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

2021/07/30

私のプレイリスト

後で見る

Kent Schroeder 58 分 前

Try this stuff with imaginary numbers.

Brian Moore 2 時間 前

Does it happen with 999999999?

guyontheblackchair 2 時間 前

"Insignificant" and "almost" don't seem to be scientific terms. Especially "insignificant".

Joseph Tanous 2 時間 前

12:33 this one actually cracked me up lol

mario gamer pro 3 時間 前

Decimals

John Onyon 4 時間 前

I just love i have the option to be the first to dislike this! 588K likes, 0 dis likes! Veritasium rules!

Duggy Dugg 4 時間 前

what problem ? you ffing know the outcome.. pi has no outcome in a 10 based number system .. wonder how pi would do in say a 12 base counting system.. if for example we were born with twelve fingers vs ten

Faique yusuf 4 時間 前

I am choosing 0 ..so we don’t end up on 1-4 loop 🤣😂😂

MaveriQ 5 時間 前

The solution seems trivial to me. Since the result of all the odd numbers after the calculation 3x+1 will be always even numbers, and half of the even numbers will be even numbers, 75 % of the results will be divided by two, 25 % will be multiplied by three. ( The +1 can be neglected for this purpose.) Although I am not sure about the exact percentages, (maybe it is 75 % against 66,66 %,) it is trivial, that the graph tends downwards. Through the definition of the calculation it will continue, until it gets caught up in a loop. It seems, that there is no other loop than 1, 4, 2, 1, 4.... I cannot prove, that there is no other loop. Sorry for my German/English. I hope, you will understand my point.

crewlj 5 時間 前

Seems true enough.

crewlj 6 時間 前

What's the problem anyway. I don't have one.

David Anderson 6 時間 前

1 divided by 3=.333 repeating. you subtract one, then multiply by two. you never reach -6 because you're always dividing a -5.99999? decimal point

David Anderson 5 時間 前

also something along the lines of negative two thirds being an even number (and/or -.6666 repeating) being an even number, thus allowing the equation to be able to repeat, based upon the fact that its allowing itself to repeat if negative two thirds is an even number, because one fraction with the same denominator will always add equally to itself as long as it has the same denominator, if -.666 repeating can be considered equal than the equation (in my opinion) is just a repeating pattern of fractions of itself, itself that never reaches -6 because of the way that it's represented WITH numbers (specifically the number one). making fractions of 1 whether negative or positive numbers still represents what the equation is doing, and maybe that correlates with how the original equation works although that's all i have to say about this.

David Anderson 5 時間 前

got me thinking the numbers you can put into the original equation is infinite but any number you put in the original equation will not result in infinity, that only the number of numbers that can be inputted into the original equation is infinity with the result of the orginal equation always being the same and never infinity

David Anderson 6 時間 前

there is no way you could reach negative six because you are never dividing -6 by three, and if you are dividing a -5.9 repeating number by three you will never get the number -two because -5.9 repeating will never divide by three to result in -two. -5.9 repeating is not -6.

TankerCT 6 時間 前

"Pick a number, any number." *thinks of five. "Seven? Good choice." Oh, alright then.

Lanc Lancc 6 時間 前

i?

Michael Kiaunis 6 時間 前

" by brute force" i love it !

z Morrell 6 時間 前

There's no end statement therefore the formula is incomplete.

Jesse 7 時間 前

I think you have to prove a number is even before you can solve this problem.

Clarens Jeanty 7 時間 前

When you have this math problem what are you trying solve, what is the objective, example your trying to find what 3+1= to.

Game Coder T 8 時間 前

zero. 0/2 = 0

Ken Surrency 8 時間 前

This really puts Newton and Einstein and people like them into perspective. They were on a completely different plane of existence than the rest of us.

Pastor Sabloon Nanda 8 時間 前

Awesome 👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻👍🏻 it's totally mind blowing.........

Nikos Pharma 8 時間 前

Question: How can you prove that 1+1 = 2 ? I do not say "know", I say "prove".

Santi Fresnel 8 時間 前

probably something at quantic level might explain it one day.

Rabia Farhan 8 時間 前

Try 3 6 9 because these are the keys to the universe then tell me

Reuben Manzo 9 時間 前

I have a number for you to try. How about 0? 0 is even, x/2=0. Ta-da. There's your independent loop.

Pola Pola 10 時間 前

E

fred 12 時間 前

What was the point of this exercise?

#JustPhilosophy 12 時間 前

Feeling sorry about this bald guy who wasted 20 years of his career in 3x+1 😂 We created the numbers and now we are suffering from these :p

Robert Cooper Jr. 15 時間 前

I'm not a mathematician, but I don't understand why this can't be solved by the logic that since numbers are infinite, this series must hit a power of 2 at some point.

Suhayl 15 時間 前

with all due respect, what exactly is the problem we are trying to solve here? could I be so dumb that I can't see the significance of any effort spent here?

Cobus Potgieter 15 時間 前

its easy, Any number??? just use 7,3 which is still a number and it keeps climbing

supercellonova 16 時間 前

To solve this, new rules may need to be applied. If you build a circle of rails, it doesn't matter what kind of train you put on them. The circle is constant until you bust it open, and lay more rails in a different direction.

SpicyCombo 16 時間 前

math

whITe dEVIL 16 時間 前

10:35 my financial goal set

Hart Matejczyk 17 時間 前

I feel like there is a solution to 3x+1. Whether a number goes up or down in a movement is binary, either up or down. If any whole positive integer is multiplied by 3 and added to 1, the result is always even. That means the next movement will be down (x/2). Multiple downward movements in a row are possible, but multiple upward movements in a row are not. If the number goes down, it always goes down by 50% (X/2). If the number goes up, it always goes up by a percent somewhere in the range of 400% (1*3+1) to 300.01% with the .0 repeating. (I can't figure out how to type a combining macron). Let's call that change 300% to make things simple. So when a number goes up, it goes up by 300%, and when it goes down it goes down by 50%. (side note: It seems that the extra possible 100% in upward movement is irrelevant because if any number is hit that exists within the sequence of an already tested initial input number, then the sequence from then on is identical. That means the initial input with the equation applied will reach the 4, 2, 1 loop. It seems the numbers that have been identified as existing within a tested sequence go high enough to make the extra 100% irrelevant. Either way, proving that won't be necessary to prove my statement.) If numbers can go up by 300% or down by 50%, then for a general downward trend, downward movements have to occur at a rate of just higher than 1.5x (or 3/2) upward movements. For example, 10*(2/3)*3/2=10. If there is a general downward trend, starting from any positive whole integer, then the 4, 2,1 loop has to be reached. So all we have to prove, to prove the equation is always true, is that downward movements will happen at a rate higher than 1.5x upward movements. Statistically, this is inevitable. Remember, upward movements will never repeat, but downward movements can. I don't know the exact percentage chance that a downward movement will repeat, but I don't think it's necessary to. If there is any chance of a downward movement repeating twice in a row, and that chance is consistent throughout the entire sequence, then eventually, given infinite opportunities, a rate of higher than 1.5x will be reached for long enough to reach the 4, 2, 1 loop. I am not a mathematician, I'm a startup founder without a college degree and I'm sure I got some things wrong here. Please tell me where my approach and assumptions are false!

Dudeguy21 18 時間 前

man this was a good video. thought it would suck

Companion of Owls 18 時間 前

5 minutes in and I don't know why I'm still watching this

Mike Sadler 18 時間 前

...but isn't '2' an odd number... and prime... because it can *only* be divided by itself AND 1?

Bob Crane 18 時間 前

Mathematicians claim fluctuations of the stock market is random, psychologists and traders tend to disagree.

MrDuran9000 19 時間 前

Very deep buttt, where is the ZERO? Everything returns from where it came. Without a space ( or zero ) nothing can exist that is. No better than the CHICKEN OR THE EGG! Hmmm?

Jose Negron 19 時間 前

I don’t what the problem is. All I see is once you hit a multiple of 4 you going down to one.

Sacredservants 19 時間 前

We can teach an AI to play dota 2 and beat all the worlds best players by playing its self for over 1mil hours over by runing countless games at the same time improving each time. But we cant stick an AI over this? ... dota 2 is more important than math ?

JDogJunior 21 時間 前

Wouldn’t you not be able to solve because you have no value for x!?!?

Scott Bros 22 時間 前

It's actually not a math problem but rather a fun puzzle unless you apply it to the depopulation dream/problem of the Elites but they have moved on to other more practical methods of depopulation as the puzzle is just dreaming not practical. 🤓

Monte Wills 22 時間 前

Amazing and mind boggling

Travis Little 23 時間 前

its just one of many algorithm used to generate the simulation

oddjobbob 23 時間 前

The problem with the idea of solving 3X+1 (did I write that correctly?) is that it’s not really a problem. Problems have solutions. Solutions are answers. Answers have questions to which they are, well, the answers. 3X+1 is not a question. It cannot, therefore, have a solution. It’s a quirky thing. It probably gets a geeky math dude laid if he meets a half-drunk skank in a bar. Or a math-dudette laid anytime she wants because, well, let’s face it, 95% of all men will put it in any hole they are offered and the other 5% are liars. But there is no solution because there is no problem. And the only people who want to talk about it as if it were a problem is that bald guy or that Terry guy who probably couldn’t solve a real problem like, why the hell does my toilet clog when my turds are hard little turdlettes, even if they are only 6 or 8, but flows smooth when they are nice and soft like fiber filled granola, even if they flow from my bowel like the mud from the Mt. St. Helens eruption?

J Modified 22 時間 前

The problem is whether the conjecture is true, false, or undecidable. The answer is a proof or counterexample.

oddjobbob 23 時間 前

Is that bald guy actually getting paid to do whatever it is that he does? Or that Terry guy?

Tandem Axle 23 時間 前

This is a perfect representation of my stock portfolio.

Kathy L 23 時間 前

What happens if this is done with numbers of a different base?

zdcyclops1 lickley 23 時間 前

Dr. Muller should know better. When pronouncing numbers and is only used to indicate addition or the decimal point.

tyrolias 日 前

12:34 made my day

rick pilotte 日 前

I've stumbled upon a very interesting math axium that might relate to this...and other number progressions: any square multiplied will always be one number more than any pair of numbers on each side of the square, and any number cubed will always progressivley gain one number when all three numbers (the number cubed, and the numbers on both sides of the number cubed) are multiplied together.

Jim Balter 23 時間 前

That's neither interesting nor an axiom, and certainly doesn't relate to this. (x-1)*(x+1) = x^2 - 1 is simple algebra.

The Comedy Pilot 日 前

But this is a math equation that was invented by a human being, it's not a legit form of math that is an existing thing in life, it's just a neat little equation that a fellow human invented, so this video is ultimately pointless.

Chaos Ordeal 日 前

Simplify the rules to: if it's odd, you multiply by 1.5 and add .5, and if it's even you halve. The problem includes an extraneous step. If 3x + 1 is true, I suspect that it implies that "infinity" is only conceptual, and not a real in any way, and that seems unlikely (but possible). How could numbers be infinite if there is no series of numbers that forever reach an odd number when multiplied by 1.5 ?

Jim Balter 日 前

That's not simpler and doesn't help.

U X 日 前

mimics the fate of most stocks lol

SPORT SKILL 日 前

But have the tried 69420

S S 日 前

So,How does one apply this in real life

steven nixon 日 前

it's easy, it's only because you're using two concepts (even and odd) that start with an odd number, always, and end there. Natural numbers vs. integers just a curiosity that's all

Dennis van Donkelaar 日 前

It's actually simple. You are looking at uneven numbers. You should look at even numbers. The *3+1 always is always even which is dived by two untill it is one, so this results again in an uneven number or an even number. Since you divided the even number with 2 and multiple the uneven number with 3 and add 1 this number will always be bigger and a loop is there making the even number bigger untill you get in the loop again that finally result in 4, 2, 1, because if an uneven number comes again it makes the next even number bigger and bigger untill it gets an even number that finally results in 4,2,1. It's just obvious thinking there is no number which not follow this rule only numbers with comma. Its a simple rule the uneven number results in an even number going to 1 or the even number is getting bigger untill it does, because an even number goes to 1 finally or it gets bigger. *2+1 wouldnt work, because it would not always result in a bigger even number when an uneven number comes. *3+1 is the first it works for. *5+1 *7+1, etc to infiniti should also work.....

Dennis van Donkelaar 22 時間 前

So I proces iT today not by proving that any number would not work, because the formula is infinite so there cannot excist any number that not work

Dennis van Donkelaar 22 時間 前

Looking for a number that doesnt work is like looking for the end ik an endless infinite loop like this is also

Dennis van Donkelaar 23 時間 前

And nobody can ever prove a number excist that would not work because your even number just would Get bigger and bigger and iT infinite so No computer can ever find a number that doesnt work iT would be really Stupid to even try because every even number would just increase infinite iT means You can never find a number that doesnt work iT could only if number were not infinite but with every number You can put a 1 or a zero after amd iT is ten times as big

Dennis van Donkelaar 23 時間 前

I did but is is simple there excist No number on which iT dont work because You just iterate through the even Numbers until iT works

Jim Balter 23 時間 前

It's not simple ... maybe watch the video to learn why.

Andril Chakraborty 日 前

2

Andril Chakraborty 日 前

4,^ 54

Andril Chakraborty 日 前

9000

Andril Chakraborty 日 前

Ah Soviets

MusicalJackknife 日 前

I learned about this through Trolley Problem Memes a while back and loved the idea...not good at math but it's cool to learn about!

Nicholas H 日 前

As a computer scientist myself, I was going to say, if anyone will solve this, it will be John Conway. I met him in once after he gave a lecture at my university in the early 90's. He unfortunately passed away in 2020. Absolutely brilliant man.

Jim Balter 日 前

If anyone will solve this, it's Terence Tao. Conway's FRACTRAN was just an illustration of a point, it wasn't directed at solving this.

TheNextGenLyrics 日 前

My friend: "Math is a easy subject" 3x+1: "Well, actually..."

Dreamthepogger 日 前

the answer is 4x

LightningSNTz55 日 前

AAAAAH I CANT STOP DOING THAT

Alex 日 前

The 3X+1 problem is innocent enough on its face value, but in reality it’s nothing short of gazing into the very nature and the workings of infinity. There is, however, an interesting observation here: As also pointed out in the video, both the 3X+1 problem and the stock market are examples of the geometric browning motion with the difference that over the long run the 3X+1 problem trends downwards while the stock market trends upwards. Mathematically, of course, we know exactly why that is, but in the case of the stock market the long-term upward trend must also have something to do with our inherent psychological desires for growth and positive returns. The rules influence the outcomes, while the humans influence the rules.

Thorsten Wiegand 日 前

If you try using 3x-1, you also get more than one loop (in essence it´s the exactly same as using negative numbers on 3x+1): 7x3-1 = 20 -> 10 -> 5x3-1 = 14->7 / 3x3-1=8 ->4 -> 2 -> 1x3-1=2->1 / usw. / Conclusion: Because of the +1, you will always land (earlier or later) on a 2 squared number (2, 4, 16, 32 usw) and because of that you end immediately in the 1/4 Loop. Using -1 instead can get you to end on a prime number.

Thorsten Wiegand 12 時間 前

@Jim Balter : I meant to write: "to end on a different prime number loop". Sorry for the confusion. Yes, you are correct. The is no "end" with this formulae, it´s always a loop at the end.

Jim Balter 日 前

With factors other than 3, e.g., 5x+1, 7x+1, 2x+1, it's easy to prove that they either loop or go to infinity. 3x+1 is special because it's right on the cusp. " Using -1 instead can get you to end on a prime number." There is no "end", for either 3x+1 or 3x-1, and prime numbers appear in both these formulae and any other formulae because they are applied to *every* integer.

Smith Hoowe 日 前

3X+1 is what cryptocurrency is made of.

myth 日 前

math is not all about complex mathematics questions, u just need basic mathematics in ur daily life to live :) if u know 1+1= 2 then that means u know math :)

Paige Crowley Jones 日 前

no? you need a lot more than that in your daily life, and way more than that for specific jobs that are very important for our lives.

Michael Hargrove 日 前

What a stupid, pointless exercise. Like betting on both red and black at roulette...

Young Chip 日 前

give me like max 3 months ill make a program and solve it hold my keyboard

Caleb Glover 日 前

It is a mathematical spiral ... the odd even rule is the polarities and the 4 2 1 loop sequence is when it returns for another loop .... should switch a 1 to a negative # if it repeats

Tomasz Solga 日 前

Guys I know it is interesting problem, it is algorithm and to be honest i know that math is just a tool, very useful tool to prove concepts we cannot imagine. And as far as i know it may be impossible to prove, but what is more important why should we prove it? Why we should lose resorces to problem that may not have any application, to problem that is made up. Anyway i think that 3n + 1 sooner or later will became 1... sooner or later going to infinity it will be for example power of 2^n - yeah i know i just stated it, it is not proven - but im sure u cant prove it. I think that this is mistake and we have to figure it out how to go around, this way around for sure would solve problem with prime nombers too i guess.

Angelos Batalakis 日 前

It becomes clear that the algorithm will hit a power of 2 at some point.

Mariamleshious 日 前

Thanks

Franck Freeman 日 前

this is not math problems, they just inventing problems for no reason

Orion Federation 日 前

So 3n+1 is the Satan in the world of mathematics.

Anders Mellgren 日 前

Damn! I just wasted 22 minutes on the infamous 3x+1 konjecture…

Dylan Paul 日 前

3x+1=3x+1

Gabriel 日 前

Never in my life have I found maths so damn fascinating.

Nugget Entertainment 日 前

I'll tell you the real problem. How can you prove if it goes to infinity if our ability to count isn't infinite? Even if infinity happened... Would we be able to prove it's really going to infinity?

Paige Crowley Jones 日 前

well that’s what proof is. showing that something is always true no matter what value it takes.

Humberto Ramos Costa 日 前

11:50 In fact any sequence that goes to 2^n formula will end the sequence...

Scott Morgan 日 前

That was three minutes of my life I will never get back.

The Crowned Crown 日 前

*Pick a number, any number.* Me: Okay, seven *Seven, good choice.*

Elliot Hansen 日 前

i told my teacher about this and they struggled so much

Marius-Daniel Țuțu 日 前

What if we would take 3^2 x 3 + 1 = 9 + 3^2 x 2 + 1 (10 + 3^2 x 2) / 2 = 5 + 3^2 Is it cheating? Even though we know that the result is fourteen but it doesn't look even so we grow the number again

Cotronixco 日 前

If 1,000 people continue to throw a ball at a basketball hoop, starting from various points, eventually they will all go in.

vineet kapshikar 日 前

Mathematics is a concept. It is intangible, our invention. It will definitely have some dead ends.

Ravi Kumar 日 前

There is nothing special in it.who told you to multiply by 3 and add1 then divide mby 2.fools can do anything

Elven Justice 日 前

3x+1=4x

Keyser Smöze 日 前

A counter example would have to be in closed form proof. You can't just chase a sequence to infinity and prove it goes to infinity just because it farther than you have ever been.

Randy Jurgensen 日 前

4

Nick Saltern 日 前

At a guess, I would say every prime number will hit another prime number just before it's final fall to 1. Don't know the significance of this. (29)x3+1=88, 44, 22, (11), 34,(17),52, 26, (13), 40, 20, 10, (5),16, 8, 4, (2), 1. I wonder what happens if you times by 3 and minus1 3x-1

Nathan Mostoller 日 前

I solved it, but I’m not telling you what the answer is…

turtle 日 前

Im sure you did

Tako Boutit 日 前

I feel like we're looking at it the wrong way - Our mathematical definition of an odd number is: n = 2k + 1 or n = 2k - 1 And even numbers are: n= 2k. First of all, if you're using a formula that uses our first two prime/odd numbers to get it to an even-enough of a number to eventually divide it by 2 until it reaches 1, of course it will eventually be divisible by our smallest even number because all even numbers are divisible by two. While, the purpose of the 3x+1 is to just get it to an even number so it can be divided. A) The better question here is, why are we so surprised that it CAN go on to infinity? No matter what number you grab, it will always come back to 1 simply by the nature of how we structured math to work. The foundation of the way we compute numbers will always make it so. Multiplying an odd number by 3 and adding 1 until it equals an even number, will never not give you an even number, and the cycle will continue until eventually it just flows right back to 1. Perhaps the question should instead be: Why is it that we structured math in the way which makes this possible? B) What actually ends up shedding a bit more light on this issue is: Why is it that when we change the formula to 3x - 1, we do not get the same loops - instead, we get the following loops: 2, 1 8, 4, 2, 1 20, 10, 5, 14, 7 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34, 17 In theory, as stated above, n = 2k - 1, to get an odd number in the same manner as the above formula, we should be able to arrive to the same number using 3x - 1 but we can't. Hope this was an interesting thought for you all! Thanks! :-)

Jim Balter 日 前

"the purpose of the 3x+1 is to just get it to an even number so it can be divided" No, it doesn't have a "purpose". "why are we so surprised that it CAN go on to infinity" We don't know that it can. " No matter what number you grab, it will always come back to 1" That's the thing to be proved. "simply by the nature of how we structured math to work" No, this is mistaken and is a rather fundamental misunderstanding of the problem.

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