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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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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

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

What is the question here actually?

I solved it in 3 miliseconds

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

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

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

Russians: Now, try 2x + 1

@Релёкс84 ackchyually lol

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

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

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

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.

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

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.

so my teacher was pretending shes smart huh

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

what about -1

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

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.

answer : don't look in 2 dimensions

That's an advice, not an answer.

Programmers know how to handle recursion 😉

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

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.

hmmm 3x+1=? 4

Looks like a penny stock chart.

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?

10?

555555555555+ 39 = 555555555594

I know the answer It’s 4 :)

PIx10^4364345 maybe

That's not an integer though

We are all one

What is the question? What is the problem?

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)

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

¶-2

Sorry radians

Try using substitution method using 2pi against 360

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

Where you at prince charming? 🐯

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

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

😆 Never ending loop

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

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

Rare number 1 Answer = ∞?

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.

@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.

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

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?

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

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

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

What about starting with pi

I'm sure you can see the problem with that

It’d be 12

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.

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

x= -1/3 duh.

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

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

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...

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

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?

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

Would this eventually work for decimals too?

Simple! 1

Arts major: 😴

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

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

I'm gonna ask my math teacher this now

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

Collatz conjecture? Completed it mate.

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.

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

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

what about imaginary numbers?

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

This is the most boring video in youtube ever.

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.

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.

Quite simple, really; Who cares?

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.

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

@J Modified Oh do they loop back to 3?

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

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

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

Like i’m pretty sure if you used any odd number as your multiple, and then just + 1, always divide when even, you will end up back at one, because the formula is essentially going through every possible number when you + 1

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.

@Africano Su internwt How odd of you to say.

1 is odd

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...

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.

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

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

ok ok…. but what about 3x-1???

the animations tho !

What about zero, theory broken, noble prize please

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...

my brain hurts

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?

So beautiful

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

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

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.

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

answers 42

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

How come the number range doesn't start at 0. That would solve the problem, wouldn't it, as 0 would be an even number?

Well, 0 is a trivial loop. But that doesn't say anything about other numbers, though.

-7 - the cycle of the number -7

Hi! Why not use negative numbers?

I found the solution. I´ll publish in an academic paper soon. Working on it.

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 possible valid value for X. 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 2 being even and prime, and 2 and 4 are both even, and 3 is eliminated entirely from the solution set, it creates a barrier from continuing the equation past 4, since the only multiples of 4 are, 1, 2, and 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 y = 3x+1. Because we can only use integers, and not all real numbers, this limits the number of possible solutions to any given value of X. So you are starting the problem with the false assumption that 3x+1 is a linear equation (which it normally is), when in reality you are limiting the values of x to positive 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. 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 accept positive integer values of y, given x, then it becomes undefined.

@KaiserAndFriends I appreciate anyone who gives me information that I don’t have. So, thank you for pointing out my error. I have edited it so that it removes any reference to 1 being a prime number. ;)

@Sima Sima so whats is your solution

@Flapjack Mollases I understood your point of view. This shows you do not have a one dimensional mind

@VatanTURK I don't like Einstein anyway I respect your idea , actually noone can feel the smart people :)

@Flapjack Mollases I mean i showed you a mistake you made in your "theses" i would concider that somethin pretty beneficial if i was you...i basicly taught you something, for free! your welcome!

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.

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

3×+1=10

3×+1 could be An infinite ammount of numbers not just 10

Simple

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 < 1: 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 < 1: 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()))

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

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