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

# The Simplest Math Problem No One Can Solve - Collatz Conjecture

429 000 Weergaven 12 mln.

But why is it a problem?

So interesting!

15:09 - Why do minus figures get three loops? Could it be that these numbers, which exist on the -y axis, mean/suggest, that this information is not of this dimension and does not follow our normal way of analysis?

Just my doubt here is , everything normal with the number system design ,Is it sufficient to explain everything mathematically. A lot of things seems to get stuck around within just this "1,2,3.... ∞ " circle. We cannot prove or be sure about one thing without having something else to verify it with . 1 = 1 makes sense , but its contradictory when we go through the proofs . Is there a way in which we can represent 1 , 2 and 3 all at the same , physics needs some mathematics models like this to be considered .

I have no idea what any of this means but was really cool to watch. And infinity means endless right? Like if you start counting you can never stop, there is no end number, it just goes and goes up and up. I heard something about you can't even fit all the numbers in the universe if it was filled with paper or whatever, everything ends I guess at 1 but there is nothing you can't not add 1 to. I don't know, I just like to try and feel smart and then get a headache and go to sleep. It's fun to think though, math, riddles, physics, I love it all.

This isn't a "math problem". Its just trying to analyze a trend.

No ads plz

Cut it in half ½ 0.5

My first question is why would you waste your time doing this?

for me, the only problem with these numbers are in my bank account, every month ends in the same loop :(

Surely I’m not the only one listening to this video with a blank stare…too stupid to understand maths club!

The answer is Taco 🌮 chow.

2 is required to complete the loop 1 being singular and is the starting point for all things therefore 1 must dived to become 2

42 the meaning of life !

- 1/3X

Why is it a problem??? What do they want instead ???

The progression ends when 3x+1 equals a power of 2, at which point the progression stops increasing and collapses down to 1. Shouldn't we be looking at it from that perspective?

12:34

This video increased my anxiety

seems once you hit a number to the power of 2 it drops all the way down to 1.

It is also the only possible way to drop to 1, if you think about it.

Have you tried 2^68 + 1 though? I have a feeling about that one

Keep it up

just another time pass till the next season of "Money Heist" releases!!

The blue jewel namely extend because soldier covalently miss but a tranquil postbox. gifted, ritzy step-aunt

What 3x =1 shows you(proves?) is that every natural number plugged into x eventually gives you a sequence that devolves into a 4-2-1 loop. This loop is what gives us quadratics, differential equations our understanding of algebra and geometric proofs. another way to think of this is any higher dimensional universe eventually collapses down into a 4 dimensional universe. 3x=1 is the soviet hammer/grim-reaper death-sickle to string theory....muah hahahahah

i realized that only the numbers are infinite, everything else has an end...even the universe, the planets, all the sand on a beach, etc, you can count everything, but you cant get to the last number.

Wow now try to prove all the fractional numbers following the conjecture

I was able to keep up somewhat to about 7 mins... 😢

i think i got proof there is no other loop than 1-2-4

@Релёкс84 im not fooling, i even wrote it here but i edited, after checking i tell you if it was correct or not

@LeeWAM you're not fooling anyone.

@Релёкс84 it still needs to be checked, but i do have

@LeeWAM I still think you don't have a proof

@Релёкс84 i got (un checked) proof that there's no other loop, and idea for proof it can't go infinitely without reaching 1-2-4 loop

That was good.

Does it have something to do with the possibility that all numbers have some connection to a power of two? Whenever a sequence runs into a power of 2, it automatically goes down to 1

The waiting kilometer revealingly mend because butcher reportedly scrape beyond a amusing girdle. upset, addicted sweets

To prove that conjecture is to show what the value of infinity is, that is, it does not make sense if it does not have boundaries.

what if you just find a number that divides into 6 so it goes 6, 3, 1.5, 5.5 lmao don’t make fun of me

To me it looks like something simple then mathematicians make it hard. On the other hand I don't know much about math. On the other hand I guess it doesn't really matter.

Dude imagine if we are just living in one computer complex that had been trying to figure out the answer to this problem for so long and so fast that it created an entire organic galaxy within itself. In the video, the 3D module looks like some sort of plant, so think about what it could create if it had been running for thousands of years on a machine infinitely times as powerful as powerful as our computers.🧠

i'm too high for this

Why is this something that people want to “crack”? Nobody cracks 2+2=4. There are tons of patterns in nature. I don’t see the big deal. But I’m not a mathematician. Or a Russian.

The sequence of kaos

You're multiplying a random number by 3, then adding 1. But also applying different rules depending on the outcome after that. So, it's just a simple eventuality that I'm sure a smarter person than myself could 'solve'. It sounds interesting on paper, "nobody can solve this". But really, there's nothing to solve. It's simple mathematics.

If someone could solve it they would. There is a (slightly over) $1,000,000 prize.

Actually, there was no problem in the first place. This equation is very ordinary, but people decided to put a rule to make it hard to solve. Why on the earth would I divide by 2 if even, and use 3x+1 if odd. There is no need to apply such rule, unless you are out of mind and want to play a stupid game.

Try 963

I know the answer its MATH

I don't understand anything about this at all, but when he said Fibonacci sequence....😓😓😓 that brings back memories

"As yet, no loop has been found" Ummm...4?

I am a bit confused. Numbers are never ending. Even 2^1000 is nothing, and still a finite searchable area (although not by current computers). Going by this the conjecture can never be proven. Only disproved (if possible). At what point do you simply assume that it is true for infinity?

@Ruhaan Burger See the Wikipedia page on "mathematical induction" for example. Most likely if Collatz is true, any proof would involve induction of some sort, though nothing as simple as the examples on that page. Another possibility is to show through a series of transformations that the problem is equivalent to one for which the solution is already known. Proof by contradiction is another possibility. For example you could show that if there is a non-trivial loop, that implies that something else is true, where that thing is known to be false.

@J Modified Pls elaborate on the many possible methods.

You can never assume it is true. A discrete proof is required. There are many possible methods for achieving that.

What does “solve” mean in this context? What exactly is the goal here?

To prove the conjecture true or false.

You forgot about the decimals. 🙃

How is this a problem rather than a pattern? Also how is it random if you're following 2 set rules? I don't get it.

in leftist ideology math ls racist

I'm an engineer who uses a lot of math, but not a mathematician. 2^32 is 4 billion, and it's also the size of an unsigned integer on a 32 bit OS. In my world, if something is true up to 4 billion, it's true. My world has heuristics, not proofs. I wrote a Woodoku optimizer for fun. I do it on multi-million dollar equipment at work.

@9:56 Laniakea?

This is impossible to solve so dont even bother trying

How would you know?

Hold my beer

I wrote a code for my Arduino to brute force this thing. I had a memory overflow at 32k proven numbers and I fixed it by changing integers to "long integers". After 36 minutes of running, it reached 338k, but it is slowly slowing down, because it has to do more and more calculations to prove that a number will end up in the 4-2-1 loop. In fact, when the result of a calculation is less than the number it's trying to prove, it jumps to the next number to prove, because it reached a number that was already proven. The program skips even numbers, because the result would be instantly less than the number it started with. Now it is at 421k proven numbers. Looks like something took a lot of time before 338k. And I don't print the calculations, only the progress, because it would slow down the process.

Oops, I missed that you are already doing (1) and (4). Anyway, even on a super slow processor your program should not be anywhere close to that slow. Maybe it is purely limited by printing speed.

That's very slow. You might want to take advantage of these: 1) You only need to test odd numbers. 2) Test for even/odd using bitwise "and", which is much faster than mod 2. 3) Divide by two by right shifting, which is much faster than a divide operation. 4) If testing in order, you can stop when the number goes below the seed. 5) Since you are only testing odd numbers, you can immediately do a 3x + 1 then divide by two before testing in a loop. 6) After a 3x + 1 you can always immediately divide by two without testing for odd again or checking the result against the seed, since the result will be even and higher than the seed if x was higher than the seed. 7) If you must print progress, use a bitwise test to print every hundred billion numbers or so, or however many it gets through about every ten seconds (I have no idea how slow the processor you are using is).

Mathematicians: "I fear no man, but that thing (3n+1) it scares me..."

Have you tried negativesss....

Yes. They said that in the video

Have you watched the entire video?

I'll bet the guy who thought up this problem solved it, he just couldn't fit the solution in the margin.

What is the objective in this video?

I saw a special Andrew Wiles* about who solved Fermat's Last Theorem. People had warned him that working on it wasn't real math, the solution did so much to advance math it was far from a waste time. *Thanks for the correction magicmulder

@Tri Dave Fermat's proof was very very likely wrong. It's highly unlikely an "elegant" proof has evaded people for 300 years (although I know of one modern case where a proof took 30 years and it was a few lines and super easy to understand).

@magicmulder thanks for the name fix, yes that was my point. It used so much modern math (all way beyond my comprehension) we still don't know how or if Fermat had an actual proof but it wasn't a waste of time or not 'real' math as Wiles had been told early on his career. Who knows what will be unraveled when someone finally figures this one out.

*Wiles Also Wiles and Taylor effectively proved a lot more than just a curious riddle from 300 years ago. Proving the Taniyama-Shimura conjecture was a lot more important to mathematics than that.

3x+1=10

It always confuses me when someone calls something a problem but does not describe what you achieve by solving it.

We do not yet know, but solving it may yield lots of knowledge on the way, just like Fermat’s Last Theorem.

My question is, why are we picking a random number and applying 3x + 1 to it? Then adding different rules based of even or odd? That is why it doesn't work.?

@_ It's more than random numbers: this is actually the simplest variation of the problem (i.e. with the smallest numbers) where the solution is not known. For things like 2x+1, 1x+1 etc the behavior is fully predictable for any number.

@Релёкс84 so its just random numbers somebody has picked that dont have a solution.

it's just how the rules are defined. You can absolutely choose a different set of rules if you want, but that would make it a different problem.

I think there are beautiful growing trees that start at 1 instead of any number and go down.

Put a decimal in the mix because nothing is a whole and problem solved.

So this is why Charles Xavier went bald.

how is this a question tho? so assuming an answer was discovered what could it look like?

The question is "Do all numbers eventually fall back to one?" and the answer is either "Yes" or "No". But to give a definite answer you'd need a proof behind it, and that's where it becomes hard. The problem itself is super easy to understand, but practically impossible to solve with current knowledge.

It´s not a problem though, it´s a simple fact. There is no solution. It´s a dumb question. Some questions don´t have answers. You´re always going to reach 1 if you use division of even, positive integers, because it´s the first positive integer, and even numbers are unavoidable using this sequence. No? What am I missing, I´m 17 and no mathematician so I have to be missing something big lmao pls someone tell me

Meh; it's like one of those tricks about how to find your age. The simple fact is that the equation has gravity. Two rules that lead any number to move in one direction. Down.

thx brillant

Does it work with irrational numbers going by if the last digit is even or odd?

@Релёкс84 my brain was afk that moment my bad

There is no "last digit" for an irrational number.

I theorize that the answer or more practically the way to "avoid" this problem lies with people or more specifically, psychology. Though sounding ironic try thinking of it with births and deaths. Theoretically it can be used more in 'living' applications.

12:35I thought it was real😂

have they tried decimals

Please read this. Hello NLrock comments I'm from Bosnia and Herzegovina so my writing in English will be bad. I know that this is not a place to ask for help but I feel like I don't have other choice. Let me explain, I'm born here in Bosnia in 1992 in a middle of war. In that war my family lost alot and I was 60 days old when I lost my father. So growing without a man to explain you alot of thing's which mother can't was harsh especially in this country where everyone is corrupt. But I did it. I think that I have grown in a solid man. After scool I got a solid job got a wife and we got 2 beautiful boys. So I had to take a loan to buy a home for us to live in and it was fine until October 2018 when I was diagnosed with diabetes then alot of changes I had to make in my life so I could be healthy and support my family. With diabetes came another sicnes I don't how to call it in English (I have to much fat in blood) and thanks God insuline is free here in Bosnia so it was not a big deal I had to spend only about 17 dollars on some pills every 15 days. Now in 2021 I got 3 heart attacks no one does not know how I made it alive but I did. The reason for heart failure is diabetis it stuck some veins and heart could not pump blood. Now the reason why I'm writing all of this is with all of this and alot of things happening to me in my journey tru life I'm in depression I have no idea what to do and how to help my family. I don't see an exit and I don't want to beg on the streets and my Country does not give a damn about me and 95% of other people here. The reason I'm searching for help is because wih heart like mine and diabetes I can't work and bring money home. We stil have 9 years of loan left. Since I'm not working now and I have to give 380 km to bank and my wife makes 500 km evry month then the medications are 120 km for me. We cannot afford nothing more. And we need to pay bills and bring food home. The reason I'm gona ask you guys to somehow help me if possible are the kids i can't explain how hard I'm feeling because I can't afford nothing to them and one of them will need to go in school next year. And I can't afford him a damn pen. I don't know what to do so I came here to ask for help. Thanks. Wish you all the best ❤️

titactoe for grownups.

If u wanna have a career lol Stop watching such problems

4x+2, two rules if it's odd multiply by 8, if even divide by 3.... yay I made a problem

3 6 9

Is there anyone know why the geometric mean at 8'37'' is calculated as in the video >.

This is the math problem that made the Unabomber go crazy lol.

Maybe with modern supercomputers we could solve those mistery problems.

Just like history if u know what i mean

Easy. 28

I hate maths

No degree here, high-school dropout. But I have a question/theory If you could somehow simulate this with a form of energy, like say a more commonly known one, electricity, couldn't you come up with a formula for infinite energy?

you don't have to brut force all numbers to one you only have to get them down or up to a number that leads to one. but the time it takes to show that one doesn't lead to one. AAAAAAAAAAAAAHHHHHHHHHHHHHHHHHHHHHHHH I hate this thanks

An idea that was not brought up in this video would be to instead of proving that any positive integer, when these two rules are applied to it, eventually ends at one, instead try to prove that you can start at one and apply the two rules in reverse to reach any positive integer. Start at one and keep applying the x2 rule, you can also apply the (-1)/3 rule whenever you want, but only if this results in a positive integer. Show that by doing this, you can reach any positive integer.

While that is somewhat faster at first, it requires a lot of memory since you need to track all the numbers encountered. Testing in order requires almost no memory. To forward-test 2^68 numbers, you'd need a memory chip the size of our solar system.

Vain foolishness.

Says the guy commenting on a computer that was once considered vain foolishness. Grow up.

18:22 "It's very hard to prove a theorem that's false."

@J Modified The intended meaning of the citation is that you'll have a really hard time proving a statement if that stamement is actually incorrect.

Which makes no sense. You could always state the Collatz conjecture as its negation. I don't see any reason to prefer it stated one way or the other.

Why can’t we just write a computer program to check these numbers?

I gnow whay and the answer 🤔maybe.. You are wrong on the way you put the problem that whay 🤔??? The bigest mistake Is that zero Is missing e without that you try to do mathematic when zero Is a part of It way????without zero you can Not talk about this

If you whant continuity the use It the (zero)if Not It ends like this

With zero everthing change

i have one question.......... Why do we care if its false or not?

3x+1 is the stock market prices. That is where all the stock prices end up because that is how much they are worth.

Solution to this I think is to wait for a bit and use a quantum computer to its highest computing powers in order to simulate lot more numbers and see if there really is a solution

If the solution is somewhere beyond TREE(3), no quantum computer in the next billion years will help us.

SCOTUS refused to acknowledge Benford's Law.

i dont get it

I would have thought that with the advent of super computers which can do calculations so fast, that we would have been able to see the pattern, despite the seeming chaos, and prove one way or the other.

Haven't watched, 8 th grade. I say 4x