#### Am Vor Monat

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

## KOMMENTARE

## Harvey the paragliding chaser

It has been revealed. All is One

Vor Stunde## Cool Conner

Just start with 0

Vor 4 Stunden## AnAnomaly 334

Me who knows 3x+1 is just a kinda steep line that passes through (0,1)

Vor 4 Stunden## Archibald Bagge

Meanwhile the rest of us are having sex with each other.

Vor 5 Stunden## TextDrivers Kill

Math won't be around for long. Blacks hate it and you know what that means.

Vor 6 Stunden## CalvinB

What is the point of this though? Not even trying to be funny but I don’t even understand what the point of this question is lol. I have pretty limited math experience sorry if that’s a dumb question.

Vor 6 Stunden## Ozan Swain

notice how at 2 minutes when the graph shows the different paths the numbers take it stops at 27 which is the number that goes above 9000

Vor 6 Stunden## Christina Brannon

0.123156 I think I did it right with the math and in 2 steps ended up with 0.

Vor 7 Stunden## Kamaliddin Sattorov

Didn’t understand anything but enjoyed watching it. Thanks

Vor 10 Stunden## Christopher John

1 is the only eligible number.

Vor 11 Stunden## Freak Freaky

This just proves that some people are dumb timewasters to the infinity.

Vor 12 Stunden## Bharat Pandey

What is the question here actually?

Vor 13 Stunden## YUGANSH ARORA

I solved it in 3 miliseconds

Vor 13 Stunden## Juan Paulo Concha

I don't understand yet why it is a problem. Is it because no one can explain why it becomes one?

Vor 15 Stunden## J Modified

It's because we don't know if it always does become one.

Vor 15 Stunden## cheng wang

Why is the title of the video the simplest math problem noeoen can solve it?

Vor 15 Stunden## Alex

Russians: Now, try 2x + 1

Vor 16 Stunden## Alex

@Релёкс84 ackchyually lol

Vor 14 Stunden## Релёкс84

That one is trivial. You can accurately describe the sequence for any number no matter how large.

Vor 15 Stunden## 777 THE TRUTH

Just curious, why does this even matter? What purpose does it serve?

Vor 17 Stunden## 777 THE TRUTH

If no one can solve it, then it's not "simple!"

Vor 17 Stunden## Dave

What this tells me is that we, as humans, probably lack the intelligence to have a conceptual model of mathematics which is sufficient to prove or disprove the conjecture. We could be thinking about numbers completely the wrong way (but the only we can as the sentient beings we are), which could have implications for many other problems about how we model and describe reality which we can't solve. Questions like whether interstellar travel could be possible, time travel, the elusive theory of everything. I think there are answers to these things and I think they can be represented in mathematical models, but not our mathematical models, because of the limits of our own capacity.

Vor 17 Stunden## Franklin 3

Numbers of old said the wise man who uses them not. 😶

Vor 17 Stunden## Cricket Rodeo

I'm not understanding why we can't just keep checking every number sequence (well beyond 2 to the 68) by running a program on a quantum computer until we find the exception. Am I overestimating quantum computing?? On another note, the discovery of mathematical equations in nature is so fascinating, and opens up beautiful possibilities to designing urban (and other) environments in a way that looks more "natural", thereby creating cognitive ease and reduced mental stress to humans.

Vor 17 Stunden## OWIL

so my teacher was pretending shes smart huh

Vor 17 Stunden## 春

people trying to solve 3x+1 for decades : me trynna play tetris to fit everything inside my fridge :

Vor 18 Stunden## Victor Onstenk

what about -1

Vor 18 Stunden## Dommie80

3x+1 is quiet usefull for mountain generators i suppose (for games)

Vor 19 Stunden## Mr. Creosote

Once it gets to 4, it's in the loop. So real question is why does it always go to 4. The way to 4 is either 8 (8 divided by 2) or 1 (3x+ 1=4). Whatever.

Vor 20 Stunden## Mola

answer : don't look in 2 dimensions

Vor 21 Stunde## Релёкс84

That's an advice, not an answer.

Vor 16 Stunden## Mohammed Imtiyaz

Programmers know how to handle recursion 😉

Vor Tag## James Wallis

I am more amazed that the narrator of this video knew I chose 7…

Vor Tag## TheSingleBean

What about positive numbers with odd decimal going by those decimals it goes up to infinity consider 1.1 3x+1 = 4.3 and 4.3 the count the .3 as the number to go of of for the use of equation. so 4.3 3x+1 =13.9 considering 13.9 has an odd decimal you go off of that and it repeats for infinity and therefore a new problem is created and number that is multipled by the tenth place has become an infinate loop of 3x+1 just repeating the same question unto infinity.

Vor Tag## Drave

hmmm 3x+1=? 4

Vor Tag## Anna Becker

Looks like a penny stock chart.

Vor Tag## Collin MacInnis

If you add to the conjecture... say after you start the sequence.... 3x+1.... the last digit works out to be a prime... you add the square root..... does 9 become a digit that will never show as the last digit? Basically only apply +1 to odds other than nine as a last digit... that would be a (+3) (1=1) (4=2) (9=3) same loop would occur... if you add the square you lock it in as a loop that ends at the prime... 16+4= 20÷2 =10÷2=5×(3)+1= 16?

Vor Tag## rachit1431

10?

Vor Tag## CHEEZ CHEEZ

555555555555+ 39 = 555555555594

Vor Tag## ♡︎Ocean.Vibes♡︎

I know the answer It’s 4 :)

Vor Tag## chau lu

PIx10^4364345 maybe

Vor Tag## Релёкс84

That's not an integer though

Vor Tag## Wooda

We are all one

Vor Tag## Wooda

What is the question? What is the problem?

Vor Tag## Релёкс84

The question is simply "Do all numbers fall back to 1?", and the problem is finding a definite answer (and justifying it of course, guesswork has no value)

Vor Tag## Harjyot Sohal

3x + 1 = 4x because if you add 1 + 3 then you get 4 so its 4x. Done.

Vor Tag## รัชต์ชยุตท์ แสงอรุณส์

¶-2

Vor Tag## shahzeb malik

Sorry radians

Vor Tag## shahzeb malik

Try using substitution method using 2pi against 360

Vor Tag## mario juric

Maybe one time when we learn physic my profesor actually write something similar , and then when I ask him why there is no number between he said we well there is 2 possibilities and that is 1,0 just that so there could be great chance that we can put 0 between

Vor Tag## Pranati S

Where you at prince charming? 🐯

Vor Tag## Mmiikkii

9:40 Math is indeed the language of nature and universe

Vor Tag## Mmiikkii

I mean. 4 is the triple of 1, + 1. And 2, is half of 4. While 1 keeps being half... Of 2. We are in the original 1. That 1, that tripled +1 keeps back to 4... And so on

Vor Tag## wolfgirl1439

😆 Never ending loop

Vor Tag## Richard C

OK, so exactly what is 3x+1? Is there something in nature that 3x+1 represents? This seems to be an exercise without purpose.

Vor Tag## Mr. Bro142

Fact: He said 1 is a odd no. But actually 1 is a unit no.

Vor Tag## Christof Apolinario

Rare number 1 Answer = ∞?

Vor 2 Tage## Xuancong Wang

Since this paper ( arXiv:2101.06107 ) has already been accepted on 2021.01.08, I presume the conjecture has already been proven, unless the authors of the paper made a mistake in their proof.

Vor 2 Tage## Релёкс84

@Xuancong Wang There are several, but the most critical one I've seen (-and I didn't bother looking much further, and even someone who's job is to review it probably won't either), is at the start of the first point of the first proof: while the conclusions overall are logically sound, there is a blunder so astronomical I had to reread it a few times to make sure I was reading correctly. It happens at the exact moment "then f(k') = k" is written: if it's not immediatly clear what's the problem with that, just read the previous two lines again until it is. There are other inconcistencies: for isntance in the second part of this proof the reasoning is sound, but it's written down in a way that's not sound and generally reeks of unprofessionalism. But I'll leave it up to you to find as many as you want, many aren't that hard to spot if you read through carefully.

Vor Tag## Xuancong Wang

@Релёкс84 So on which page and which paragraph does it make the mistake? Can be more specific? Thanks!

Vor Tag## Релёкс84

I read through the first proof, and right off the bat there's a math mistake that invalidates it, and that's not even mentioning all the typos. Is this paper a joke?

Vor Tag## Pasta Sauce

the answer is 4 3 plus one is 4, 4 multiplied by one is 4 so the answer is 4

Vor 2 Tage## FollowIfYouSwallow

Paul Erdos's ghost responded, watch the lights in the background. 20:32

Vor 2 Tage## Undervain

So wait if im right doesn't that mean 3x+1 is kinda every number almost. This broke my brain

Vor 2 Tage## Lucas Strange

What about starting with pi

Vor 2 Tage## Релёкс84

I'm sure you can see the problem with that

Vor 2 Tage## Bloop

It’d be 12

Vor 2 Tage## oncall2day

It is solved! 4,2,1. 4000 years for the Jewish nation, 2000 years for the gentiles and of course 1000 years for the millennial kingdom. Yes God's fingerprints are everywhere.

Vor 2 Tage## Kaios- Ken

It’s impossible that any number that shoots up into infinity because there are infinitely many multiples of 4

Vor 2 Tage## scott hermelin

x= -1/3 duh.

Vor 2 Tage## Fabian Pascal Abt

At least, assuming there is no loop, there are Infinite numbers that "dont fit in".

Vor 2 Tage## Emily Offenhauser

Hol’ up- I remember my 3rd grade teacher making me do this

Vor 2 Tage## ChevaliersEmeraude

I mean, it's impossible to go up to infinity or anything, but it seems like you can accelerate the process by eliminating any number below your starting number, assuming you already tried that number. Like, for example, if you know that 27 works for this, then 54 is also eliminated because you start by dividing it by 2, reaching 27, and you know that one works, so you know 54 works as well. Now that I think about it, ain't that the solution? Anytime you start with an even number, you divide by 2, reaching a number you already know works. And if you start with an odd number, you do the 3x+1, which will always give you an even number, which you divide by 2, and since you've already established that dividing by 2 will reach an even number you already determined to work, then that one works too! I don't know; of course it assumes you already tested earlier number, but as you test more odd numbers, you also reach higher even numbers to test. So you're really also testing higher even number while doing lower odd numbers. Again, I don't know, but it feels like there's something there...

Vor 2 Tage## Lygi

Its a amazing video but pls: Whats the name of the music which starts at 1:03 ?

Vor 2 Tage## rand debiel

To me, it makes perfect sense that you will always end up in the 4-2-1. When starting with negative numbers, you obviously should use 3X -/- 1 instead of 3X +1. Anyway, what if i, without any math education other than highschool, would proof this theory to be true, on 1 sheet of paper with more words than numbers?

Vor 2 Tage## Релёкс84

Your proof wouldn't be valid, but you're welcome to try

Vor 2 Tage## Bloodfetus

Would this eventually work for decimals too?

Vor 2 Tage## TriPyramid Pictures

Simple! 1

Vor 2 Tage## Mostafizur Rahman

Arts major: 😴

Vor 2 Tage## SB PRODD.

What if the only reason this happens is because one is also described as a perfect number

Vor 2 Tage## Robot Plays

All 9 years olds be like: this hurts my brain I'm losing brain cells.

Vor 2 Tage## 10 Points

I'm gonna ask my math teacher this now

Vor 2 Tage## Valentine Pantoja

Every number is just a representation of breaking down things to an individual being and/or a group of representatives of being (a single unique quark) (an inch/mile) (shapes), this equation could just be an Easter egg in the universe, or maybe we still have false ideas about math that are stunting us in understanding truth

Vor 2 Tage## Luke Hyde

Collatz conjecture? Completed it mate.

Vor 2 Tage## 738polarbear

It is NOT a problem at all What a waste of time this whole video is.I find it quite PATHETIC that people would waste time on this NON problem.

Vor 2 Tage## Africano Su internwt

Than why are you wasting time commenting and watching a video you define as a waste of time?

Vor 2 Tage## MJM

I don't understand. Why is this a problem?

Vor 2 Tage## Ivo Popov

what about imaginary numbers?

Vor 2 Tage## Glenn Clark

Well congratulations Derek. I think you've inspired a whole lot of people to waste a whole lot of time. :D

Vor 2 Tage## kajamix

This is the most boring video in youtube ever.

Vor 2 Tage## Africano Su internwt

If you cant understand it that does not mean it is boring. It just means you are to stupid for this and you should go play pokemon instead.

Vor 2 Tage## Fi Sophia

All number and calculations back to 1.because they come from 1.only 1 Almighty of God created all and u like or not for him u will returned.

Vor 2 Tage## ً

Quite simple, really; Who cares?

Vor 3 Tage## BJ Culpepper

Let me tell you the problem with this - academics only think academically so that usually miss the very picture they are describing. They say numbers are unusual. NO THEY ARE NOT. The FORMULA is the challenge - NOT numbers. You can create other RIDICULOUS formulas to create other snazzy questions - but it is the formula - not the number. Divide by zero creates an error. That is not a number issue, that is a formula issue.

Vor 3 Tage## Aaron Waterman

Ngl i didn’t even finish the video. Paused at 7:53 to test it out for myself. I think it will always end in 1 because when you add 1, you are essentially going through every single possible number till it is divisible all the way back to 1. Idk just an idea

Vor 3 Tage## J Modified

@Aaron Waterman 13 and 17, and for some seeds it apparently goes to infinity.

Vor 11 Stunden## Aaron Waterman

@J Modified Oh do they loop back to 3?

Vor 12 Stunden## J Modified

@Aaron Waterman No, 5x + 1 has loops that don't go to 1.

Vor Tag## Aaron Waterman

@J Modified Yes. Loops is the operative word. It will always return to 1, it'll just take longer

Vor Tag## J Modified

@Aaron Waterman That is not the case. 5x + 1 has higher loops.

Vor 2 Tage## Afterlife Voices

As a idiot, I would say 1 is an even number, in that other odd numbers are part of a multiple, and 1 is a singular.

Vor 3 Tage## Afterlife Voices

@Africano Su internwt How odd of you to say.

Vor 2 Tage## Africano Su internwt

1 is odd

Vor 2 Tage## jumpieva

I am curious if the disparity between looping patterns (the 1 so far with even, and 3 so far with odd) could have some bearing as to why there was an unbalance of matter/anti matter at the creation of the universe. It shows mathematics and physics could follow some strange rules or patterns we don't fully grasp yet. Maybe I'm not quite articulating this right though...

Vor 3 Tage## Arguan Modeth

I am not good at math but what is the probability of running into an exponent of 2 with 3x+1 added to the probability of running into a number smaller than x. Given that every 3x+1 meets a 1/2 and every 1/2 is either odd or even ... and if even it's 1/2 has a 50% chance of being even shows there are more 1/2s than there are 3x+1s so the number has to decrease over time given the law of large numbers. Since you only need about 2 1/2s to best a 3x+1 and your probability of getting a 1/2 is 100%+50%+25%+1/8+1/16. Added to the probability of running into an exponent of 2. Why did Collatz use 3x+1 anyway? Why not nx? Screw it I am giving a thumbs down.

Vor 3 Tage## J Modified

That same probability applies to 3x - 1, but for that there are higher loops.

Vor 2 Tage## nathan bullock

soooo if its not possible say its infinite. if you haven't guessed I'm stuped sooo...

Vor 3 Tage## Kanoosh YT

ok ok…. but what about 3x-1???

Vor 3 Tage## Vince Oh Myyy

the animations tho !

Vor 3 Tage## Alan H

What about zero, theory broken, noble prize please

Vor 3 Tage## Zacarriss

What about the Number 0 which is hypothetically below 1 but not a -negative number? 0 can Link with infinity by it's very nature as a closed loop... Indeed though this is likely outside the scope of the rules of the equation...

Vor 3 Tage## Xylofreeze

my brain hurts

Vor 3 Tage## Colby Chase

So like, I'm a nerd and find this very interesting. But I have to ask, what would we get out of proving/disproving the conjecture? Singularity?

Vor 3 Tage## Betta TV

So beautiful

Vor 3 Tage## beau dreher

The callous employee extragingivally afford because hill directly reduce aboard a hoc camp. voiceless, silky map

Vor 3 Tage## Jesus Suadi

this is a masterpiece of a video holly smokess... appreciate the music and sfx

Vor 3 Tage## Quapadople

0:21 It's not that "Mathematics is not yet ripe enough for such questions"...it's the fact that we can't measure or quantify something by lacking the cognitive capacity of doing so. On a case basis it might also be a useless and/or limited question...asked from the point of view and/or understanding of an individual that is unable to asses its own limitations. Humans have this tendency to think they know more they actually do...and we see that throughout human history from the consequences of our predictable actions. Understanding we cannot understand some things...it's a monumental achievement. I simply know that I don't know...and I'm okay with that. Edit: One may most probably be the universal balance just like everything around a nucleus core will find itself at position 1(one as in the prime order(measured numerically by humans) of universal balance). As an example...let's absurdly assume you could just poor water on earth enough to cover everything. The surface of the water will be perfectly balanced around the gravitational force coming from the center of the planet...and so everything will balance to the universal order of 1. Through force/energy you can change that order...but no matter what it will always come back to 1. Edit: 2 "The Simplest Math Problem No One Can Solve" can simply be a mathematical representation through numerical explanation of what we call and/or perceive as universal balance.

Vor 3 Tage## Saarphire TTV

watching this video at 11PM and literally feeling my brain melting... love it.

Vor 3 Tage## michael wilkie

answers 42

Vor 3 Tage## Picholas Cage

Pff easy, he said there were rules, no restrictions, just add 103 everytime you get to one

Vor 3 Tage