Publisert 30. juli. 2021

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

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.

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

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

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

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

▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

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

## Kommentarer

do it with 350

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?

Isn’t it 10

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!

I LOVE THIS PROGRAM!

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.

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

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

Another great video

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

Humans can live happily ever after without 3x+1

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

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.

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.

lol fractrum what a load of rubbish

the logo of the video

its 10

Does it only work with whole numbers tho

Just gonna throw out square root of 421 Billion

AH... My head, hurts

The number q is not in this loop 😎

Me: Going Try with a decimal

the problem is in the rules...

POV You discovered NOwine’s algorithm

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

Isnt the answer 4 or 6?

Pointless, like all human thought!

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

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

My brain stop working

This conjecture is never ending and will always end up in the loop because between all of the number you could choose (1-infinity) you can find a sequence to land on a specific number. This way of finding numbers in the sequence is why it will always lead back to 1, because any number can be found with the right number chosen to be "n".

when the algorithm lands on any power of 2 it will reduce to the loop. The real problem is whether or not the algorithm will always get to a power of 2.

i watch these at night to go to sleep because i don't know math :)

Well this is all well and good. But what I don't understand is, why is it even a problem that requires solving?

This formula looks like 3 dimensional space plus 1 time.

maybe just remove number 9 in number system. Try to convert 10 to 11. I think no one try that. maybe I sound stupid but I'm just trying to help.

I have a counterexample, its Bannana sundae Bannana sundae to ??? To bannana sundae. *This is huge*

The answer is 1. Why is this the issue? What am I missing?

The simplest problem that is impossible to solve This dude: *shows how to solve* Me: you said it was impossible

I don't understand why it's so hard for anyone to solve. What's 3x+1? It's addition. People always overthink things.

Yare yare

whats wrong with a loop? many things in life repeats itself in one form or another. life still goes on.

I like math, but this is the dumbest thing I have ever seen. LOL

Well... "why is using negative numbers giving such different results?" Basically 3n+1 with negative numbers is the same as 3n-1 with positive numbers. (at least if you compare based on absolute values) And that is trivial - it is the example in math lectures/... for "why is Collatz's problem so hard? Others are trivial!".

Isn‘t the solution simply the second law of thermodynamics? As we start with a finite number (think of potential energy), there is no additional energy to be gained by a simply oscillating process such as 3x + 1 / divide by 2, robbing the system from any external energy input, therefore the process must ultimately come to a halt (4 2 1 sequence). Applying an oscillator just adds random noise to the system that masks the inevitable loss of energy, confusing everyone.

I think its pretty obvious that if you keep making every odd number even and diving the evens by 2 you'll always get to 1, thats isn't surpring at all

@Mood Swings But by your logic, 3x-1 should get to 1 even faster than 3x+1, since you're still halving more often than multiplying, and after multiplying you go down 1 instead of up 1. It does not though - there are several higher known loops in 3x-1.

@J Modified if you only multiply by 3 the odd numbers then you’re halving more often than you’re multiplying, which would eventually decrease it to 1

Not if you multiply by three before making it even. I mean, if you make it even by subtracting one instead of adding one, it doesn't necessarily get to one.

This is so interesting! it seems that any number that is a power of 2 will eventually lead to the loop of one. So with infinite opportunities, you will eventually come across a number that is 2 to the power of x and cause the loop 4-1 loop!

What is there to work on?...

I chose 5 :(

because we dont know 3x+1 all the things we can solve are miracles ? he lost me on that one lol becuase we cant solve an "easy" problem? why does he think its so easy if it hasnt been solved .

they should have a computer running all the numbers forever or until we find something

The problem is mathematicians. They are too lazy, unmotivated, and underpaid to keep calculating into infinity ☺️ The “loop” cannot be “dismissed” until we run it an infinite number of times.

Wait but you said there are 3 loops instead on 1 for negative numbers? That’s an asymmetry, just like one thats supposed to have happened in the universe so the matter “defeated” the antimatter

You must have an olympic medal in jumping to conclusions

I feel like I wanna study this more but I'm a dumbass. Why does math have to be so hard.

Ha ha.......this is why I hate math

Question,¿this sequence can mean the multiverse right? how all starts at one and reduced to one, like also oposites, this could be a type clue that says that owr own universe can have infinite big bangs and big crunch. But I dont have all information to say its real, it just an observation. Becouse, Reading my comment again, also ¿whats zero?, there are many unknowns, its really interesting all what means science.

Me: *watches video about basically unsolvable problem* Veritasium: young mathematicians are warned not to try to solve this Me, a ten year old: *solves after 5 weeks of thought* I posted a proof on my channel!

It's up!

@A math geek with way too much time on his hands I'll gladly hear from it. The proof, not the rugby game, of course.

Will update video later. Got a football game to watch

@Релёкс84 but I did prove the first thing. That’s all I actually did in the video.

@A math geek with way too much time on his hands "See, all I need is that all odds reach a point below themselves." This is true. If someone shows that, they will have proven the Collatz conjecture. "I can say without a doubt that all evens make it if you just keep dividing." ...And this is where you take a leap of faith. You say "without a doubt" when really you should have a doubt. Look at the secodn half of your answer again. You basically say "no matter what, it must be true" without showing any concrete proof, and that's the cricital part - there currently is no proof of that statement, and you're using it as if it were obvious. Let me remind you that what you're saying is, any even number goes down to zero. Except, every number, no matter if odd or even, will have at some point an even number in its sequence: this is extremely easy to prove. So you're missing something, and the thing you're missing is the thing everyone on Earth who's ever worked on this problem is also missing. You will not find it, but you're welcome to try, though.

Me at 1:12 am and i dont even freaking like math.

If + = positive then the answer would be 3 Edit: or +3

How about decimal numbers?

@Релёкс84 Thank you for your explantation :)

@Dave Gilmour The rules of this problem rely on the parity of the number: multiply by 3 and add 1 if odd, and divide by 2 if even. But the problem is, only integers can be odd or even - that concept makes no sense for decimal numbers. Now you could adapt the rules to include decimal numbers, but how exactly to do that, and whether or not it's mathematically interesting, is up to you.

@Релёкс84 Err, does this work if we use decimals and not just whole numbers? I thought that was pretty obvious in my question. I'm not a mathematician, so I would like to know - does the number always come back to the 4 2 1 loop if the numbers are decimals or even power numbers)?

What about them?

On the negative side, are they testing 3n-1 or 3n+1? It seems like you get 3 separate loops because you’re adding one rather than subtracting; maybe you’d get the same 3 loops if you tested 3n-1 is positive integers?

Whats the problem here?

In the beginning math was created to explain physics. All physics. But to create something useful required an unsolvable equation. The wheel, or a perfectly round cylinder. The equation to get there is pi. The math will only get Us close. There is no conclusion to the math yet we can make things perfectly round, or at least it seems to be

I found the meaning of life while watching this.

let's s assume that you find a number that doesn't loop to 4-2-1 then WHAT ?

I guess you get a million dollars. The rules on the prize are a bit unclear on that though. I skimmed them and couldn't tell if a single counterexample would be an acceptable proof for prize purposes.

this has cryptographic uses.

If there are many higher loops and a way to find them, then it could.

1:26 My mind: it's just like saying 1 2 ka 4 , 4 2 ka 1 ,, my name is Lakkhan 😂😂😂 Only Indians who watches bollywood songs and movies will understand 😂

To solve it you must include the value of Pi divided by 2 in all sequences and loops, and that will show you a path to enter the Matrix...Then just follow the white rabbit...eventually you will see it when you reach infinity...simple!

How about Pi? Is that fraction finite?

I'm not sure what that would have to do with the above video, but the decimal expansion of pi is indeed never ending, if that's your question.

me as a 9yr old but my iq is 144 and i do yr 9 math (not bragging lmao)

@J Modified OK

@Eimai Xazos Why? It's not that unusual - 1 in 300 or 1 in 600 people depending on the IQ scale are at or above 144.

Doubt it

Not watching the vid but the thumbnail equals “Y”

Because Y=mx+b

By simply adding one, the row 1..9 is shifted and we get more even numbers in this row, if there are more of these numbers, then more often there will be division by 2, which will tend to decrease rather than multiply. That is, by adding one, you actually first chose not 3, but 4, and so on from 2x to 10. Adding one always gets more even numbers than odd ones.

Soviets: Yes ! we broke him! He is now a mathematical nihilist

Still wrong tho. You've applied a ÷2 that doesn't exist. So 3x +1=y (3×X)+1= ?+1=Y. Thats the correct answer. You don't have x. And you never said to suddenly ÷ by 2 or continue to repeat the formula for your set of new unwritten rules. Therefore the question is click bait!!!!

Mathematical analog to life. People live for very different lengths of time, but they all eventualy die.

😐

time waste

This is NOT a problem. A statement of how a Particular mathematical function works. And it can’t be proven for every number even N plus one can’t be proven because not all numbers can ever be tested

A proof does not require testing all numbers.

computational equivalence 😄

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I like the comment where the guy says other proofs are actual miracles we don't deserve

💩 + 💩 =s Liberals brains 🤯👍🏼

I got lost in the middle, but I'm here to look at smart ppl, dreaming of becoming one.

Does this still hold true outside of base 10?

Nothing in the process depends on the number base or representation.

so do these numbers include . values or just whole values so 1.5 > 5.5 > 17.5 > 53.5

Oh it's simple you just add 3 + 1 which is 4 so x is 4

I got into a programming logic challenge where I was supposed to explain this thing... Then it clicked! "I saw this problem before!!!" And that was the video I heard about de 3x+1 for the first time. Thank you my dude!

i picked 7, im nervous now

My head ached from watching this video... Nice job!

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if you can solve this problem then i think you are one step closer to eternal life because all cicluses end where they started just like life and death

Does he explain why you can't use negatives?

Since negative seeds don't end up in the 4-2-1 loop and since there are several loops at low negative values, I suppose it's just to make the conjecture "clean".

I would say 3x1+1 =4 but its rly Not that easy .

“Don’t talk about this if you want a career” Veritasium:

Soviets are smart, huh?

I don't see a problem here

Please tell me what is the unsolved part in collatz conjecture and if I solved it where should I submit it and get the prize Please

It is unknown if the conjecture is true or false. A solution would be a proof either way. To have a solution considered for the prize, you must have it published in a respectable mathematical journal. If you actually have a proof, that will not be a problem.

13:02 me on my scientific calculator at school for some reason:

Very simple. 3X=1. Since X =0, you have 000+1= 1. Don't need to go thru loops.