The most unexpected answer to a counting puzzle

Solution: • Why do colliding block...
Even prettier solution: • How colliding blocks a...
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/clacks-...
New to this channel? It's all about teaching math visually. Take a look and see if there's anything you'd like to learn.
NY Times blog post about this problem:
wordplay.blogs...
The original paper by Gregory Galperin:
www.maths.tcd....
Evidently, Numberphile also described this problem (I had not known):
• Pi and Bouncing Balls ...
You'll notice that video has an added factor of 16 throughout, which is not here. That's because they're only counting the collisions between blocks (well, balls in their case), and they're only counting to the point where the big block starts moving the other way.
Thanks to these viewers for their contributions to translations
Bengali: Prayas Sanyal
Hebrew: Omer Tuchfeld
Italian: @Deye27, @hi-anji
------------------
These animations are largely made using manim, a scrappy open source python library: github.com/3b1...
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...
Stream the music on Spotify:
open.spotify.c...
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 KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brow...
Twitter: / 3blue1brown
Reddit: / 3blue1brown
Instagram: / 3blue1brown_animations
Patreon: / 3blue1brown
Facebook: / 3blue1brown

[@Jouzou87]

Physicists: "Noo! You can't have ideal collisions make a sound!" 3B1B: "Haha, blocks go brr"

[@midlanismail416]

Don't you mean clack

[@pixelartkid7965]

@@midlanismail416 in the 100000kg one it went brrrrrr

[@aa01blue38]

The sound also goes hypersonic because the frequency of clacks is so high

[@thelukedankwalker]

@@aa01blue38 lmao what, that’s not how that works dude

[@jamesorendorff2284]

@@aa01blue38 "Hypersonic" means "faster than sound"... You literally just said "the sound goes faster than sound".

[@poopcatapult2623]

Teacher: "gimme some digits of pi" Me: "clack clack clackclackclackcla... clackclack clack clack... Wait for it" Teacher: "what on earth is that supp...?" Me: "... clack"

[@RubyPiec]

3.1415926535897932 There take that

[@mediding7001]

No, it's: clack clackclack claclaclclcl reeEEEEEeee clacla clack clack... clack

[@mikeymcmikeface5599]

I demand immediate satisfaction!

[@chikoroblox9734]

I baked you a pie

[@skyspects1211]

Damn, this made my day! 😂😂😂

[@ikkocan]

Originally discovered in 1995, published in 2003. maybe he DID count the clacks?

[@mikeymcmikeface5599]

lol

[@fifofuko1864]

Alem adamsın

[@ViratKohli-jj3wj]

Lmfao😂😂😂

[@diogoandre756]

lol

[@Choquetel]

Probably a computer can do this simulation and count the clacks

[@andrewdavis1138]

2:37 I was watching in the middle of the night and got absolutely flashbanged by the sudden swap from dark coloured example to bright white paper.

[@redelf8052]

same

[@Swingylad]

the spherical cow

[@Chair_On_Water]

Wth same 😭🙏

[@nicholasstanton9575]

r.i.p

[@Mr.supergamer20]

I just have my brightness up all the way so...

[@antiscribe4150]

Pi has no business showing up literally everywhere in math.

[@onebeets]

Wait, it's all pi?

[@gruncho8227]

@@onebeets always has been...

[@bignicebear2428]

What goes around comes around and voila: pi.

[@Saturnares]

Nobody expects the Pi inquisition!

[@dathaniel9403]

@antiscribe it’s like that one guy who always seems to be at every party but no one knows who invite him. He just shows up no matter where you are.

[@The-Rest-of-Us]

Highest quality KZbinr out there. And I mean that in every dimension.

[@ariqahmer]

Make that in concurrent parallel dimensions 😉

[@jacobkleeman8546]

Even the fourth?

[@ganaraminukshuk0]

@@jacobkleeman8546 All spatial and abstract dimensions up to infinity, even the temporal dimensions (if 2D time and up makes any sense). Even the fractal dimensions where you can have non-integer values (like 1.3425 dimensions). Even any system, existing or to-be-discovered or purely nonsensical for the sake of argument, that calls for negative value dimensions, or imaginary number values, or complex values, or quaternions, or octonions, or sedonions, and beyond.

[@papi1050]

@@ariqahmer we need to talk about parallel universes

[@ariqahmer]

@@papi1050 Agreed 🤔😎😍

[@evank3718]

1:40 Me opening the door at 1:43 am

[@orvillevroemen3956]

3:14 am

[@jcgongavoe337]

PRODUCES SLAP BASS MELODIES，SO TRUE

[@appiusssss]

hahahahah

[@rolanddogna921]

@@orvillevroemen3956 3:14

[@colinjava8447]

Sounds like a sound effect on a zx spectrum game

[@coolguydata]

This is why I love math. You always look at a problem, read it out loud, then discover something about that problem. It's like there is always a hidden puzzle in math equations. For example, in 7th grade, we were learning about circumference. My teacher showed the class a video which said that if you take the diameter and try to wrap it around a circle, there's a tiny bit left, to which I realized that that tiny bit looked EXACTLY like pi, or 3.14. It's so cool finding small details that make so much since!

[@user-rc1mv2zy3r]

🤓🤓🤓

[@coolguydata]

@@user-rc1mv2zy3r thanks :)

[@berserk9968]

​@@user-rc1mv2zy3ryou didn't come to a mathematical video to just spot "nerds" now didnt you

[@user-rc1mv2zy3r]

🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓🤓

[@Selicre]

That animation of the spherical cow actually made me wheeze. That was unexpected

[@chobyriley417]

Selicre [Hyper] it’s my favorite picture on wikipedia

[@NickiRusin]

that's a great image

[@Selicre]

@@NickiRusin I really, really, really like this image.

[@NickiRusin]

@@Selicre a long time ago my dad told me a great physics joke. I don't have the patience to translate it from Russian, but the punchline boils down to "a spherical horse in a vacuum". For some reason I never tried to visualize that, but now it's crisp in my mind thanks to this video.

[@AlRoderick]

ASSUME THE SPHERICAL COW!

[@ZzSlumberzZ]

*gets this on recommendations for the 10th time* Brain: click on it. Me: but I've already wa- Brain: *do it.*

[@SpaceSeal64]

this is probably like my 6th time

[@andy-kg5fb]

Its my 5th time

[@boldizsarfiser9894]

its my 4th time

[@Game_Sometimes]

This is my 12th time…

[@seanjericdioquino197]

Ah yes.. 𝗥𝗲𝗹𝗮𝘁𝗮𝗯𝗹𝗲.

[@rishujeetrai5780]

Pi is a creep. I'm gonna file a restraining order on him. He has started to show up on my integration problems now. He's gone too far.

[@oblivion2755]

pi is a cursed number, way more cursed than 13 or 666.

[@iqbaltrojan]

@@oblivion2755 whats wrong with 13 lol it's my lucky number

[@thewall4069]

@@iqbaltrojan oh the irony

[@notyepdranel961]

@@oblivion2755 *4* is the worst

[@akasakasvault7597]

@@oblivion2755 indeed four, or, in Japanese, shi, which us also the Japanese word for death, is terribly cursed

[@erinc129]

the clacking sound is so satisfying i want it on repeat forever in my brain

[@EnglishNijisanji]

I don’t speak English. So I don’t get it well. But when I got that the collisions number turned near π, I was like “！？.” It was so beautiful phenomenon.

[@anselmschueler]

cool

[@dxrpz1669]

Hi (sorry for my bad english)

[@IQuickscopeCA]

!!コメントだけで登録者10万人を目指す すうがくはすばらしいですね

[@EnglishNijisanji]

Blakee Yes, I believe math is beautiful and also amazing

[@ARlELATOM]

!!コメントだけで登録者10万人を目指す You write English wonderfully! ❤️

[@andrewmoonbeam321]

2:32 ' Credit to the viewer Henry Cavill.' Of course Superman would know the answer. He's brilliant at math. And physical education.

[@andrewzhang8512]

*kavle

[@mono6359]

@@andrewzhang8512 guess who didn't get the joke

[@andrewzhang8512]

@@mono6359 ohhhhhhhhhhhhhhhhhhhhhhhhhh mb

[@hansondesa188]

Lol

[@jedlehwiodiewalteh]

Silver for physics, steel for maths

[@bmschech]

I thought your video on relating the Basel Problem to the circle was simply gorgeous, astonishing and unforgettable. These three surpass even that! Thank you so very much!

[@Indomat64]

Small brain: Memorizing digits of pi Galaxy brain: Having blocks of precise mass on hand and counting the collisions

[@froggymine5003]

Also me: destroying earth to find Pi to the 20th digit

[@torezcoasters6043]

Don’t forget about removing friction!

[@bobbyp4025]

Sliding off to infinity, never to be touched again- so sad

[@Kido336]

*adding sad comment about you and your ex

[@ah2522]

my dad's hand and my crtoch

[@noddye1764]

r/cursed comments

[@tarzbaow]

@AssmasterFlex69 until the big crunch, and everything will be once again, at once place, at one time

[@maxtreme2901]

Bobby P the blocks will forever remember how their first and last touches were

[@epochthekid]

3:56 Why is this cow your idea of "way over-idealized" and why do I agree?

[@different_stuff]

i don't know about a spherical cow, but there is popular joke about spherical horse: Some man hired physicists to calculate which horse will finish first in the upcoming races. They gave him their results. And that horse didn't win. Angry man asked physicists why is that so, and they replied, that they calculated race results for spherical absolutely black horses in vacuum. So this is a joke about over-idealized conditions that physicists use in their calculations.

[@fordsquared537]

DifferentStuff Yeah, that’s pretty much it. Both my engineering teachers in the past enjoyed the phrase “spherical cow in a vacuum,” which just says how engineers and physicists would assume the cow is a sphere so that calculations are much easier

[@georgiykireev9678]

@@fordsquared537 In my language it's a horse.

[@infintiyward]

in topology, you count holes. A cow (assuming it had no holes) would be the same as a sphere from a topological perspective since they both have no holes.

[@josephnimal953]

It's from a Neil deGrasse Tyson's joke. It's about how physicists love to see the things in universe to be a perfect sphere. If u ask a farmer what is a perfect cow, he will answer a perfect cow is the one which will give lots of milk, a butcher will answer a perfect cow is the one which is fat. But a physicist will answer that a perfect cow is a cow which is spherical.

[@Lefty7788tinkatolli]

"Sailing off into infinity never to be touched again" I felt that.

[@mrsaussissonsec1054]

Saw this at least 5 times. Still amazed at the quality of the explanations and the correlation itself. You are truly one of the best out there.

[@marcgonzalez5628]

Watch it again

[@mrsaussissonsec1054]

@@marcgonzalez5628 aight

[@Adarsh_amd]

again

[@destroyer100onblitz]

Again

[@mrsaussissonsec1054]

@@destroyer100onblitz ayo it's only been 4 days

[@DynestiGTI]

2:31 when the actor for Superman helps out 3Blue1Brown

[@ToxicTerrance]

LUL

[@kasperjoonatan6014]

But Gregory Galperin is the real Superman here ❣

[@EayuProuxm]

When you need to move blocks weighing 10^(20-1) kg, you'd better call Superman to help out!

[@duckymomo7935]

Is it really the same person ie superman?

[@thanostitan.infinity]

True LMAO

[@stevenvanhulle7242]

Originally discovered in 1995, published in 2003. Galperin (for 8 years): "Unbelievable. Unbelievable. Unbe... Well, time to publish then, I guess."

[@TechSupportDave]

that's what i was wondering. why wait almost 10 years to publish it?

[@hiransarkar1236]

The time he took to count the no. of collisions

[@Proccito]

@@hiransarkar1236 Gralperin: "956...957...958..." His wife: "Honey. Dinner is ready!" Gralperin: "Sure, I am comming soon...fuck...1...2...3..."

[@matejpesl1]

Xd

[@derwastl]

no

[@ItsNekotaku]

i love coming back to this video every once in a while because it's just so mind boggling that it reblows my mind every time

[@kyriakos02]

Clack.

[@AmadeuShinChan]

[ I donot know. Let us ask someone. ]

[@finalftl732]

At some point the clacks would be so rapid that the frequency created would be too high pitched for human ears lmao

[@Torthrodhel]

@@finalftl732 so ideally you'd find the ideal magnitude that over a 10 hour total period would at some point get closer to the highest averagely humanly perceivable pitch than the next magnitude, which breaks that barrier. And there's your 10 hour video.

[@alexniggins1799]

MrBeast can, ofc.

[@danielreed540]

Just loop the system - add an opposing wall the other side of the masses/objects; masses between 2 walls. Then that system cycles, repeating (to infinity). Under the special ideal conditions. A truly closed system, with only total motion & energy transfers, where all is constant & only velocity can change.

[@myukunigunde8372]

Any maths/science/engineering problem: *exists* Pi: aight imma head *in*

[@siddharthabbanerjee]

Another interesting observation : When the masses colliding are powers of some other base (say 3), the number of collisions still equal the digits of Pi, but in the same base. Eg : Pi in base 3 is 10.010211012222010211002111110221222220111201212121... If you run the simulation with masses of 1, 3^(2 * 1), 3^(2 * 2), 3^(2 * 3),..., then the number of collisions will be 1 (base 3), 10 (base 3), 100 (base 3), and 1001 (base 3) respectively. Number of collisions for 1,3^(2 * 50) will be first 50 digits of Pi in base 3 : 10010211012222010211002111110221222220111201212121 , or 2255343044159619899886237 in decimals.

[@JeffMTX]

Now that is very very cool!

[@vijayrajendran4006]

this is what I thought... awesome!!!

[@Mimaloodak]

yes

[@partyboy4121]

Good thinking

[@mark-karm]

🤓

[@SCP--sf3fu]

3blue: Quick! I need some visual way to show the audience how over-idealized this simulation is! 1brown: Cow sphere 3blue: w h a t ? 1brown: _c o w s p h e r e_

[@shadesmarerik4112]

cowsphere is one word actually

[@alx-lm3kg]

@@shadesmarerik4112 hiw do you know that???!?!?

[@jasongusta7419]

did he stutter 3blue?

[@maxnolife_]

@@achtsekundenfurz7876 Oh so it’s just the humor for simplified models such as a spherical cow because the shape of a cow is too complex lmao

[@IanBLacy]

“Assume a spherical cow in a vacuum”

[@iamsopro4115]

Doctor: it’s not gonna hurt! The kid in the next room: 2:22

[@hellothere-pf9qu]

Lmao

[@FacultyofKhan]

I always COUNT my blessings whenever 3b1b uploads an UNEXPECTED video. As usual, great work!

[@ParthKohli]

We feel the same way about you. :)

[@SheikhEddy]

I love the videos you put up! Great content that is nearly impossible to find elsewhere :D

[@PhysicsMath]

I am also making video on physics

[@FacultyofKhan]

@Just A Random Dood Shhhh let me make my lame puns on the title

[@vikranttyagiRN]

I Subbed to Ur channel and really glad to have found you

[@underscoredfrisk]

Me: Hey that looks like Pi lol what a coincidence Me: Ah

[@IAmNotASandwich453]

Exactly what I was thinking 😂

[@djsalad5752]

“Like a satisfying game of breakout.” Is my favorite analogy on this channel so far.

[@o_o............]

The ideal cow takes the cake for me

[@howdareyou-cs4qn]

This is amazing, physics teachers never taught this, and everything we learned was designed to solve problems, from conservation of momentum to conservation of mechanical energy, and we even came to a conclusion E（lose）=1/2 (M1•M2)/(M1+M2) • V（relative） how fascinating physics is now

[@BobbyDukeArts]

What the what!!!!! That's so cool

[@godchild7545]

Wow I didn’t know you were interested in this kind of stuff!

[@damianh.1429]

Yo big fan dude keep it up

[@colewilkie]

Fancy meeting you here, woodsman.

[@lasmalux]

What the what

[@xilnoi]

wood

[@tyriekovco711]

Okay how many collisions if it was 10^1,000,000 times the weight of a 1kg object? Me: C L A C K

[@batzal9459]

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000^10 years later " ... Clack.. I freaking finished to count that, oh boi!!"

[@full5339]

It will become part of it.

[@full5339]

Imagine gravity.

[@full5339]

Or it would get so hot. May be it could melt.

[@oliverholm3973]

@@full5339 I prefer to read this comment without the context of your other two. Hello. _Imagine gravity._

[@jotarokujo5849]

Wrong The 100²⁰ would have destroyed our slippery floor

[@carltonblend]

And our tiny cube, either that or the bigger cube itself collapses into a black hole lol

[@afoxwithahat7846]

@@carltonblend And eats the Tinny cube

[@thfFromRussia]

​@@carltonblend and even if it has no enough mass for a black hole its gravity influences purity of the experiment )

[@Voidmoth1]

what about a 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 kg mass

[@asfi637]

@@carltonblend What if the cube is made out of bedrock?

[@witherphoenixlp6476]

Im not here for the math stuff. Im here for the colliding noise..

[@jessywinters9666]

And im here for cow sphere

[@hisham1111]

I really, REALLY, appreciate you leaving in that last bump at 3:53

[@EntergeticalakaBot]

Its just so amazing when the 1 kg block is moving slightly faster than the other block and you are just like ooooooooooooooooooooooooooooooooooOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOHHHHHHHHHH AAAAAAY

[@EntergeticalakaBot]

@DON'T I WONT 😶

[@iammrbeat]

Woah. Freaking woah.

[@aidenmcubing]

@Laquelectro woah

[@sunitakrishna3864]

s

[@ignacio6851]

mr beast give me moners

[@fakeuber8254]

@@ignacio6851 this is not Mr Beast, but Mr Beat. Instead of giving you money he gives you a beat down

[@osirisapex7483]

Mr Beat on a 3b1b vid? Worlds are colliding

[@Shubham-qk8fw]

Content - 💯 Editing - 💯 Voiceover - 💯 That's the definition of 3 blue 1 brown. Keep up the good work. U will definitely hit 10M subscribers soon

[@guilhermegondin151]

I would give another 💯 for the colision sound kkkkk

[@enverko]

I wouldn't say soon, because for some reason most people hate math for no apparent reason. If they were to see the true beauty of math I believe there would be a lot more people loving math!

[@Shubham-qk8fw]

@@guilhermegondin151 true, how could I forget that?

[@Shubham-qk8fw]

@@enverko Yaa you are absolutely right

[@totoxahc]

Date format - 0

[@PersonWithAFunnyBone]

3Blue1Brown never fails to make me question reality!

[@toothandsticks]

Dr. Galperin was my geometry professor at University. I have never enjoyed geometry so much in my life. The man knows and can prove an incredible number of astounding, non-obvious facts. Thanks for sharing his work!

[@jademonass2954]

1:40 what a cool sound effect

[@hishykot]

Kinda sounds like the beginning of that one Crystal Castles song🤔

[@uvuvwevwevweonyetenyevweug6314]

Sounds like a radio

[@Geotrax2]

the beginning sounds like a geiger counter

[@megaman4354]

sounds like a creaking door

[@_._---.]

​@@hishykotCrimewave? Yeah I thought the same thing.

[@MatematicaRio]

Pure poetry! ❤️

[@ChaosAtlantis33]

How has noone noticed you lol?

[@jonathasdavid9902]

Olha, brasileiro na área

[@jgperes]

O RAFAEL

[@redswordcalice235]

Krlh a lenda aqui.

[@nikhilnagaria2672]

yo

[@lehamsterhollandais9908]

i truly understand why people loves so much mathematics, all makes sense and everything is explained , thats just magical 😊

[@humanguy1643]

Everybody gangsta till the blocks start pullin’ out autotune

[@intravine]

supertone lol

[@feynstein1004]

Ah so 3blue1brown is blueballing me. Figures 😂

[@dreamer097]

you mean 3blue1brownballing you..?

[@GewelReal]

3blueballing

[@lankymcgainsplease1236]

Would you rather they brown balled you?

[@PrincessEev]

tfw you came into the comments to make this exact joke xD

[@aidanroberjot7469]

Well at least you can see where the circle comes from coz the equation of a circle is (x-a) ^2 +(y-b) ^2 = r^2 so 1/2m1v^2 + 1/2m2v2^2 = const. is clearly a circle equation

[@GoldenGamerFox7272fromYT]

me: **moving frame by frame at **2:29** and seeing 59 instantly become 313,979** **doing the same for **3:12** and seeing 941 become 314,159,265,136** this looks so fast... gotta know how fast it was...

[@AspectOfTheBlade]

So true. I wanna know too. I could not even pause it between 100 and 3 hundred trillion

[@marzipug5439]

@@AspectOfTheBlade He literally said the rate in the video.

[@xxromerocksxx2889]

literally did the same thing bruh

[@GoldenGamerFox7272fromYT]

@Fernando García salazar i already knew that

[@BritishEngineer]

Unfortunately it will be faster than the frame rate of the video. You would need a 314 million fps youtube viewer

[@Ashishbro]

0:38 the best decision ever took place on the planet

[@PhoenixLive_YT]

3:56 me after watching this video....

[@pi2590]

YEAH PHOENIX

[@acerrolaXD]

hehe 69 likes

[@blackpenredpen]

WOW!!

[@herlock2]

Well hello there :)

[@giladu.6551]

Hi BPRP! Love your videos

[@wontpower]

Isn’t it?

[@yash5879]

Congrats for the 200k man 1/1-x

[@blackpenredpen]

me thank you!!!!

[@dustinsanders5780]

3Blue1Brown: "We have 2 sliding blocks and a wall" Me: "I'm sorry could you repeat that, I'm already lost."

[@X_w45ey89]

Never have I ever thought that the small explanation bit with words sliding away at 3:59 from a math video would get me giggling like a kid at six in the morning

[@basimansari6759]

As a 6th grader, I don't understand the terms you use in your video but I so much hope to learn them throughout my coming years. I find your videos very intriguing, keep up the amazing work!

[@pritamdavis]

Keep being curious bro😎🔥

[@e2532e]

@@pritamdavis until being killed by the education system of the country

[@pritamdavis]

@@e2532e agreed bro.. it really sucks at times

[@rajan8516]

@@e2532e well 😔

[@silverseacow]

@@pritamdavis im 7th grade rn

[@XxJoe1101xX]

That "clack" sound you added is apparently ASMR to my ears, so it's very appreciated.

[@sharbel9624]

Reading comments section: . . Expectations: people discuss math reality: clack clack clack

[@timwhite1783]

0:19 Love the spherical cow animation btw.

[@Mason-25]

Me too

[@PriyabrataHalder]

2:30 when superman passes his time doing maths.

[@paglobal]

Was about to comment something like this😂

[@user-ov2fc5sd1e]

I don't get it. Yes I'm not the sharpest tool in the shed

[@examination4088]

@@user-ov2fc5sd1e henry cavil plays superman in the dceu

[@the4spaceconstantstetraqua886]

@@user-ov2fc5sd1e The Big Mass?

[@dulearning8256]

@@user-ov2fc5sd1e the youtuber says henry cavil lol henry cavil is the actor name for superman movie

[@lezhilo772]

Solved :D The circle in question is a circle in the momentum phase space. Specifically say m2 is the heavier block, if you rescale the momentum variable p1 and p2 by sqrt{m}, then you have a very nice circle equation. The entire process amounts to turning by angle pi from the point (p1,p2)=(0,1) to (0,-1) so that the heavier block is reflected. Every collision between the two blocks can be written down as a mapping between points on the circle in the phase space(ie old momentum determining new momentum completely, while obeying energy conservation so staying on the circle). Every collision at the wall serves to reflect the point about the p2 axis so that the direction of p1 is flipped. Every step leaves a point on the circle, and each point corresponds to a click sound. So the total number of points on the circle(minus the starting point) is the same as the total number of clicks heard. Since all collisions between two blocks is followed by a collision at the wall, we can consider these two events forming a single step. It turns out that as the ratio r=m1/m2 gets smaller, this step approaches an infinitesimal rotation generator, with each step’s rotation angle being 2sqrt{r}. If we let k to be the number of rotation steps to execute the full pi angle rotation from (1,0) to (0,1), then 2sqrt{r}=\pi/k, or 2k=\pi/\sqrt{r}. We identify 2k as the total number of points on the circle(involving both the rotation and the reflection, thats why there’s a factor of 2), then if sqrt{r} is 10^-2n, we have 2k=\pi * 10^n, which is exactly what we have. The only thing left is the round off but I need my beauty sleep now XD. Can’t wait to see Grant’s solution and animations! :D

[@coconutflour9868]

That would only be the case if the sum of the squares of the momenta (p1^2 + p2^2) is conserved, which is not the case. Conservation of energy demands that p1^2/m1 + p2^2/m2 is conserved, which is the same as what you tried only if the masses are equal

[@sauravchauhan4172]

Energy is also conserved here

[@coconutflour9868]

@@sauravchauhan4172 How so?

[@sauravchauhan4172]

@@coconutflour9868 he said that in the video too , energy is also conserved , and I think circle is not the solution, it can be an ellipse also. Maybe wrong

[@lezhilo772]

@@coconutflour9868 I rescaled the momentum variable by 1/sqrt(m). That means given old p^2/m, I define P=p/sqrt{m}. The conservation of energy equation p1^2/m1+p2^2/m2=E then becomes P1^2+P2^2=E. If we use the rescaled momentum variables as the axis of the phase space, then states of constant energy form a circle.

[@zerokelvin3626]

It comforts me that there is this abstract, interesting, mathematical world full of independent truth, no matter how dire our situation in the real world may be. A sacred place.

[@hhoopplaa]

@Ron I'm assuming he means its mathematical parts of it. In reality, there would (probably) be also physics and biology and stuff, ...right? Honestly this sounds very philosophical.

[@Improdeter]

Bro you forgot something, the gravity is just gonna not allow them to "slide to infinite" or might not even continue clacking because neuton said that every object is not moving unless a Force is aplied to it, wich is gravity in this case, it can slide to infinity if there is no air and no gravity then we will have no Force aplied to the object, physics is cool!

[@iwillspam5985]

3B1B's homework best homework

[@alessandroceloria]

The next day... Math teacher: Have you done your homework for today? Me: No, but I know why if you shoot a moving object to a still one with a mass ratio of 10^k under no friction conditions you get the digits of pi!

[@avgchz9444]

I know, I'm gonna do this instead of my actual homework

[@juliangoulette7600]

Perhaps you could set up analogous situations for any other base: 0. You have two blocks with masses of 1 and b^2^(d-1), where b is the base of the number system and d is the number of digits in that base you want to compute. 1. Count the number of collisions in that base. 2. You now have an approximation of pi*b^(d-1) in that base.

[@kauboy9816]

I was watching this from the preview, and I would swear the counter of "314 clacks" hit exactly at 3:14 left in the video. Well done.

[@h3xty]

@@RIPToot it was, it says 3:14 *left*

[@DavidG2P]

This channel is of suprahuman intelligence

[@ojasvikamra6835]

At 1:58 if youre curious

[@johnchessant3012]

I love how the pi creature goes "this doesn't seem like me..."

[@brawnstein]

If you see this Professor Sanderson . It came to my mind that the equation for a circle at the origin is x^2 + y^2 = r^2 where r is constant And 1/2Mu^2 + 1/2mv^2 = constant . Is it related to solution ? P.S 99% comments- other things 1 % comments- the puzzle itself

[@SKO_PL]

Hey... I think you're onto something! If x is the velocity of the bigger mass, y of the smaller, then we're pretty much going around a semicircle. Every momenntum transfer is a chord on that semicircle (since energy is conserved). I suppose every such transfer would have length related to the radius of the circle. All the transfers combined approximate half a length of a circle and I think that's where π shows up.

[@thomasagu761]

@@SKO_PL I think every clap has to do with a rotation of 1/10^n rad and the experiments could end when a pi rotation has been made. We have then (Nb_of_claps)/10^n > pi, (Nb_of_claps) minimum. And by definition this is Nb_of_claps = floor(10^n*pi)+1. This is not a proof, just a little intuition (and it doesn't even give the right result lol).

[@nazishahmad1337]

Yeah bro I too was thinking the same

[@amalantony8594]

@@SKO_PL Why is that a semicircle? It should be a whole circle,right?considering both positive and negative directions for u and v... Anyway ,that was a nice geometrical interpretation of the given problem.

[@SKO_PL]

@@amalantony8594 Because the big mass starts by moving to the left and ends by going to the right. x has to have opposite sign in the end

[@aa0000]

3:58 I like this representation of how this is just theoretical, not practical. Creative.

[@solt4r1]

Let’s give a moment of silence to the blocks that are still sliding towards infinity :

[@mmmDaber]

They are us

[@sparecreeper1580]

@@mmmDaber that explains the endless abyss dragging me away from a white wall

[@Scrubique]

@@mmmDaber Still waiting for my first collision in this middle of the abyss

[@DJB3lfry]

Now the real question is: If the blocks are moving at a constant pace along an infinite path for an infinite amount of time, how many infinities would it take for the blocks to reach the end of the infinite path?

[@hadisakho848]

@@DJB3lfry end and infinite are not looking fine in the same sentence haha

[@yyy76yyvhxxffb32]

It probably appears because the velocity gained and lost between both cubes which then represent aceleration diferences which then represents the variation between sen and coss

[@fishtard9347]

purpose of this video: hey everyone someone discovered this really cool thing about blocks colliding and the number pi and it's digits! comment section: OMG THE CLICK SOUNDS WHEN THEY COLLIDE SOUND COOL

[@EpochIsEpic]

Fishtard I play a rhythm game where every time you hit a note it makes a very similar sound.

[@EpochIsEpic]

Tech GO! Yep

[@Philgob]

if he didn’t want people to talk about the clacks he shouldn’t have made the sound so good 😔✌️

[@Novasky2007]

May i just say lets all take a moment to pay respect to the sheer mass of the wall needed for them to clack off of undamaged.

[@daviddavis-vanatta1017]

... and undamped!

[@joeljobi6149]

it has to be infinite or it won't work

[@Jake-hy6xs]

omg he literally said it wouldent be possible in real physics because there is no such wall

[@skyhui3412]

It's midterm and I'm sending this puzzle to everyone in my class to distract them from exam revision. Ha.

[@diegosanchez894]

well I have midterms and this falls quite nicely with the subject so who's laughing now!

[@operator8014]

Gotta get a jump on that curve!

[@skyhui3412]

@@diegosanchez894 good for you but I'm just a junior-highschooler and it isn't gonna be very helpful to those poor fellas lol

[@diegosanchez894]

@@skyhui3412 if you're planning on doing a stem degree later on it will be useful.

[@skyhui3412]

@@diegosanchez894 yeah of course for the long-term, but for the short-term it may not be a good idea to try solving a math problem before the geography exam

[@cheeseburgerpro69]

Thank you for letting me learn what “ 20 to the power of 5” is

[@Misha-775]

Я в шоке! Не только от потрясающего эксперимента, но и от огромной культуры физики за рубежом! Thanks! It was so great!

[@zeddown]

denada

[@MidnightBloomDev]

2018: I'm gonna count all of em 2019: Don't worry, still counting

[@soyoltoi]

2020: Yep, still counting.

[@-wicked-eharmonywithme7185]

I'll be back in 2021

[@underscoredfrisk]

I think you missed one! its that half a millisecond where I am sure you counted 312 but there are actually 314 clacks

[@khushiamin9307]

It's 2021 I hope ur still counting

[@jamesgodden7760]

U there yet?

[@somebodylikesbacon1960]

3:43 The sound of my brain figuring out the puzzle.

[@RPDC-ng8ej]

100,000,000,000_000,000,000_000,000,000_000,000,000 (10^38) (100 tripodecillion)

[@Cardalisius]

1:47 my fav part 3b1b:"wait for it..."

[@dancoulson6579]

1:33 - It's really satisfying how the clacks were so rapid, but they actually made the effort to create a believable sound.

[@InsertName5015]

1:34 The sound is perfect

[@Caterblock]

134 rearranged is 314

[@devsus]

bambi fantrack

[@user-ke7ki6mx2z]

Duck

[@an9em0n]

So, Superman shared this with you? 🤯 Then this is actual kryptonian knowledge

[@micahbradford2629]

I'm surprised more people aren't talking about this.

[@richiegibbons7103]

@@micahbradford2629 i know right?

[@roymuerlunos2426]

Common core lessons in a billion billion billion billion nutshells, Clacks is in session

[@xyz.ijk.]

Actually, there have only been about 50 views. The other 539,542 were me.

[@jayvyas1827]

😀😁😂

[@pudy2487]

The video is 5:12 long (312 sec), and you watched it 539,542 times, for a total of 168,337,104 sec, or 2,805,618.4 minutes, 46,760.3 hours, or 1948.3458 days. The video was published 1 day before you commented this, so you are a liar.

[@Eltaurus]

@@pudy2487 He might have been watching it in parallel on different screens.

[@dnzssrl]

@@Eltaurus So he has 1949 monitors :D

[@mehieltube]

@@dnzssrl interesting and he can't answer now cause he suffers from epilepsy. We have a case here!

[@3kbschannel288]

1:42 "Did you just fart?" "No, mom. I'm watching a physics video"

[@blazeguruz8989]

did you just farted

[@oblivion2755]

@@blazeguruz8989 have you did farted?

[@Ixions]

*plot twist* you did fart while watching a physics video

[@cubeofmeat4982]

Who tf farts like that

[@craetydonutkey1348]

Sounds like your bedroom’s door hinges need some lubricant XD

[@muffincherry9895]

me and my crush belike : Sliding off to infinity, never to be touched again

[@dragon-614-nest]

1:38 open the 0.25x speed

[@kgsr_x2]

Nice

[@AbCat4]

I like how the speed of the last collision is an expression of the remaining digits. So when it's 314(15...) collisions it juuuust reaches the line, but when it's 31415(92...), it gives the moving block a proper final spank to send it on its way.

[@vijayrajendran4006]

wait, if what you're saying is true, we can get more digits by analysing the speed more intuitively?

[@jonasmarcell]

2:03 when you approach the forgotten sandwich in your school with a radiation detector in your hand

[@hyperplayer22]

Ahaha good comment

[@craiglee6461]

You diserve more likes on that

[@craiglee6461]

You diserve more likes on that

[@uxleumas]

deserve?

[@nod_jawk]

Why do you put your nose in your hand ? ;)

[@lucasguedesdossantos4142]

hey guys i will say what i discovered here, but i REALLY dont know if it is the right way . 1.CONSERVATION OF ENERGY 2.CONSERVATION OF MOMENTUM 3.SHOCKS (graphical algorithm) As a lot of comment here, the conservation of energy equation( mv^2/2 + MV^2/2 = cte) remember the equation of ellipse (x^2/a^2 + y^2/b^2 = 1), maybe the circle we need is here... the energy in the beginning is E = Mx^2/2 (x = inicial velocity ), and at any time, E' = mv'^2/2 + MV'^2/2. E=E' => Mx^2/2 = mv'^2/2 + MV'^2/2. In the video, he talks about the ratio between the masses, so let M = km. we get kmx^2/2 = mv'^2/2 + kmV'^2/2. Dividing both sides by m/2 : kx^2 = v'^2 + kV'^2. Dividing now by kx^2 we have: (v'^2)/(kx^2) + (V'^2)/x^2 = 1. (Ellipse!! :D ) now, in a 2d plane, with the x axis representing V, and y axis v (arbitrary), we have a ellipse, that the dots represent possible combinations of V and v, that keeps the energy constant. (Semi axis x) = x, (semi axis y) = x sqrt(k). Ok, now we need to look at the conservation of momentum(Q) : MV + mv = MV' + mv' => Q = kV + v = KV' + v'. In the plane described before, this equation represents a line, that crosses the Y axis on Q, and the ellipse on (V,v) and (V',v'). This means that for any shock between the blocks , we need to draw a line that cross Q on y axis, and (V,v), and the other point that this line crosses the ellipse will represent the new velocities! when the block toutches the wall, v is multiplied by -1, graphically, it is turned up/down around the x axis. We know that ends when the big block is faster than the other one (to the right), it will happen in the points below the line y = x, for x >0 with this, we can (graphically) predict the shocks!!! i can see it turning into a "3b1b thing" with some beautifull maths in the end, and coming up with the circle.

[@QuakeJoz]

so close to the answer, try squishing the graph so the elipse is a circle and do some geometry.

[@DarioSterzi]

To add to Joseph's suggestion: you found what collusions with the wall represent geometrically; starting from what you've already fond what do the collisions between the two bodies represent?

[@lucasguedesdossantos4142]

@@QuakeJozim trying to find a logic way to squish the y axis by 1/sqrt(k), but this value is not common

[@lucasguedesdossantos4142]

@@DarioSterzi the collision between two blocks is found, geometrically, in the line that crosses (0, Q) and (V,v). But Q is the momentum of each collision, maybe squishing the y axis make the Qs of the collisions progress with some pattern

[@QuakeJoz]

​@@lucasguedesdossantos4142 You can just transform everything, i.e. define z= y/sqrt(k) (or y sqrt(k), not sure which way round your k is) and plot z instead. Don't forget to transform the y=x line and the momentum lines too.

[@xyz.ijk.]

I didn't think there could be a better 3B1B video than some of the others ... but this is it. Yes, pi is here for comprehensible reasons, but this is sooo beautifully presented and so much fun. Thank you. (Poor cow ...)

[@hemanthkotagiri8865]

I see 3 blue 1 brown upload a video, I drop everything.

[@PaulPaulPaulson]

Whatever you dropped, how often did it bounce off the floor?

[@Sachin27071998]

@@PaulPaulPaulson based on ideal physics for coefficent of restitution e, it'll still bounce infinitely although the height after the bounce decreases by factor of e^2 everytime

[@Vancha112]

That's how supermarket employees lose their job.

[@ayushshukla1438]

@@PaulPaulPaulson You must be given a medal I suppose

[@deepak3006]

Hope you don't become a carpenter

[@pauldacus4590]

2:31 apparently Superman has a lot of downtime now that the Justice League cinematic universe is dead

[@cancermelon2155]

I remember this problem! It was at the entrance test at the Scuola Normale Superiore in Pisa some years ago! I also remember i couldn't solve it :^(

[@katphisH11]

Did you get in though?

[@Wild4lon]

Damn. I applied to Cambridge for physical natural sciences and got in and I would be deceased if I got this at the entrance test

[@cancermelon2155]

@@katphisH11 This problem was not from the year i tried to enter, I tried to solve it to practice for the test. This one was actually from the physics test and probably one of the hardest. Btw i was able to pass the written tests but got rekt at the orals

[@cancermelon2155]

@@Wild4lon It's for sure the hardest math/physics entrance test in Italy and imho one of the hardest in general after high school...

[@Adomas_B]

1:41 that sounded really good in minor chords, probably someone could create a song out of this

[@tesnun2852]

It sounds like crimewave by crystal castles tho

[@prateekpanwar646]

It sounds like my stomach when I'm hungry

[@aloysiuskurnia7643]

My assumption why powers of 10 is shown there is because we're looking for *decimal* representation of π; if we used other powers, say, powers of k, we'll get the base-k representation of π. Just guessing tho.

[@3blue1brown]

Indeed. For example, here's what it looks like when the ratio is 16: kzbin.info/www/bejne/lWGvhWmVlq-Hetk 12 in binary is 1100, which is the start of pi: 11.001001000011... Likewise, for a mass ratio of 64:1, it's 25 which is 11001 in binary.

[@joefaber4397]

Why don't we use base near 1 to calculate pi easier.

[@ralusek]

That's what I was expecting too. I was watching this being like BUT WHY IS IT THE DECIMAL DIGITS OF PI?

[@gaiarocks5863]

Not for all bases. For example with base=6, and a mass ratio m2/m1=36, we get 19 collisions, which is written "31" in base-6. However, the base-6 digits of pi are "305033...", which does not start with "31". For some nasty reason my other reply to Aloysius Kurnia (with the results of my analysis) is not visible (except to me when I am logged in)...

[@ORGOrange]

another way to count the pi digits: b1=1kg. b2=10^x kg. x=2y. increase y for more digits. the amount of digits of pi is y+1. (sidenote: if y is less than 0, you still get 1 digit.)

[@maximilianlavendel3130]

I absolutely love this format, where you share a mind blowing mathematical question and callenge us to solve it. As for my part i‘d love to see more of these videos!

[@ridhomblr]

0:49 i saw this on tiktok, they did not give credit or anything no link to the original just sped it up, glad i found the original

[@quarksgluons]

A 5-minute 3b1b video can teach me more than an hour lecture at college

[@Sachin27071998]

more than a semester in college, because his questions makes me wonder.

[@Peter_1986]

I swear Matt Parker will make a real-life "calculate Pi" video very much like this one some time in the future.

[@xenogenesis9635]

I love how you caused at least 3 "oooooh" moments in 5 1/2 minutes to me, especially the "hidden circle" part. I'm not much of a maths guy, but I got what you meant by that concerning this example without watching the follow up video. So cheers for actually explaining things. Love your channel already.

[@revan1139]

1:50 wait for it.... WAIT FOR IT... love it