023*10^23) anti hydrogens so 1 gram is (3.0115 *10^23) which requires you to prepare a particle accelerator that is as efficient as that it need to create 3.0115 *10^23 anti-protons( I m using the word proton instead of hydrogen because it doesn't matter hydrogen can be called a proton as it has has only one proton and we count the proton and neutron for mass number) so the particle accelerator should work at an efficient 3.0115 *10^17 times the one currently being available. So in plain word's if we have the materials we would have done that long ago, unfortunately we don't have the materials we require. A gram of anti matter costs around 62 trillion dollars which means you need to sell your country for that!

@saaramb3452 4 года назад

@@ganeshprasad9851 ok thanks, i'll consider invading another planet

@grieferjones2237 4 года назад

haha get it it’s funny because school is bad please give me likes

@KishorKumar-tp3zu Год назад

Problems like this is so amazing they teach you so much , about your approach to problem and hoe you see a problem, Many People will find math difficult due to complexity of the given circumstances in a problem but it can be broken into smaller less complex parts ,,,,,every fking problem Thats The Beauty of math

@arthurso333 2 года назад

one of my favourite videos on the internet

@rushilkumar5258 6 лет назад

I never thought I would be able to understand how to solve a putnam problem... i am completely overwhelmed with joy... your style of explaining the problem just blew my mind.... i have been a subscriber since last year or so, thank you for making me realise the elegance in math

@Fr1nc3sc41 6 лет назад

Wow! Your 'simple' presentation makes one wonder why so many billions of us can't even begin to grapple with the problem. It takes special talent for making the nearly impossible accessible. As a teacher, you must be unforgettable and irreplaceable. As a student, your teachers must have felt humbled to have met you.

@coolnoah8183 6 лет назад

Fr1nc3sc41 I had one and only one teacher like this in math who was almost this good at explaining difficult problems, and even he doesnt hold much of a candle to this guy

@ABaumstumpf 6 лет назад

The thing is: this doesn't actually help you solve those problems. They are designed in such a way that you have to have the right understanding and intuition to solve them. Anybody can solve them if they know the correct method - but nearly nobody does and there are so many problems and variations that can be given that no person can ever hope to come even close to remembering them all. hat is needed is the ability to come up with the solution on your own - and that is hard.

@z_vizo4594 5 лет назад

Not surprising you're an Indian (its a compliment, no negativity intended)

@stevanhaire5300 5 лет назад

@@z_vizo4594 Smart Bois

@dizzy212 4 года назад

Dawg why did the RU-vid algorithm send me here I’m dumb asf

@ez-boys4268 4 года назад

Literally was watching highlights of football and the next vid was this

@kayymini 4 года назад

Felt this😭

@stfuernie 4 года назад

fax im only tryna listen to Carti snippets

@lij9724 4 года назад

@@ez-boys4268 i was watching highlights of football to

@localdreamer13 4 года назад

They know that, so they're recommending you videos to make you smart, or at least pretend that you are to a group of nerds.

@mortaldev4999 2 года назад

this is probably the best problem explaination thankyou

@nourbelcaid8850 2 года назад

Great explanation and easy to understand 👍 🙌💯

@cheesiesticks 4 года назад

He’s literally solving the meaning of life and I’m trying to make sure that 3+4 is 7 just in case

@ea8455 4 года назад

Using two calculators because you don't trust the first one

@ahmedkhaled581 4 года назад

@@ea8455 😂

@jonathan0225 4 года назад

@@ea8455 was looking for this comment, and was not disappointed. 😆. A man of culture indeed.

@karlspasst2235 4 года назад

Enes Akbulut you cracked me up HAHAHHAHAHAHAHAH

@raydarable 4 года назад

Same.

@non123y5 5 лет назад

I probably watched this 100 times Now i have started to understand what an tetrahedron is

@nusretakbulut7159 5 лет назад

😂😂😂😂

@iqbaltrojan 4 года назад

a 3d triangle

@non123y5 4 года назад

Ahhhh!This was suppose to be a joke

@malikhafez8056 4 года назад

HAHAHAHAHA

@the_rover1 4 года назад

I still wonder. what is it actually? :|

@polishethan 2 года назад

I don’t even understand basic algebra but I still watch videos like this. Haven’t got a scoob as to what you’re doing.

@jimi02468 Год назад

Awesome. Always try to find a clever way to re-frame the question.

@Darkmage1293 2 года назад

"Some of you with some multi-variable calculus under your belt..." "...uh....yeah, I got tons of that. Need a bigger belt even."

@zentempo7902 2 года назад

Yeah same i need a huge belt for all of the multivariable calculus

@lamarjackson9736 2 года назад

😭😭😭😭

@jamespeck10 2 года назад

@annablue1237 4 года назад

me : * barely scraping maths* also me: i bet i could do this

@adriannanad4675 4 года назад

haven't done maths in 10 years also me: i bet i can do this with my pure primary school mathematics

@tone-deafbard3247 4 года назад

Tamanna Kareem lol same

@a.bagasm.7253 4 года назад

OnO

@holctomaz2562 4 года назад

@@adriannanad4675 A thing i like abaut this probelm is that you can "oraly" solve it with just almost no understanding in mathematics.

@prayogaliong5395 Год назад

Thank you, this video gave me an extended essay topic

@shravansinghrathore8600 2 года назад

I watched this video a while ago and now when I was preparing for jee advanced I saw the 2d version of this problem and suddenly remembered your video

@iceborneuprize3998 3 года назад

I must be doing something right to have gotten this on my reccomended.

@OKayD3N 3 года назад

Nope you have anime pfp

@james68279 3 года назад

For sure it ain't spelling 😏

@OKayD3N 3 года назад

@@jesse406 ok.. you’re subscribed to “dloow”

@christiancy2514 3 года назад

@@OKayD3N wtf do u have against people who put anime pics as their pfp? they just like anime

@OKayD3N 3 года назад

@@jesse406 What’s the issue with me being subbed to Carson?

@joechen8082 4 года назад

Math questions be like: Jimmy has 21 oranges He is 2 hours late for school Solve for the mass of the sun

@kkaaii177 4 года назад

Not really

@cmdamien 4 года назад

@@kkaaii177 it be like that

@dingyworm 4 года назад

okay but we can actually calculate the mass of the sun despite the unrelated information given

@josephikrakowski6526 4 года назад

@@dingyworm I mean, with the information we've been given? No, but if they've given us the uniform density and radius of the sun, it should be a piece of cake, even that's not necessary, if we're given the gravitational force between earth and sun, with the distance between then and earth's mass, we can easily find the sun's mass

@jouljoker9684 4 года назад

@@dingyworm yes of course

@Xanderj89 Год назад

That way of thinking will help you in life too. When you get an insight there’s a temptation to rest on the solution, but using your new understanding of the overall pattern if you then try to use that insight to recontextualize the problem thru its lens, you can simplify your approach instead of ending up with a complicated solution. Often it can shift your perspective from seeing many problems with many different solutions that are hard to manage all at once, to seeing that they were all a connected problem with a simpler overall solution, because you stop needing to bandaid symptoms when you can work on the root cause

@keyboardwarrior3242 2 года назад

Why is it so interesting and soothing listening to people explain complex math questions when you don’t have any idea what’s going on. “How do you like them apples”

@ADeep007 4 года назад

50% chance. It either does or it doesn’t. Problem solved.

@nikj1178 4 года назад

That's not how this works, that's not how any of this works!

@thevigamerpixerlator 4 года назад

Possibility. Not probability. Possibility of you finding a 100 dollar build the next time you go for a walk? 50/50. Probability of you finding one? Probably near 0. (including the probability that you ever go for a walk in the first place lmao)

@krakatoadude4231 4 года назад

Let's imagine a world where everybody gets the irony

@ADeep007 4 года назад

Krakatoa Dude congrats on taking a post too serious. You win.

@lightskinche 4 года назад

@@krakatoadude4231 And then we have that one guy...

@victorvila1056 3 года назад

Bruh why am I so addicted to videos that I can't understand???

@disadadi8958 3 года назад

And what part didn't you understand on this video? I mean the math wasn't that complicated at all.

@eduardoizquierdo309 3 года назад

Because you have an anime profile pic

@daraobongisonguyo 3 года назад

@@disadadi8958 stfu

@mikeztecha3808 3 года назад

Because people have always had an interest in understanding things they can't, that's why we're where we are today

@tb_mrpanda1908 3 года назад

@@disadadi8958 😳 you good bro

@jman9049 2 года назад

Just took the Putnam last Saturday, I probably wouldn't have even know it existed without this video

@Tapecutter59 Год назад

Love your work. When you explained the 2d case, I guessed the 3d case would be 1/2^3. I assume, (rather than prove), the 1/2^d pattern continues into higher dimentions. eg: 4d case would be 1/2^4 = 1/16 Interestingly if you take the limit of the 1/2^d series (sum the probabilities 1/4,1/8,1/16,...), you get p=0.5 So if you genralise and make d a random integer >1. the odds of the center point of an n-dimentional circle being enclosed by a n-dimentions triangle is 50/50. I'm sure others have noticed that and stated it more formerly, but it gave me a buzz to find it myself :)

@MaxMiesen 5 лет назад

This guy is amazing at explanations. If he can teach the hardest problem this well, imagine what he would be like as a high school teacher.

@monsieurbobblehead3854 5 лет назад

Yeah

@connorschultz9022 5 лет назад

Hed probbaly be a better professor, most hs students disrespect or ignore the teacher then ask for them to re explain it or to have special tutoring over subjects they didnt listen to

@frostflakejr6324 5 лет назад

@@alycias7345 yes it does, atleast for me.

@frostflakejr6324 5 лет назад

@@connorschultz9022 I know it's really sad to see, and to top it all off the teachers most of the time do.

@GoonCommander 5 лет назад

Why teach one class, When he can teach the world with his RU-vid channel.

@georgefarah9214 6 лет назад

Fastest 3 things in the universe: 1- light 2- universe expansion 3-me clicking on a 3blue1brown notification

@SupLuiKir 6 лет назад

the universe expansion between two points is faster than the speed of light provided the two points are far enough apart. The chance two random points in spacetime are not far enough apart to outpace the speed of causality is infinitesimal, assuming an infinite universe, therefore your list should be 1-universe expansion 2-light 3-the joke flying over my head

@Kelly_Jane 6 лет назад

· 0xFFF1 I was going to make a similar, but less considered point.

@AtticusFinch65 6 лет назад

· 0xFFF1 why does it depend on how far apart two points are? And what is the semantics of your hexadecimal username?

@alexnope2180 6 лет назад

Kyle Horne pick two points within your eyesight, the expansion of the space between those points is not the speed of light or anywhere close to it However the further apart the points are the faster they move apart up to (and at a great enough distance) exceeding the speed of light This here is a little strange to imagine but arises because the universe is expanding from every point including points between points which brings us this acceleration