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 -

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30 Jul 2021



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Comentários 39 484
Blake Bormuth
Blake Bormuth Hora atrás
I doubt no one noticed this pattern but when it's falling into the 4-2-1 loop it's going down the long thread of binary. If any starting number eventually hits any number that can be created by 2^n it will follow along this giant chain. I find that very fascinating and would even change the visual aspect as a single pillar in the middle with those changes funneling into it the binary numbers. I then wonder if this has to do with the fact that 3x+1 is the simplest multiplication and addition pattern that can give you odd or even numbers of different origins, and if we change the division of 2 into 3 and the 3x+1 into a slightly differentbut similar formula, will the new pillar in the middle turn into a 3^n loop that always leads to one instead of a binary 2^n loop. Finding a pattern there would be interesting too
Mikki Carbo
Mikki Carbo 2 horas atrás
EL Mohel Castorena
EL Mohel Castorena 3 horas atrás
In that we are dealing with odd and even numbers. It's final state will be as 1,2 respectively saying the numbering sequence is the result of only odd and even (1,2) therefore 4,2,1. For negative you will have to take into account of a double negative becoming a positive with odd and even numbers.
Vanshaj Bindlish
Vanshaj Bindlish 3 horas atrás
Maybe this equation shows that no matter how hard you try to overcome nature and decipher everything around you, but something's always out of human hands.
suhani shah
suhani shah 4 horas atrás
The Turing machine part broke me...
AOV Turtle
AOV Turtle 5 horas atrás
alright you've earned my sub
GetWooked 6 horas atrás
But I didn’t want to pick 7
Titanus Gojiusours
Titanus Gojiusours 8 horas atrás
Well I already suck at math
Normal Vector
Normal Vector 9 horas atrás
13:40 it's not me, it's Patricia
anony 9 horas atrás
The video: 3x + 1 The simplest math problem no one can solve Me: *4x*
Rich Despiseus
Rich Despiseus 9 horas atrás
Basic problem is abiding by 1+1=2, the root of mathematics. The universe doesn't use this and neither does your mind. Maths is a guide to uncovering the face of reality - bit like dot-matrix porn, and it requires contempt for it's limitations.
Ryder Golde
Ryder Golde 14 horas atrás
when I saw the problem as "3x + 1" my mind went to "solve x" but seeing it in the video as "3N + 1" instead makes more sense to me I actually set 3x + 1 to = 0 and solved that I might not be the best at math I still love it so shush
Is Ad Tree
Is Ad Tree 16 horas atrás
Why do You divide by 2? I don't get it. Just keep applying the fórmula
Релёкс84 14 horas atrás
Dividing by to when even is part of the formula. I'm sure you're smart enough to understand why it's not interesting if you don't do that.
Buster Dafydd
Buster Dafydd 16 horas atrás
I came to find this video again because I thought I stumbled on this unique sequence, but Instead I found something that I think works similar... in that you divide any number by 2 and if it has a remainder square it, then divide by 2, and then repeat
Rory Galusha
Rory Galusha 16 horas atrás
Its funny how a educational video can make you feel stupider
Momo S
Momo S 16 horas atrás
What’s the problem that they are trying to figure out...maybe you left that part out?!?
Humblegrenade 17 horas atrás
It’s the difference between reality and the illusion of reality
Mr. Mattis
Mr. Mattis 18 horas atrás
369 is the answer !
Андрій Довбуш
Luis Ortiz
Luis Ortiz 19 horas atrás
I can see the foreseeable future and someone on twitter will tweet "how about you 3x+1 some bitches bro".
SUP! FPV 19 horas atrás
imagine the universe growth is following this theory.
TheSalad Guy
TheSalad Guy 20 horas atrás
its just how the universe builds planets and height maps.
omni D.
omni D. 20 horas atrás
I wonder if they tried this with 5x +1
Quirkey_Spoon67 20 horas atrás
If the last number (unit) is 1-9 the outcome will always be the same no matter the other numbers it will obviously be larger but it will only matter about the unit, the size of the number would be irrelevant
Larry Tinsley
Larry Tinsley 22 horas atrás
I don't understand the problem?
SINAVAHID 22 horas atrás
That's my internet speed graph dude
Jreison 22 horas atrás
My dumbass was sitting like "the answer is 4x"
Nic Risc
Nic Risc 22 horas atrás
can you use a computer code that will do it automatically?
aydint89 22 horas atrás
Why is this a problem ?
Kamiel Draulans
Kamiel Draulans 22 horas atrás
1 = 3x+1 JA DIT IS WAAR
Devin 23 horas atrás
if only numbers were words...
Georg S
Georg S Dia atrás
Obviously adding something (1) that cant be divided by 2 will end at exactly this numer (1) lol
DutchGamerX Dia atrás
can we use decimal
Soul Theory
Soul Theory Dia atrás
Isn't this proof of infinity? The answer to the equation is infinity.
What if we take a negative number?
A Vessel
A Vessel Dia atrás
To me, the Collatz Conjecture is a beautiful analogy that we are all random numbers in a Supreme Sequence. We will go through seasons of x3+1 where life flourishes and also /2 where life cuts us to the core... we will eventually peak at a point and then fall downwards until reaching One. The One. The Creator. This is when we enter "The Loop" also knows as Eternal Life. 2,000 years ago God became Man and was brutally murdered on the Cross for the rebellion of mankind in order for you and I to enter The Loop of Collatz' Conjecture. Repent and Trust in Jesus. Amen.
omputer fan
omputer fan 21 hora atrás
when you get to heaven can you ask big G what the number is
Gian Carlo Bata
Gian Carlo Bata Dia atrás
Just a comment from someone not in a math major... why are we so fixated with multiplying it by 3? Why not multiply it by 9 or 2 or anything else? Many mathematicians may have wondered before what if they changed 3, which is an odd number into 2, an even number. p.s. Don't hate me for this. Again, I'm someone not in a math major.
J Modified
J Modified 22 horas atrás
For some other multipliers the result is easily provable.
Arun Maiti
Arun Maiti Dia atrás
great video! btw, which model/brand of the phone is he using at 6:41, anyone?
JerzeeeDevil Dia atrás
Whatever goes up must come down.
Gamer Tayhong
Gamer Tayhong Dia atrás
If we let skynet solve the problem then we have solved another.
Jerry Lowrance
Jerry Lowrance Dia atrás
So what about "3X+1" where X=PI?
Orange Crush
Orange Crush Dia atrás
Binfords law can detect irregularities in ballots, except in the 2020 election that is………..
Why_r_ u_gai
Why_r_ u_gai Dia atrás
Amin Assadi
Amin Assadi Dia atrás
timothy kearns
timothy kearns Dia atrás
Where's the "="?
Don Cooper
Don Cooper Dia atrás
There's nothing random about the stock market. It is highly manipulated by central banks, governments, corporations and financial whales.
Supertracker007 Dia atrás
I'm sorry I can't get over this... how the hell do most numbers start with 1? This is breaking my brain and I feel like it shouldn't work, because there's always just another number and it's not like the number of potential options is changing. There's only 9 options so shouldn't it be 1/9th for each option with a slight exception in zero?
J Modified
J Modified 21 hora atrás
They don't, except in certain situations. For city sizes in the US, for example, there are many small towns, fewer mid-sized cities, and very few large cities. When you have a distribution like that where the higher the number is the less frequently it occurs, then you can get a Benford's Law distribution or something close to it. This is what you would expect because there are more 10,000-19,999 population cities than 20,000-29,999, and more 100,000-199,999 than 200,000-299,999. Many things have distributions like this. City/town populations are a very close fit. Anything closely related to that, like city land area or number of voters per city will also be a good fit. Others are company employee numbers, personal income and wealth, and animal species sizes. There are many small companies and few huge ones. There are many poor people and few ultra rich. There are many small animal species and few large ones. Of course none of those is a perfect fit because there is some clustering - the size distributions are not entirely smooth. Other things have other distributions. When rolling two dice, you get a lot of 7s and not many 2s and 12s. Male heights in inches almost all start with 6 or 7.
Dakota McKnight
Dakota McKnight Dia atrás
If from 1 every other number is odd, and from 2 every other is even, 0 is nether even or odd and both even and odd. 0÷2=0 is infinite, 0×3+1=1 falls into 4,2,1 cycle
Roger Gallebo
Roger Gallebo Dia atrás
Ephects Dia atrás
X = 1
Michael Berru
Michael Berru Dia atrás
There is no problem.
TechnoMageB5 Dia atrás
Extrapolating from the end - the 4-2-1 result - if the conjecture is correct, we only need to prove that at some point the iterations of 3x+1 eventually hit a power of 2, at which point it reduces to 1 no matter what. Since powers of 2 are as infinite as counting, this is possible. We can ignore all even numbers as an initial seed because a divide by 2 often enough will lead to either an odd number eventually or 1, if the even number is a power of 2. So we can focus our initial seed as an odd number. (Conversely, any even number that would disprove this theory first reduces down to a smaller odd number before taking off, hence using even numbers as an initial seed to try to disprove this is a waste of time. For example, let's say the number 102 would disprove it. First step, divide by 2, we get 51. So, if that were true, we would have discovered at an initial seed of 51 that it disproved the conjecture, and would not have reached 102 ever - there would have been no need to.) My question now becomes, is there a mathematical way to prove that, for any given odd number, the function (3x+1)/2 has a maximum application iteration before it reaches a power of 2 and is then forced down to 1? Note: given that powers of 2 geometrically increase with each step up, it wouldn't surprise me if the number of times the function has to be applied to reach a power of 2 also increases geometrically as the numbers go up. If this is so, logically the solution would involve a power of 2 calculation to extrapolate the maximum iteration for this to happen. At this point I would want to generate samples to observe and see if there's a pattern that can be extrapolated and translated into a formula to demonstrate this. If supercomputers have already computed that all numbers up to 2^68 resolve down to 1, we have plenty of samples to work with. Side note: I was that kid that extrapolated the quadratic formula in Algebra class in 8th grade several days before it was taught to me, in an effort to "solve for x" more efficiently than using "completing the square" and such. I just chose to go into engineering instead of mathematics as a career. I may come back to this when I have more time. Back to work for me...
Conrad Hickok
Conrad Hickok Dia atrás
I feel like it can't be proven because it's not a truth, it's a tendency, and a proof would disprove outliers. Try to prove a tendency, you're in tough water. Demonstrate one, sure.
Conrad Hickok
Conrad Hickok Dia atrás
I meant to add the equation itself is naturally reductive so eventually it reduces itself into the smallest amount it can realize.
Jay Ray
Jay Ray Dia atrás
I don't know if you read these, but I have a theory and I'm curious as to whether you think it would work, and if not, why? Begin with a very very large number, and allow a computer to make these calculations, and save every number it ran into. Now have this computer restart at 1 higher, and any time it strikes a saved number it terminates, knowing that outcome will be the same from where it landed, and restart one higher. At the same time have a second computer begin making calculations from the bottom, filling in any holes the first computer does not. These two computers can share a database of numbers, and repeat this process until almost any smaller number is already accounted for. This would allow for a shortcut to calculation, allowing a computer to calculate significantly more numbers in a given time frame, and also significantly lower the "playing field" of possible numbers over time.
J Modified
J Modified 21 hora atrás
Recording tested numbers is not feasible. To test the seeds already known, you would need a memory chip the size of our solar system. What they do is test in order, stopping when the value is below the seed, and pre-filtering numbers known to reach 1 based on the highest previously tested (though testing a number is so fast that using the strictest known filtering would probably make it slower).
Soorena Aslani
Soorena Aslani Dia atrás
I have found the solution. I will release it for 100 bitcoins.
adam Dia atrás
Lion Pride Music
Lion Pride Music Dia atrás
This is how we go back in time
Andu Raaru
Andu Raaru Dia atrás
First of all they're looking at it backwards, they should pay attention to 1,2,4 not 4,2,1, and another hint is 1,2,4,8,7,5. Good luck
Andu Raaru
Andu Raaru Dia atrás
Again, this is not a "problem" and does not have a "solution". The visible patterns and Fibonacci distribution should tell anyone with understanding this is not something you can break or introduce chaos into.
Andu Raaru
Andu Raaru Dia atrás
In the tree graph, you may notice, it begins with 1,2,4,8,16 (1+6 = 7) and then to either side of 16 you have 32 (3+2=5) and 5. You will after notice two different types of branches. One type of branch will follow this infinite pattern of 1,2,4,8,7,5. The other type of branch will repeat the alternating pattern of 3,6,3,6,3,6....... There is no problem to be solved here, only things to be realized. Anyone who truly wishes to understand what is going on here may start by researching Nikola Tesla and 3,6,9
Purple Knight
Purple Knight Dia atrás
Tony Stark and Shuri probably solved this as children for funsies
Tyrrian Dia atrás
I’m pretty sure the answer is 1*1=2.
Mafort Dia atrás
Did they test one Gazillion? I have a feeling about this one.
kamran pervez
kamran pervez Dia atrás
Interesting but seems useless. Math is full of such weird looking phenomena but most of these phenomena are form due to the very nature of math.
LeadCodpiece 2 dias atrás
I still don't get why is that a problem
TNQR_Warrior 2 dias atrás
3 + 1 = 4. 3 x +1 = ?. The x is what's messing people up. x is the 24th letter of the alphabet. 3 + 24 = 27. 27 + 1 = 28. Answer: 28. (Please take this comment as a joke)
Georgia's Library
Georgia's Library 2 dias atrás
I solved it! No, because of 0. If you divide 0 by 0, you get undefined and even if you still got 0, you would be stuck at 0 forever.
Pranoy Kumar
Pranoy Kumar 2 dias atrás
lol, I still don't understand what they are trying to solve.
Fatema Tasnim
Fatema Tasnim 2 dias atrás
Lucky I have my calculator 😁
Jesse 2 dias atrás
Wieh Ahuo
Wieh Ahuo 2 dias atrás
rule Even/2 represent At present, the double number of 2^68 is not exhaustive... The double number of 2^68X2 is not... 2^68X2X2 is not twice the number... Infinite loop 規則 偶數/2 代表 目前窮舉2^68的兩倍數都不是... 2^68X2的兩倍數也都不是... 2^68X2X2的兩倍數也都不是... 無限循環
Wieh Ahuo
Wieh Ahuo 2 dias atrás
3x+1 1 Why is it so special? 1 cannot be reversed In this rule, 1 is 0 1 is an odd number, presumably some N*3+1 N*3+1=1,N*3=1-1 N*3=0,N=0*3,N=0 So 1=0 1 is produced by 2 even numbers~ 2/2=1, 1*2=2 1
Oliver Freier
Oliver Freier 2 dias atrás
Did any math hero tried to find a function / rule for the negative case that leads to a similiar loop developement as in the positive numbers ?
Pastiche 2 dias atrás
What if we include decimal numbers ? Like 1.85 or like 18.092 ? Thanks.
Debabrata Nayak
Debabrata Nayak 2 dias atrás
This video is a gem 💎 , although the problem is mind boggling but to express us in this way it's really a gem.
Xtrri 2 dias atrás
What would happen if you reversed it, so you half odd numbers while apply 3x+1 to even numbers? My guess is that the graph will just get inverted.
omputer fan
omputer fan 22 horas atrás
ok lets try it 10 31 15.5
Dr noBrain
Dr noBrain 2 dias atrás
4 2 1? did Jupiter copy pasted this for it's moons orbital resonance?😁
Δημήτρης Σταυρόπουλος
Well, infinity is not a number so you have to specify what infinity means to any given problem. If you say that after 10000 is infinity then the 3x+1,/2 number if of 21 is close to infinity.
J Modified
J Modified 2 dias atrás
In this case infinity means that for any number you chose in a Collatz sequence, a higher number will follow.
Fabian Wichelhaus
Fabian Wichelhaus 2 dias atrás
This seems to be a problem for an algorythm, isn't it?
freeNode5 2 dias atrás
think of this, not as the question, but the answer ;)
Nazeeh .v
Nazeeh .v 2 dias atrás
Humans creates numbers to understamd stuff and then doesnt understand it. Some problems might be small but are actually really hard.
Can 2 dias atrás
what about imaginar number?
Андрей Николаев
3x+1 1x3 3
Robel Moreno
Robel Moreno 2 dias atrás
1/2 the time you can divide by 2 to get to the next odd number, 1/4 of the time divide 4 times... What about infinity times? Infinite times to get to an odd number?
Alain Coetmeur
Alain Coetmeur 2 dias atrás
Is the a way to prove it cannot be decided as Gödel defines ? I imagine, non, as it would be done ? For me, it seems like a halting problem. What is seducing people is what I finally understood of NP problems, the asymmetric behabior: for NP, hard to solve, easy to check. For 3x+1, easy to state, hard to check.
J Modified
J Modified 2 dias atrás
Possibly, but it doesn't "feel" unprovable to me.
Kornel 2 dias atrás
Rahul Swami
Rahul Swami 2 dias atrás
This reminds me of a sloka from Upanishads: पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते |
pranav r
pranav r 2 dias atrás
Who came here from meme.
Darko Dominik Dy
Darko Dominik Dy 2 dias atrás
its 4 right?
Lio Unique
Lio Unique 2 dias atrás
Solved this in 3 minutes and bought a fake Noble prize for myself.
pete mchardy
pete mchardy 2 dias atrás
The speed of light may be the serlusion
ApoteOtiK 2 dias atrás
Can you please make a video on 3, 6 and 9? Cheers
Kristoffer Sigerstam
Kristoffer Sigerstam 2 dias atrás
WTF Did i wake up to?
Ice13 2 dias atrás
I have a slightly above average IQ but somehow, math always gives me a headache even just looking at complicated questions
aqil iman
aqil iman 2 dias atrás
It always go back to "one".
Релёкс84 2 dias atrás
Prove it?
Nope Nada
Nope Nada 2 dias atrás
Probably being a idiot, but could a algorithim follow these rules and cross check with numbers on the chart to see what is or isn't there
OGX HUB 2 dias atrás
3X +1 just an equation with any input numbers… which through x3 and +1, then come with an extra rule applied on the output (odd/even).. what’s going on here maybe is the probability generating the output, and how you treat the output with extra rule on group. Question is why the extra rule needs to be /2 ?
Isaac Tai
Isaac Tai 2 dias atrás
Can't you just do aleph null square root aleph null?
Lukelini Gaming
Lukelini Gaming 2 dias atrás
The reason why the problem has three loops when x is negative is because the problem works differently. When x is positive and an odd number, it’s absolute value is multiplied by 3 and then added by one. When x is negative and an odd number though, it’s absolute value is multiplied by 3 and then subtracted by 1. To get the proper results you would be looking for is to do 3x-1 when x is negative and 3x+1 when x is positive.
Релёкс84 2 dias atrás
Of course. But the questions remains of why 3x-1 behaves so differently from 3x+1, even though that 1 quickly becomes negligible in comparison to the 3x. A lot of attempts at on-the-fly justifications for 3x+1 always going back to 1 are so imprecise that they almost always apply just as well to 3x-1, except the latter doesn't always go back to 1 making their reasoning invalid.
Gargon 2 dias atrás
Since its 3x+1, 1 is the starting point of the y axis. If we go the other way, in the negative x, we have a tiny portion that has the x as negative but y as positive. It goes to only -0.33333333..., then both y and x are negative and don't obey the same rules. But -0.33333333... is infinite. we never quite get back to our 1 y axis. So the only way that there would be another loop or wouldn't end in the 4-2-1 loop would be with fractions, decimals, or any number that doesn't in some way have a positive 1 in it. Any number has a +1 in it..... except 0.... and the negatives... so that is my answer... idk if what i had in mind could be properly said through this message.
Michael K.
Michael K. 2 dias atrás
I guess 96 pecent of the viewers including me dont even know what mathematic proof means exactly lol
J Modified
J Modified 2 dias atrás
I once graded a computational theory course. 60% of those CS Master's degree students did not seem to know what mathematic proof means.
Michael K.
Michael K. 2 dias atrás
Mathematicians may say it's a problem, but I'm fine with it
William Grosbach
William Grosbach 2 dias atrás
Ever hear of a twentieth-century Austrian-American logician-mathematician-philosopher named Kurt Gödel?
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