The Simplest Math Problem No One Can Solve - Collatz Conjecture

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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.

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

Published on


Jul 30, 2021




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Comments 39 117
Ismael Sosa
Ismael Sosa 26 minutes ago
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!
Theliftingdutchman 4 hours ago
How about Pi? Is that fraction finite?
Релёкс84 3 hours ago
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.
Victor Zheng
Victor Zheng 4 hours ago
me as a 9yr old but my iq is 144 and i do yr 9 math (not bragging lmao)
Eimai Xazos
Eimai Xazos 47 minutes ago
Doubt it
Yxng_Itxchi07 6 hours ago
Not watching the vid but the thumbnail equals “Y”
Yxng_Itxchi07 6 hours ago
Because Y=mx+b
Stanislav Serov
Stanislav Serov 7 hours ago
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.
Thuo Kagiri
Thuo Kagiri 8 hours ago
Soviets: Yes ! we broke him! He is now a mathematical nihilist
Jester Kemble
Jester Kemble 8 hours ago
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!!!!
Ranger Smith
Ranger Smith 9 hours ago
Mathematical analog to life. People live for very different lengths of time, but they all eventualy die.
Funky Friday Songs
Funky Friday Songs 9 hours ago
Ahmad Alshamsi
Ahmad Alshamsi 9 hours ago
time waste
Jeff Evans
Jeff Evans 9 hours ago
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
Connor Kearley
Connor Kearley 11 hours ago
computational equivalence 😄
Alvin Jiang
Alvin Jiang 12 hours ago
The small january distinctly offend because nepal feasibly permit pro a reminiscent risk. tight, workable amusement
Apsteronaldo 12 hours ago
I like the comment where the guy says other proofs are actual miracles we don't deserve
- 12 hours ago
💩 + 💩 =s Liberals brains 🤯👍🏼
Dawid Dettlaff
Dawid Dettlaff 13 hours ago
I got lost in the middle, but I'm here to look at smart ppl, dreaming of becoming one.
Tommy Two Tacos
Tommy Two Tacos 13 hours ago
Does this still hold true outside of base 10?
Private Dead
Private Dead 13 hours ago
so do these numbers include . values or just whole values so 1.5 > 5.5 > 17.5 > 53.5
Cyntrix 14 hours ago
Oh it's simple you just add 3 + 1 which is 4 so x is 4
César S.
César S. 14 hours ago
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!
Holken 14 hours ago
i picked 7, im nervous now
Thuong Phung
Thuong Phung 16 hours ago
My head ached from watching this video... Nice job!
Winter Bird
Winter Bird 16 hours ago
The married taiwan frustratingly bat because pants tinctorially interfere excluding a expensive venezuela. muddled, sedate waitress
Pietje Bell
Pietje Bell 17 hours ago
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
theniceobz 17 hours ago
Does he explain why you can't use negatives?
J Modified
J Modified 15 hours ago
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 .
John Weisbrod
John Weisbrod 18 hours ago
“Don’t talk about this if you want a career” Veritasium:
Asadbek Khudayberganov
Soviets are smart, huh?
William Thomas
William Thomas 18 hours ago
I don't see a problem here
Karim Sibai
Karim Sibai 18 hours ago
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
J Modified
J Modified 15 hours ago
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.
Legomach 19 hours ago
13:02 me on my scientific calculator at school for some reason:
Bonnie Hawkins
Bonnie Hawkins 20 hours ago
Very simple. 3X=1. Since X =0, you have 000+1= 1. Don't need to go thru loops.
Avary GameCorner
Avary GameCorner 21 hour ago
I do not get where dividing by 2 came from... Did they just said... when its even now we divide by two to CREATE the problem? Ifc ull once hit number that is a 2 on N in the end cause you just are trying ods to get here :D the f
Релёкс84 20 hours ago
The rules were chosen arbitrarily because they are simple to understand and their behavior is still interesting.
Avary GameCorner
Avary GameCorner 21 hour ago
Thats not odd thats dumb... its same as Labirint go always left tactic... ull eventualy end up in the end
Rishabh Kayastha
Rishabh Kayastha 22 hours ago
how about we try 3x-1 instead of 3x+1. we always get the numbers we started with. Food for thoughts
nadia bairamis
nadia bairamis 23 hours ago
looks just like the branches of evolution tree
mysticxiii Day ago
I take issue with the operation. Why is there a rule of odds/evens here? What requires us to perform these functions each time?
Релёкс84 23 hours ago
It's just how the rule is defined. You're completely free to make up your own rules and formulate your own problem if you want, but that wouldn't be the Collatz conjecture anymore.
Len Taclof
Len Taclof Day ago
Mathematicians go insane. Physicists don't. That's why I'm a physicist.
Len Taclof
Len Taclof 21 hour ago
@Релёкс84 I'm 66 and have been studying Quantum Mechanics for 16 now and the University I was planning on attending shut down because of Covid. Ironically when I turned 65 I was eligible to attend for free BUT you're absolutely right.
Релёкс84 22 hours ago
Well yeah, except you're just someone who's interested in hearing about physics, but that does not make you a physicist.
keppa Day ago
12:35 I couldn't control myself😂
If the negative numbers get you to 3 possible loops, does that mean that 3x-1 applied to them should give the - 4 - 2 - 1 routine ? And if so if you apply 3x-1 to positive numbers it shall lead you to 3 loops?
Mikael Olsson
Correct. The offset needs to keep moving away from 0 axis, otherwise it's a completely different pattern/fractal.
Tien Dat Nguyen
Do we have math to calculate the steps to make a number become 1?
Vladimir Kovalev
I like maths and programming, but I don't see why this is interesting.
MUR motivation & music
Rutmer Hoitema
But why is it a problem?
Expulsion Science
So interesting!
GHOST5663 Day ago
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?
Mikael Olsson
Try changing the rule to 3x-1 and stay in the positive space. Positive number: +1 moves away from 0. Negative number: +1 moves TO 0. Positive number: -1 moves ....
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 .
Ee J
Ee J Day ago
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.
D R Day ago
This isn't a "math problem". Its just trying to analyze a trend.
Jake Kim
Jake Kim Day ago
No ads plz
Sid M
Sid M Day ago
Cut it in half ½ 0.5
Pepsi Max
Pepsi Max Day ago
My first question is why would you waste your time doing this?
X-style Day ago
for me, the only problem with these numbers are in my bank account, every month ends in the same loop :(
neonpeacocks Day ago
Surely I’m not the only one listening to this video with a blank stare…too stupid to understand maths club!
Anti -
Anti - Day ago
The answer is Taco 🌮 chow.
Aquea Ventus
Aquea Ventus Day ago
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
Aquea Ventus
Aquea Ventus Day ago
42 the meaning of life !
-.- Day ago
- 1/3X
A D Day ago
Why is it a problem??? What do they want instead ???
Thomas Hoens
Thomas Hoens Day ago
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?
Romano Okudi
Romano Okudi 21 hour ago
I guess the real question is why that happens. why does the equation eventually reach the power of 2?
Fahim Montasir
Adu Balu
Adu Balu Day ago
This video increased my anxiety
Sean Chandler
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.
Cosmo Thompson
Have you tried 2^68 + 1 though? I have a feeling about that one
Day ago
Keep it up
Kalyan Boro
Kalyan Boro Day ago
just another time pass till the next season of "Money Heist" releases!!
ariel krische
ariel krische 2 days ago
The blue jewel namely extend because soldier covalently miss but a tranquil postbox. gifted, ritzy step-aunt
Hazbin Hotel
Hazbin Hotel 2 days ago
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
Robert M
Robert M 2 days ago
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.
itszian 2 days ago
Wow now try to prove all the fractional numbers following the conjecture
Jimbob Robob
Jimbob Robob 2 days ago
I was able to keep up somewhat to about 7 mins... 😢
LeeWAM 2 days ago
i think i got proof there is no other loop than 1-2-4
LeeWAM Day ago
@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.
LeeWAM 2 days ago
@Релёкс84 it still needs to be checked, but i do have
Релёкс84 2 days ago
@LeeWAM I still think you don't have a proof
LeeWAM 2 days ago
@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
Bob Zim
Bob Zim 2 days ago
That was good.
Justin TimberFlake
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
Dwight Gutierrez
Dwight Gutierrez 2 days ago
The waiting kilometer revealingly mend because butcher reportedly scrape beyond a amusing girdle. upset, addicted sweets
Joseph Martillo Palacios
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.
vqynszn 2 days ago
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
Brian Colbert
Brian Colbert 2 days ago
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.
Milky1L 2 days ago
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.🧠
Noé de Cominges Arce
i'm too high for this
Laramie On Location
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.
Andre 2 days ago
The sequence of kaos
Brandon Brouillette
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.
J Modified
J Modified 2 days ago
If someone could solve it they would. There is a (slightly over) $1,000,000 prize.
Saoud Alzoabi
Saoud Alzoabi 2 days ago
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.
TGC 2 days ago
Try 963
Abylssent 2 days ago
I know the answer its MATH
Sticks 2 days ago
I don't understand anything about this at all, but when he said Fibonacci sequence....😓😓😓 that brings back memories
Dale 2 days ago
"As yet, no loop has been found" Ummm...4?
Ruhaan Burger
Ruhaan Burger 2 days ago
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?
J Modified
J Modified Day ago
@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.
Ruhaan Burger
Ruhaan Burger 2 days ago
@J Modified Pls elaborate on the many possible methods.
J Modified
J Modified 2 days ago
You can never assume it is true. A discrete proof is required. There are many possible methods for achieving that.
James Targin
James Targin 2 days ago
What does “solve” mean in this context? What exactly is the goal here?
J Modified
J Modified 2 days ago
To prove the conjecture true or false.
Karlee TDM
Karlee TDM 2 days ago
You forgot about the decimals. 🙃
Whatisthisthing 2 days ago
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.
Koon Troll
Koon Troll 2 days ago
in leftist ideology math ls racist
Karl Davis
Karl Davis 2 days ago
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.
Anton Novikov
Anton Novikov 2 days ago
@9:56 Laniakea?
Bad News Broadcast
This is impossible to solve so dont even bother trying
Релёкс84 2 days ago
How would you know?
shredx81 2 days ago
Hold my beer
Bálint Éliás
Bálint Éliás 2 days ago
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.
J Modified
J Modified 2 days ago
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.
J Modified
J Modified 2 days ago
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).
Dani_Stars230 2 days ago
Mathematicians: "I fear no man, but that thing (3n+1) it scares me..."
Vasu Gupta
Vasu Gupta 2 days ago
Have you tried negativesss....
Hi Hi
Hi Hi 2 days ago
Yes. They said that in the video
Релёкс84 2 days ago
Have you watched the entire video?
mwilliams6464 2 days ago
I'll bet the guy who thought up this problem solved it, he just couldn't fit the solution in the margin.
Mark Watney
Mark Watney 7 hours ago
He was just following Pierre De Fermat's legacy.
3 Perplexing Physics Problems