The Simplest Math Problem No One Can Solve - Collatz Conjecture

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 -




  • When you find yourself back at 1 and change the rules and divide 1 by 2 and divide every subsequent number by 2 because 0 is a even number in the spectrum it brings you to the number 7.450581e-9 according to my calculations. Which could possibly mean nothing..... Prove me wrong

    chris2005fchris2005f36 dakika önce
  • The number 12 ends odd

    Jacquan JonesJacquan Jones53 dakika önce
  • ı hate you !!!

    Ahmet BoranAhmet BoranSaatler önce

    Param Veer Singh BhatiaParam Veer Singh BhatiaSaatler önce

    Param Veer Singh BhatiaParam Veer Singh BhatiaSaatler önce
  • I made it to 8:35 before I started to get confused

    Desbug fanDesbug fanSaatler önce
  • Of all of your videos, this one blows my mind the most. Whenever I think I am smart, I remember this video and the people in it, and I feel stupid again. Thanks for that. :) Great video!

    Marty BMarty BSaatler önce
  • I watched for ten minutes, still never heard anyone say what problem they were trying to solve. I get the arbitrary rules of multiply if odd, divide if even, but what is the actual problem?

    Gary DarlingGary DarlingSaatler önce
  • 4....................................................................................................................................

    Bobby TectalabyssBobby TectalabyssSaatler önce
  • “Pick a number” Me: *Picks 0* 😎

    Memes_ConfirmedMemes_Confirmed2 saatler önce
  • No wonder why stock and crypto market will crash at the end :)

    vor texvor tex3 saatler önce
  • The answers 35

    Brandon_DuhgoatBrandon_Duhgoat4 saatler önce
  • 3x+1=3x+1 ez

    Hamster GrinderHamster Grinder4 saatler önce
  • Zufall ist es nicht haha... Chaos steht über Raum und Zeit 3,6,9 als Flussdiagramm? schauts eher aus wie ein Kreis, ist es aber nicht^^

    shivanoxshivanox4 saatler önce
  • Looking for a counter example, PI comes to my mind. According to the definition only integers are even or odd. I guess if a number is odd when there is a remainder after being divided by 2, then every real number not part of integers would be odd. With 3 x + 1 algorithm, some of these number would grow until being eventually big enough to be integers themselves... For those who are infinite fractions, it means they would grow until infinity, and thus, like PI, would be on different trees. And the conjecture would be false. Interesting topic!

    jokesterfrjokesterfr5 saatler önce
    • That's not surprizing nor interesting, as all irrational numbers, which stastistically speaking are 100% of real numbers, would trivially diverge to infinity under those rules. The interesting part is when limiting your view to integers, and asking yourself if they all follow the trend of falling down to 1, or if exceptions exist.

      Релёкс84Релёкс844 saatler önce
  • "this stupid little problem, how come we cant solve it", bitch i dont even get the problem, my brain is smoking i dont think its a stupid little problem or simple

    Eliana PetEliana Pet5 saatler önce
  • Finally a way to stall a lesson.

    OddElephantOddElephant5 saatler önce
  • Math does not have a proof for all true statements. I does not have a proof for all false statements. I would expect - numbers to be different after all -x- =+ but a +x+ = + the 2 sides of the number line are not equal in operations. What about complex numbers?

    Kenny BravermanKenny Braverman5 saatler önce
  • If you end up on a square number you always end up on 1

    finlayyearsley Guitarfinlayyearsley Guitar5 saatler önce
  • 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

    Blake BormuthBlake Bormuth7 saatler önce
  • Infinite

    Mikki CarboMikki Carbo7 saatler önce
  • 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.

    EL Mohel CastorenaEL Mohel Castorena8 saatler önce
  • 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.

    Vanshaj BindlishVanshaj Bindlish9 saatler önce
  • The Turing machine part broke me...

    suhani shahsuhani shah9 saatler önce
  • alright you've earned my sub

    AOV TurtleAOV Turtle11 saatler önce
  • But I didn’t want to pick 7

    GetWookedGetWooked11 saatler önce
  • Well I already suck at math

    Titanus GojiusoursTitanus Gojiusours14 saatler önce
  • 13:40 it's not me, it's Patricia

    Normal VectorNormal Vector14 saatler önce
  • The video: 3x + 1 The simplest math problem no one can solve Me: *4x*

    anonyanony15 saatler önce
  • 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.

    Rich DespiseusRich Despiseus15 saatler önce
  • 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

    Ryder GoldeRyder Golde20 saatler önce
  • Why do You divide by 2? I don't get it. Just keep applying the fórmula

    Is Ad TreeIs Ad Tree21 saatler önce
    • 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.

      Релёкс84Релёкс8420 saatler önce
  • 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

    Buster DafyddBuster Dafydd22 saatler önce
  • Its funny how a educational video can make you feel stupider

    Rory GalushaRory Galusha22 saatler önce
  • What’s the problem that they are trying to figure out...maybe you left that part out?!?

    Momo SMomo S22 saatler önce
  • It’s the difference between reality and the illusion of reality

    HumblegrenadeHumblegrenade22 saatler önce
  • 369 is the answer !

    Mr. MattisMr. MattisGün önce
  • 10958¹⁰⁹⁵⁸

    Андрій ДовбушАндрій ДовбушGün önce
  • I can see the foreseeable future and someone on twitter will tweet "how about you 3x+1 some bitches bro".

    Luis OrtizLuis OrtizGün önce
  • imagine the universe growth is following this theory.

    SUP! FPVSUP! FPVGün önce
  • its just how the universe builds planets and height maps.

    TheSalad GuyTheSalad GuyGün önce
  • I wonder if they tried this with 5x +1

    omni D.omni D.Gün önce
  • 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

    Quirkey_Spoon67Quirkey_Spoon67Gün önce
  • I don't understand the problem?

    Larry TinsleyLarry TinsleyGün önce
  • That's my internet speed graph dude

  • My dumbass was sitting like "the answer is 4x"

    JreisonJreisonGün önce
  • can you use a computer code that will do it automatically?

    Nic RiscNic RiscGün önce
  • Why is this a problem ?

    aydint89aydint89Gün önce
  • 1 = 3x+1 JA DIT IS WAAR

    Kamiel DraulansKamiel DraulansGün önce
  • if only numbers were words...

    DevinDevinGün önce
  • Obviously adding something (1) that cant be divided by 2 will end at exactly this numer (1) lol

    Georg SGeorg SGün önce
  • can we use decimal

    DutchGamerXDutchGamerXGün önce
  • Isn't this proof of infinity? The answer to the equation is infinity.

    Soul TheorySoul TheoryGün önce
  • What if we take a negative number?

  • 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.

    A VesselA VesselGün önce
    • when you get to heaven can you ask big G what the number is

      omputer fanomputer fanGün önce
  • 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.

    Gian Carlo BataGian Carlo BataGün önce
    • For some other multipliers the result is easily provable.

      J ModifiedJ ModifiedGün önce
  • great video! btw, which model/brand of the phone is he using at 6:41, anyone?

    Arun MaitiArun MaitiGün önce
  • Whatever goes up must come down.

    JerzeeeDevilJerzeeeDevilGün önce
  • If we let skynet solve the problem then we have solved another.

    Gamer TayhongGamer TayhongGün önce
  • So what about "3X+1" where X=PI?

    Jerry LowranceJerry LowranceGün önce
  • Binfords law can detect irregularities in ballots, except in the 2020 election that is………..

    Orange CrushOrange CrushGün önce
  • 0

    Why_r_ u_gaiWhy_r_ u_gaiGün önce
  • Awesome

    Amin AssadiAmin AssadiGün önce
  • Where's the "="?

    timothy kearnstimothy kearnsGün önce
  • There's nothing random about the stock market. It is highly manipulated by central banks, governments, corporations and financial whales.

    Don CooperDon CooperGün önce
  • 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?

    Supertracker007Supertracker007Gün önce
    • 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.

      J ModifiedJ ModifiedGün önce
  • 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

    Dakota McKnightDakota McKnightGün önce
  • 10

    Roger GalleboRoger GalleboGün önce
  • X = 1

    EphectsEphectsGün önce
The Simplest Math Problem No One Can Solve - Collatz Conjecture