The Simplest Math Problem No One Can Solve - Collatz Conjecture

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    Veritasium

    The Collatz Conjecture is the simplest math problem no one can solve - it is easy enough for almost anyone to understand but notoriously difficult to solve. This video is sponsored by Brilliant. The first 200 people to sign up via brilliant.org/veritasium get 20% off a yearly subscription.

    Special thanks to Prof. Alex Kontorovich for introducing us to this topic, filming the interview, and consulting on the script and earlier drafts of this video.

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    References:
    Lagarias, J. C. (2006). The 3x+ 1 problem: An annotated bibliography, II (2000-2009). arXiv preprint math/0608208. - ve42.co/Lagarias2006

    Lagarias, J. C. (2003). The 3x+ 1 problem: An annotated bibliography (1963-1999). The ultimate challenge: the 3x, 1, 267-341. - ve42.co/Lagarias2003

    Tao, T (2020). The Notorious Collatz Conjecture - ve42.co/Tao2020

    A. Kontorovich and Y. Sinai, Structure Theorem for (d,g,h)-Maps, Bulletin of the Brazilian Mathematical Society, New Series 33(2), 2002, pp. 213-224.

    A. Kontorovich and S. Miller Benford's Law, values of L-functions and the 3x+1 Problem, Acta Arithmetica 120 (2005), 269-297.

    A. Kontorovich and J. Lagarias Stochastic Models for the 3x + 1 and 5x + 1 Problems, in "The Ultimate Challenge: The 3x+1 Problem," AMS 2010.

    Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values. arXiv preprint arXiv:1909.03562. - ve42.co/Tao2019

    Conway, J. H. (1987). Fractran: A simple universal programming language for arithmetic. In Open problems in Communication and Computation (pp. 4-26). Springer, New York, NY. - ve42.co/Conway1987

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    Special thanks to Patreon supporters: Alvaro Naranjo, Burt Humburg, Blake Byers, Dumky, Mike Tung, Evgeny Skvortsov, Meekay, Ismail Öncü Usta, Paul Peijzel, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Juan Benet, Ruslan Khroma, Robert Blum, Richard Sundvall, Lee Redden, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson, Sam Lutfi, Ron Neal

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    Written by Derek Muller, Alex Kontorovich and Petr Lebedev
    Animation by Iván Tello, Jonny Hyman, Jesús Enrique Rascón and Mike Radjabov
    Filmed by Derek Muller and Emily Zhang
    Edited by Derek Muller
    SFX by Shaun Clifford
    Additional video supplied by Getty Images
    Produced by Derek Muller, Petr Lebedev and Emily Zhang

    3d Coral by Vasilis Triantafyllou and Niklas Rosenstein - ve42.co/3DCoral
    Coral visualisation by Algoritmarte - ve42.co/Coral

Flapjack Mollases
Flapjack Mollases

If you notice, there is a pattern where every turn, or equation change in the sequence is signaled by a prime number. It doesn't just simply reduce from an even number to a simple odd number. Each odd number encountered in any path is also a prime number. The "chain" will continue to rise in value until a prime number lower than the starting value is encountered, and then it will reduce down to 1, the lowest prime number. That was my observation doing about 20-30 samples, if anyone found anything different, I'm open to hear it, I'm not saying this is concrete, just what I have observed. The looping effect stems from the fact that 1 and 2 are both prime numbers. Since 2 is both prime and even, it creates a barrier from continuing the equation past 4, since the only multiples of 4 are, 1, 2, and 4. Taking the smallest prime number (1) and applying it to the "growing" equation of 3x+1, then the result is 4. For instance, you will never encounter the number 27, unless you start with this number. Why? there is no integer value that can satisfy the equation 3x+1 = 27. There are also a multitude of even numbers that will never be encountered, unless that number is used as a starting value. What about 54? Since 54/2 is 27? There is also no integer value that satisfies 3x+1 = 54. I haven't done enough tests to prove anything, but I observed that any even number that reduces to an odd number that is not prime is also not present. Even though this appears to be a linear equation, this is very misleading. In a linear equation, we assume any value along that line can be a solution that satisfies X. Because we can only use integers, and not all real numbers, this limits the number of possible solutions to any given situation. So you are starting the problem with the false assumption that 3x+1 is a linear equation (which it normally is), when you are limiting the values of x to integers. When you begin a problem with false assumptions, you can't possibly hope to have a real answer. Point? You can say with pretty strong confidence that any multiple of any number that satisfies the equation 3x+1= 0 is not defined by the problem. Therefore, the "solution" to this problem does not lie in the range of possible values for X. So the problem itself becomes undefined. Take 0 for example. 3(0)+1 returns 1. The smallest prime number. If we then use the rules to apply this 3(1)+1, we again get 4. If we assume x = 0 for each equation, then the first returns 1, and the second returns 0. We would assume then that the answer would lie somewhere between 1 and 0, and since we can only have integer values of x, then it becomes undefined.

2 uur geleden
Azuleslight
Azuleslight

it makes sense to me, its basically like a tree. where the roots eventually invert to make a tap root so to speak. while the positive numbers branch out like a tree. also like a tree the tap root goes deep and slims down eventually. the divergence happens at zero.

4 uur geleden
Vikram Mardi
Vikram Mardi

One two ka four Four two ka one My name is Vikram

6 uur geleden
dreesee
dreesee

3×+1=10

6 uur geleden
dreesee
dreesee

Simple

6 uur geleden
mraksoll
mraksoll

This will go to infine , and you can it check :) ( try chenge first and see what hapen if give result to 2 :) def col(n): sp = [n] if n return [] while n > 1: if n % 2 == -100: n = 3 // n - 1 else: n = n * 2 sp.append(n) print(n) while True: col(int(input())) check via this def col(n): sp = [n] if n return [] while n > 1: if n % 2 == 0: n = n // 2 else: n = 3 * n + 1 sp.append(n) for i in sp: print(n) while True: col(int(input()))

8 uur geleden
J J K
J J K

We're nothing but smart apes. I hope we go extinct soon.

9 uur geleden
J J K
J J K

Some problems were never meant to be solved. Stop trying to be God, humans. Just die.

9 uur geleden
Rob Adkins
Rob Adkins

I believe the issue here is that we are trying to using decimals. There is a solution described dividing by 16, by 8, by 2....that's hex, octal, and binary. What if these numbers were translated to binary and then worked through? Just a thought.

9 uur geleden
Релёкс84
Релёкс84

@Rob AdkinsThe representation method you use changes nothing about the properties of numbers. Also, the binary representation was explicitly used in the video 17:37, and while it's definitely interesting there's nothing that can easily be said about it that couldn't otherwise.

Uur geleden
Rob Adkins
Rob Adkins

Interestingly, 4,2,1, are the first 3 weights of binary numbers. A decimal 4 written in binary is 100. Again, interesting that the whole number is 100, which is 3 variables plus 1 logic high..3x+1.

9 uur geleden
Joe Seabert
Joe Seabert

Not 3/2, it’s 1/6th

10 uur geleden
Joe Seabert
Joe Seabert

-1/6th

10 uur geleden
Joe Seabert
Joe Seabert

Click and clack the tappit brothers said a long time ago, the answer is always 1.

10 uur geleden
Antoine Bilbord
Antoine Bilbord

The Universe is like this math problem

12 uur geleden
Mrmace51
Mrmace51

I dont see what is signifigant about patterns set by arbitrary formulae

12 uur geleden
Humbleaction
Humbleaction

Why is it a +1 though? Where or what is that 1 and why not any other number? Or are all the other problems solved and this is the only 1? Lol.

13 uur geleden
Релёкс84
Релёкс84

You could also add -1 or +3 if you want, as long as it's an odd number. If it's an even number, including 0, then the answer is really simple. 3x+1 is arguably the simplest variation of the problem for which we have no definite answer.

11 uur geleden
Nikola Tesla
Nikola Tesla

Well, I don't know if this means anything but the premise here is strongly related to that of a pyramid... Think of how a pyramid is laid out - 1 stone is at the top, then you have 2 stones below that, which is 3 in total, then below that you have 4 stones, which is 7 stones in total then the layer below that has 8 stones, which is 15 total stones, then below that is 16 stones, which is 31 total stones and so on - but there is always that 1 stone on the very top which is why any number of stones you chose always ends up at 1.. Of course a pyramid is a geometric object so this math riddle obviously has ties to geometry, weather or not if that means something, I don't know? Below is a simpleton "math diagram" of what I'm trying to imply er say here, lol. 1 2 ----3^ 4 ----7^ 8 ----15 16 -----31 1 1+2=3 3+4=7 8+7=15 16+15=31 32+31=63 It's the total sum of stones needed to construct a perfect pyramid.. It doesn't matter what number of stones you choose because you will always end up at the top of the pyramid... It's basically the reverse engineering of a pyramid.. Look, I'm not a math nerd, lol... It's difficult trying to explain abstract ideas but I was just watching a video where pyramids where discussed, a light went off in my head and it made me think of this video and I associated the sum of stones it takes to build a perfect pyramid with this math riddle and it matches perfectly. Now I don't know if geometry has anything to do with this or if this was a coincidence - or even an answer - but I thought it was fascinating to say the least..

14 uur geleden
Bee Lynn
Bee Lynn

This guy likes saying arbitrarily.

14 uur geleden
Lalitha Janghamaiha
Lalitha Janghamaiha

An awesome equation which will be solved by future guys soon or an AI 😉

14 uur geleden
Landon Cady
Landon Cady

imagine how you would solve this if the numbers were negitive - hmmm how's that for an math problem :D

16 uur geleden
Landon Cady
Landon Cady

i wonder would you get the same result or diffrent outcome :D ???

16 uur geleden
Christo du Plessis
Christo du Plessis

What happens if you do this with Grahams number? Wouldn't that go to infinity?

16 uur geleden
Angieghost UFORick
Angieghost UFORick

4

16 uur geleden
Andrey Z
Andrey Z

How is that sequence, if it takes random quantity of operations of dividing the number by 2 to bring it to odd number?

16 uur geleden
Charles
Charles

A even number divided by 2 can be an odd number, 6/2 = 3

3 uur geleden
Wild Malibu
Wild Malibu

It’s just a function that tapers down into a pool. That pool being a cycle that math itself created and perceived by humans of this age are not capable of breaking down and seeing further into, once math becomes evolved enough to break the cycle down, we as humans will discover that the exact same thing will take place, it is only in comparison to another intellectual age that we have evolved, in its own system function perceiving from within from a mind of its own era there will seem to be no answer, but on the brink of evolution, inspiration and awe fuels change and new perception on the very material existence of consciousness itself.

16 uur geleden
mohammed ikram
mohammed ikram

Try Pi

17 uur geleden
Релёкс84
Релёкс84

integer

11 uur geleden
Arlen Gonçalves Martins
Arlen Gonçalves Martins

what about zero? 0/2 = 0

18 uur geleden
tim watley
tim watley

If you look at the graph upside-down it will go up instead of down. Problem solved.

18 uur geleden
Chip
Chip

the name of this channel is based off the Latin word which means truth

18 uur geleden
Gergely Roboz
Gergely Roboz

What is the most fascinating thing about this video is that I understand nothing of it, but I still find it super interesting and entertaining. You (and your team) are (an) amazing presenter(s)!

19 uur geleden
Daisy Do
Daisy Do

Jeremy Bearimy

19 uur geleden
Flapcow
Flapcow

So this is calculus

20 uur geleden
Guts
Guts

3x+1=life

21 uur geleden
Rohan Preis
Rohan Preis

Wait, what’s the problem?

21 uur geleden
Chandra Saud
Chandra Saud

Really amazing. It's like organic math😅

22 uur geleden
Ananya Srivastava
Ananya Srivastava

Your fuel for science is so contagious, Derek! Absolutely love your work.. one of a kind creator. Hands down!!

22 uur geleden
idk anymore
idk anymore

answer is 21

22 uur geleden
John Sessa
John Sessa

Numbers are interesting on their own. But why are humans obsess with them, the numbers?

23 uur geleden
Wacow
Wacow

Somewhat understanding these videos makes me feel like a mathematical genius when I'm getting loaded with my friends.

23 uur geleden
nostaWrD
nostaWrD

If ever I believe my work is done Then I'll start back at one !

23 uur geleden
mrfm1704
mrfm1704

Nah nah, my question is, why does 3x+1 even exist in the first place?

23 uur geleden
Vansh Sharma
Vansh Sharma

I ve always hated maths and now I hate it even more.

23 uur geleden
MornThunder
MornThunder

"communism is the riddle of history solved, and it knows itself to be this solution. "

23 uur geleden
Sunshine
Sunshine

I don't need it, but it's mildly interesting

Dag geleden
Rocco Castagnola
Rocco Castagnola

beautiful

Dag geleden
Elyes Bens
Elyes Bens

10:36 you're welcome people

Dag geleden
Matthijs Koelewijn
Matthijs Koelewijn

lowest number must be zero

Dag geleden
Владимир Пронин
Владимир Пронин

hello. I no every write and speak English. You give 1,57!!!

Dag geleden
•Oficxllymxli•
•Oficxllymxli•

10:38 is he okay??

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m&m _kanal
m&m _kanal

this stuff hurts my brain

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Angga Hadi
Angga Hadi

But what is the problem?

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Tamunotokini Pokubo
Tamunotokini Pokubo

MATHEMATICIANS: 3X +1 NOT POSSIBLE TO SOLVE ME: X=-1/3 PROBLEM SOLVED * DROPS PEN*

Dag geleden
TheRealLori
TheRealLori

The simplest way to avoid spending your life trying to solve this is by only applying it to the numbers 1 through 10 or technically 0 through 10 but zero doesn't do anything. any number that comes before 1-10 doesn't matter eg: 438 only the last digit matters while the 430 is completely meaningless because being an odd or even number is defined by the last digit.

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Релёкс84
Релёкс84

@TheRealLori What doesn't matter for what?

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TheRealLori
TheRealLori

@Релёкс84 but that doesn't really matter

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Релёкс84
Релёкс84

Sure, but the value of the other digits come into play as soon as you start dividing by 2, which starts happening no later than on the second step.

Dag geleden
Kazus
Kazus

You can't prove it true because it's not math

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Dank
Dank

Now I'm wondering. What if plants are more smarter than us humans? I feel like it's somehow connected with math. Okay blame the Soviets for this question.

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Eriberto Acedo
Eriberto Acedo

Most interesting when I feel like a challenge I will definitely try this until then I'll work on my Hot Rod, customizing is a challenge on its own because you can't buy the parts you have to create them which I've done to every inch of my hot rod. Measure two or three times then cut shape and fit all other pieces interlock with each other makes it stronger than what you can buy on the showroom floor and looks much better because it's Customized! Better than original did the same with my Harley Davison 1950 vintage totally customize it road around on it, 22 years without opening up the cases just a valve job and replace the rings which I did in the kitchen with my ex-wife helping me worked out great! some people have an Eye for customizing most people don't majority don't. that's what I do along with electrical wiring with no mistakes! From San Diego California ( Customindesign.)

Dag geleden
Sean Gray
Sean Gray

this seems similar to the halting problem (proven to be "undecidable" by Alan Turing) Edit: nevermind, you touched on that later lol. I need to remember to watch the whole video *before* I comment

Dag geleden
Katherine Wan
Katherine Wan

my math teacher told me to watch this lmao-- thanks for the homework fjsfakllfsdflksjf

Dag geleden
David Roman
David Roman

Even in trying in Base 8 it still gets to 4, 2, 1 cycle But, if you ignore actual numbers and think about even distribution of stuff without numbers then if follows logic path.... The sequence of "stuff" are measured in discrete units of Z starting with nothing (0) and increasing by Z, with a repeating sequence Where Z-Z is nothing and X is cumulative of all Z, and Y is half of X it looks like: Whole numbers only X is the start value in sequence, X can be even or odd Y is always even (proof) if X is even and X=2Y then Y is even Z is always odd and a set value of 1, so Z-Z = nothing and Z+Z=2 if X = Y+Y then X is even (Xeven) If X = Y+Y +/- Z then X is odd (Xodd) If Xeven then Y becomes new X in series. if Xodd then X+X+X+Z becomes new X in series. The sum of X+X+X+Z will always be Xeven, then the disposition of Xodd's will always become even. Even numbers can be expressed as the disposition of two equal lower value Xodd's Xeven = 2Y can be expressed as (3X+Z)+(3X+Z) or 6X+2Z. If Xeven numbers can be expressed as the disposition of lower Xodd, and the disposition of lower Xodds are even, and Xevens are always halved (divided by 2) then all Whole numbers 0->infinity will eventually reduce to the 8, 4, 2, 1, 2, 1, 2, 1, 2, 1 loop unless we introduce imaginary numbers into sequence. then again I am Kinda tired and my eyes are blurry and burning....

Dag geleden
gordon b
gordon b

boycott Beijing 2022 Olympics ! Countries engaged in committing genocide don't deserve the honor of hosting an Olympic Games.

Dag geleden
sedisvacantia
sedisvacantia

Mathematicians waste their time with this kind of nonsense, and people in Humanities are the ones everyone makes fun of.

Dag geleden
L Ruddy
L Ruddy

ok but whats the point, wheres the real world application of multiplying odd numbers by 3 and adding 1 and dividing even numbers by 2

Dag geleden
LordGaben
LordGaben

Learning, expanding knowledge. It might lead nowhere, but the next problem solved might be the gateway to humanities future of abundance. Point being to ignore the problems of they seem pointless means sticking with the status quo and as we know that is the ultimate in destructive conservative futility.

Dag geleden
Nikola Tesla
Nikola Tesla

This is hilarious, it's like a riddle for math geeks in their language, lol.. It's like watching Rain Man trying to figure out "Who's on first?" lol

Dag geleden
Trapster
Trapster

what if i pick the initial number to be Zero? Zero is a even number, so the rule says we divide it by two. So, 0 / 2 = 0. Hence we break the loop of odd number and the end result will always be zero.

Dag geleden
Melvin Shelton
Melvin Shelton

I wonder how many students have quit mathematics in dispair over not being able to solve problems like this.

Dag geleden
the warion
the warion

My smaet ass "its x4"

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Rob
Rob

Gravity

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Joshua Banda (Jkaksb)
Joshua Banda (Jkaksb)

In other words it's true

Dag geleden
wAdjwakdjawkidj
wAdjwakdjawkidj

I created another loophole for Collatz Conjecture. Which when you start off from 21 you end up still on 21, I also apologize for creating another loophole.

Dag geleden
yunus emre akca
yunus emre akca

isnt infinity a loop?

Dag geleden
Markus Müller
Markus Müller

Great video but what happened here 13:55 ?

Dag geleden
Stefano
Stefano

Would decimals make a difference?

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R Dodds
R Dodds

I suppose someone has tried this with decimal numbers?

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Релёкс84
Релёкс84

Since when are decimal numbers odd or even?

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Bool False
Bool False

19:46 - *We have no right to have solutions to all of these other problems*

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BananaKingdum248
BananaKingdum248

do it with 350

Dag geleden
Callibor 31
Callibor 31

This equation sounds like a law of infinite. And unlike the laws that humankind made, we broke trillions of them in a matter of seconds under 3x+1. Would everyone be sinners then if this equation for some reason is a law of infinite justices that even our own justice system violated?

Dag geleden
AMV FROM- Mr. Editzz
AMV FROM- Mr. Editzz

Isn’t it 10

Dag geleden
WolfY
WolfY

So, why 7 and not other nine numbers? Why do I always see such thing when people are choosing a number from 1 to 10 and it's most of the time 7? I DON'T NEED SLEEP I NEED AN ANSWER!

Dag geleden
Moreno Franco
Moreno Franco

I LOVE THIS PROGRAM!

Dag geleden
Moreno Franco
Moreno Franco

Amazing. My father showed me this when I was 10. That was 59 years ago. Now I instantly see patterns - in life and in peoples behavior. I see it in the traffic, and in crowd movements. Thanks, Verisatium Team.

Dag geleden
Sophistic History
Sophistic History

@DJ-murlock Animation ............GIGGITY...........

10 uur geleden
DJ-murlock Animation
DJ-murlock Animation

You're 69 now? Nice

Dag geleden
Fabio Müller
Fabio Müller

Is it possible to proof any prime number comes back to the original 4 2 1 sequence? Because of that is true we'd only need to proof any starting number eventually comes to a prime

Dag geleden
omp199
omp199

That doesn't sound any easier.

Dag geleden
Hayate Azekura
Hayate Azekura

Sometimes I wish I wasn't such a dumbass, So I too can understand anything

2 dagen geleden
omp199
omp199

Surely you can take comfort from the fact that when it comes to this notorious unsolved problem, nobody understands it. Here, even the world's greatest mathematicians are "dumbasses".

Dag geleden
ankokuraven
ankokuraven

Another great video

2 dagen geleden
Manny Norton
Manny Norton

The alleged sort arespectively heap because gemini psychophysically employ as a bad resolution. guttural H habitual, heavenly heavy hellish colt

2 dagen geleden
Tobi L
Tobi L

My question is: why is that a “problem”? It is just a patrern

2 dagen geleden
Renato Gabriel
Renato Gabriel

Humans can live happily ever after without 3x+1

2 dagen geleden
Máté Kurucz
Máté Kurucz

it makes me so proud that most mathematicians mentioned from the same country were hungarians in this video. such a small country yet there are many smart people who pave the path going forward, even though i lowkey didnt understand anything in the vid :'D

2 dagen geleden
Ryan Betker
Ryan Betker

Seems like someone made it up as a joke to have people spin their wheels. If you can continually divide an even number result by 2, and the setup has a propensity to produce an even number, of course it's going to get smaller. More simply, look at the last digit. Any odd number x3+1 is going to create an even number. Any even number x3+1 will make an even number in the next round anyways.

2 dagen geleden
Релёкс84
Релёкс84

It's not that simple. You could very well have a sequence of increasing odd numbers separated by even numbers. Some starting numbers reach thousands of times their initial value that way, and there's no reason to think it's impossible to grow to infinity.

2 dagen geleden
koyou33
koyou33

lol fractrum what a load of rubbish

2 dagen geleden
hassan abdeen satti
hassan abdeen satti

the logo of the video

2 dagen geleden
hassan abdeen satti
hassan abdeen satti

its 10

2 dagen geleden
John Westfaul
John Westfaul

Does it only work with whole numbers tho

2 dagen geleden
Oisin McCaul
Oisin McCaul

Just gonna throw out square root of 421 Billion

2 dagen geleden
M Najib I
M Najib I

AH... My head, hurts

2 dagen geleden
Renu Singh
Renu Singh

The number q is not in this loop 😎

2 dagen geleden
Renu Singh
Renu Singh

Me: Going Try with a decimal

2 dagen geleden
Moey Misselmany
Moey Misselmany

the problem is in the rules...

2 dagen geleden
David Collins
David Collins

POV You discovered YouTube’s algorithm

2 dagen geleden
LAkers
LAkers

Because zero isn't the end only starts at 1.

2 dagen geleden
Skeletron Prime
Skeletron Prime

Isnt the answer 4 or 6?

2 dagen geleden
just dev
just dev

Pointless, like all human thought!

2 dagen geleden
BloxGots
BloxGots

I like the fact that 15 sky rockets and then goes to 1

2 dagen geleden
Andy Pandy
Andy Pandy

I thought the whole math's concept completely useless and a waste of time problem solved

2 dagen geleden

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