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 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. -

Lagarias, J. C. (2003). The 3x+ 1 problem: An annotated bibliography (1963-1999). The ultimate challenge: the 3x, 1, 267-341. -

Tao, T (2020). The Notorious Collatz Conjecture -

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. -

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. -

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 -
Coral visualisation by Algoritmarte -

Richard Yates
Richard Yates Minut sedan
Well this is all well and good. But what I don't understand is, why is it even a problem that requires solving?
S135tu Di97P
S135tu Di97P 8 minuter sedan
This formula looks like 3 dimensional space plus 1 time.
Jake Bisnar
Jake Bisnar 27 minuter sedan
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.
Equal 99
Equal 99 Timme sedan
I have a counterexample, its Bannana sundae Bannana sundae to ??? To bannana sundae. *This is huge*
Niners4Lyphe Timme sedan
The answer is 1. Why is this the issue? What am I missing?
:D 2 timmar sedan
The simplest problem that is impossible to solve This dude: *shows how to solve* Me: you said it was impossible
Mithindis 2 timmar sedan
I don't understand why it's so hard for anyone to solve. What's 3x+1? It's addition. People always overthink things.
Babyobitowrlddddd 2 timmar sedan
Yare yare
Acc Huwgf
Acc Huwgf 2 timmar sedan
whats wrong with a loop? many things in life repeats itself in one form or another. life still goes on.
Ed 3 timmar sedan
I like math, but this is the dumbest thing I have ever seen. LOL
Ciddyism 3 timmar sedan
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!".
Matthias Zbinden
Matthias Zbinden 4 timmar sedan
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.
Mood Swings
Mood Swings 4 timmar sedan
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
J Modified
J Modified 2 timmar sedan
@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.
Mood Swings
Mood Swings 2 timmar sedan
@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
J Modified
J Modified 2 timmar sedan
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.
Samantha Beder
Samantha Beder 4 timmar sedan
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!
Mood Swings
Mood Swings 4 timmar sedan
What is there to work on?...
Mood Swings
Mood Swings 4 timmar sedan
I chose 5 :(
Slimie Slug
Slimie Slug 4 timmar sedan
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 .
Slimie Slug
Slimie Slug 4 timmar sedan
they should have a computer running all the numbers forever or until we find something
HomeOpenly 5 timmar sedan
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.
Luisito no comunica
Luisito no comunica 7 timmar sedan
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
Релёкс84 5 timmar sedan
You must have an olympic medal in jumping to conclusions
Rita Sushiro
Rita Sushiro 9 timmar sedan
I feel like I wanna study this more but I'm a dumbass. Why does math have to be so hard.
Tsatan 321
Tsatan 321 9 timmar sedan
Ha ha.......this is why I hate math
UNITSOLAR 9 timmar sedan
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.
A math geek with way too much time on his hands
A math geek with way too much time on his hands 10 timmar sedan
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!
A math geek with way too much time on his hands
A math geek with way too much time on his hands 4 timmar sedan
It's up!
Релёкс84 8 timmar sedan
@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.
A math geek with way too much time on his hands
A math geek with way too much time on his hands 8 timmar sedan
Will update video later. Got a football game to watch
A math geek with way too much time on his hands
A math geek with way too much time on his hands 8 timmar sedan
@Релёкс84 but I did prove the first thing. That’s all I actually did in the video.
Релёкс84 8 timmar sedan
@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.
Hellion 11 timmar sedan
Me at 1:12 am and i dont even freaking like math.
Lamp Ter
Lamp Ter 11 timmar sedan
If + = positive then the answer would be 3 Edit: or +3
Dave Gilmour
Dave Gilmour 11 timmar sedan
How about decimal numbers?
Dave Gilmour
Dave Gilmour 9 timmar sedan
@Релёкс84 Thank you for your explantation :)
Релёкс84 10 timmar sedan
@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.
Dave Gilmour
Dave Gilmour 10 timmar sedan
@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)?
Релёкс84 11 timmar sedan
What about them?
Adrian LeCesne
Adrian LeCesne 12 timmar sedan
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?
2 Stroke Diaries
2 Stroke Diaries 12 timmar sedan
Whats the problem here?
jefmatttab 13 timmar sedan
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
thotchuiyo achui
thotchuiyo achui 13 timmar sedan
I found the meaning of life while watching this.
HUNTER 14 timmar sedan
let's s assume that you find a number that doesn't loop to 4-2-1 then WHAT ?
J Modified
J Modified 10 timmar sedan
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.
I Am Not Amused
I Am Not Amused 14 timmar sedan
this has cryptographic uses.
J Modified
J Modified 10 timmar sedan
If there are many higher loops and a way to find them, then it could.
Ananya Srivastava
Ananya Srivastava 14 timmar sedan
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 😂
Ismael Sosa
Ismael Sosa 15 timmar sedan
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 18 timmar sedan
How about Pi? Is that fraction finite?
Релёкс84 18 timmar sedan
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 19 timmar sedan
me as a 9yr old but my iq is 144 and i do yr 9 math (not bragging lmao)
Eimai Xazos
Eimai Xazos 10 timmar sedan
@J Modified OK
J Modified
J Modified 10 timmar sedan
@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.
Eimai Xazos
Eimai Xazos 15 timmar sedan
Doubt it
Yxng_Itxchi07 21 timme sedan
Not watching the vid but the thumbnail equals “Y”
Yxng_Itxchi07 21 timme sedan
Because Y=mx+b
Stanislav Serov
Stanislav Serov 22 timmar sedan
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 22 timmar sedan
Soviets: Yes ! we broke him! He is now a mathematical nihilist
Jester Kemble
Jester Kemble 23 timmar sedan
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 Dag sedan
Mathematical analog to life. People live for very different lengths of time, but they all eventualy die.
Funky Friday Songs
Funky Friday Songs Dag sedan
Ahmad Alshamsi
Ahmad Alshamsi Dag sedan
time waste
Jeff Evans
Jeff Evans Dag sedan
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
J Modified
J Modified 10 timmar sedan
A proof does not require testing all numbers.
Connor Kearley
Connor Kearley Dag sedan
computational equivalence 😄
Alvin Jiang
Alvin Jiang Dag sedan
The small january distinctly offend because nepal feasibly permit pro a reminiscent risk. tight, workable amusement
Apsteronaldo Dag sedan
I like the comment where the guy says other proofs are actual miracles we don't deserve
- Dag sedan
💩 + 💩 =s Liberals brains 🤯👍🏼
Dawid Dettlaff
Dawid Dettlaff Dag sedan
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 Dag sedan
Does this still hold true outside of base 10?
J Modified
J Modified 10 timmar sedan
Nothing in the process depends on the number base or representation.
Private Dead
Private Dead Dag sedan
so do these numbers include . values or just whole values so 1.5 > 5.5 > 17.5 > 53.5
Cyntrix Dag sedan
Oh it's simple you just add 3 + 1 which is 4 so x is 4
César S.
César S. Dag sedan
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 Dag sedan
i picked 7, im nervous now
Thuong Phung
Thuong Phung Dag sedan
My head ached from watching this video... Nice job!
Winter Bird
Winter Bird Dag sedan
The married taiwan frustratingly bat because pants tinctorially interfere excluding a expensive venezuela. muddled, sedate waitress
Pietje Bell
Pietje Bell Dag sedan
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 Dag sedan
Does he explain why you can't use negatives?
J Modified
J Modified Dag sedan
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".
GermanLokomotiveDriver Dag sedan
I would say 3x1+1 =4 but its rly Not that easy .
John Weisbrod
John Weisbrod Dag sedan
“Don’t talk about this if you want a career” Veritasium:
Asadbek Khudayberganov
Asadbek Khudayberganov Dag sedan
Soviets are smart, huh?
William Thomas
William Thomas Dag sedan
I don't see a problem here
Karim Sibai
Karim Sibai Dag sedan
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 Dag sedan
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 Dag sedan
13:02 me on my scientific calculator at school for some reason:
Bonnie Hawkins
Bonnie Hawkins Dag sedan
Very simple. 3X=1. Since X =0, you have 000+1= 1. Don't need to go thru loops.
Avary GameCorner
Avary GameCorner Dag sedan
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 Dag sedan
The rules were chosen arbitrarily because they are simple to understand and their behavior is still interesting.
Avary GameCorner
Avary GameCorner Dag sedan
Thats not odd thats dumb... its same as Labirint go always left tactic... ull eventualy end up in the end
Rishabh Kayastha
Rishabh Kayastha Dag sedan
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 Dag sedan
looks just like the branches of evolution tree
mysticxiii Dag sedan
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 Dag sedan
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 Dag sedan
Mathematicians go insane. Physicists don't. That's why I'm a physicist.
Len Taclof
Len Taclof Dag sedan
@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 Dag sedan
Well yeah, except you're just someone who's interested in hearing about physics, but that does not make you a physicist.
keppa Dag sedan
12:35 I couldn't control myself😂
SmokingCoffee Dag sedan
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
Mikael Olsson Dag sedan
Correct. The offset needs to keep moving away from 0 axis, otherwise it's a completely different pattern/fractal.
Tien Dat Nguyen
Tien Dat Nguyen Dag sedan
Do we have math to calculate the steps to make a number become 1?
Vladimir Kovalev
Vladimir Kovalev Dag sedan
I like maths and programming, but I don't see why this is interesting.
MUR motivation & music
MUR motivation & music Dag sedan
Rutmer Hoitema
Rutmer Hoitema Dag sedan
But why is it a problem?
Expulsion Science
Expulsion Science Dag sedan
So interesting!
GHOST5663 Dag sedan
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
Mikael Olsson Dag sedan
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 ....
ScienceXplorer Dag sedan
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 Dag sedan
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 2 dagar sedan
This isn't a "math problem". Its just trying to analyze a trend.
Jake Kim
Jake Kim 2 dagar sedan
No ads plz
Sid M
Sid M 2 dagar sedan
Cut it in half ½ 0.5
Pepsi Max
Pepsi Max 2 dagar sedan
My first question is why would you waste your time doing this?
X-style 2 dagar sedan
for me, the only problem with these numbers are in my bank account, every month ends in the same loop :(
neonpeacocks 2 dagar sedan
Surely I’m not the only one listening to this video with a blank stare…too stupid to understand maths club!
Anti -
Anti - 2 dagar sedan
The answer is Taco 🌮 chow.
Aquea Ventus
Aquea Ventus 2 dagar sedan
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 2 dagar sedan
42 the meaning of life !
-.- 2 dagar sedan
- 1/3X
A D 2 dagar sedan
Why is it a problem??? What do they want instead ???
Thomas Hoens
Thomas Hoens 2 dagar sedan
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 Dag sedan
I guess the real question is why that happens. why does the equation eventually reach the power of 2?
Fahim Montasir
Fahim Montasir 2 dagar sedan
Adu Balu
Adu Balu 2 dagar sedan
This video increased my anxiety
Sean Chandler
Sean Chandler 2 dagar sedan
seems once you hit a number to the power of 2 it drops all the way down to 1.
Релёкс84 2 dagar sedan
It is also the only possible way to drop to 1, if you think about it.
Cosmo Thompson
Cosmo Thompson 2 dagar sedan
Have you tried 2^68 + 1 though? I have a feeling about that one
2 dagar sedan
Keep it up
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