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 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 Golde3 ชั่วโมงที่ผ่านมา
  • Why do You divide by 2? I don't get it. Just keep applying the fórmula

    Is Ad TreeIs Ad Tree4 ชั่วโมงที่ผ่านมา
    • 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.

      Релёкс84Релёкс842 ชั่วโมงที่ผ่านมา
  • 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 Dafydd4 ชั่วโมงที่ผ่านมา
  • Its funny how a educational video can make you feel stupider

    Rory GalushaRory Galusha5 ชั่วโมงที่ผ่านมา
  • What’s the problem that they are trying to figure out...maybe you left that part out?!?

    Momo SMomo S5 ชั่วโมงที่ผ่านมา
  • It’s the difference between reality and the illusion of reality

    HumblegrenadeHumblegrenade5 ชั่วโมงที่ผ่านมา
  • 369 is the answer !

    Mr. MattisMr. Mattis6 ชั่วโมงที่ผ่านมา
  • 10958¹⁰⁹⁵⁸

    Андрій ДовбушАндрій Довбуш7 ชั่วโมงที่ผ่านมา
  • I can see the foreseeable future and someone on twitter will tweet "how about you 3x+1 some bitches bro".

    Luis OrtizLuis Ortiz7 ชั่วโมงที่ผ่านมา
  • imagine the universe growth is following this theory.

    SUP! FPVSUP! FPV8 ชั่วโมงที่ผ่านมา
  • its just how the universe builds planets and height maps.

    TheSalad GuyTheSalad Guy8 ชั่วโมงที่ผ่านมา
  • I wonder if they tried this with 5x +1

    omni D.omni D.8 ชั่วโมงที่ผ่านมา
  • 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_Spoon679 ชั่วโมงที่ผ่านมา
  • I don't understand the problem?

    Larry TinsleyLarry Tinsley10 ชั่วโมงที่ผ่านมา
  • That's my internet speed graph dude

    SINAVAHIDSINAVAHID10 ชั่วโมงที่ผ่านมา
  • My dumbass was sitting like "the answer is 4x"

    JreisonJreison10 ชั่วโมงที่ผ่านมา
  • can you use a computer code that will do it automatically?

    Nic RiscNic Risc11 ชั่วโมงที่ผ่านมา
  • Why is this a problem ?

    aydint89aydint8911 ชั่วโมงที่ผ่านมา
  • 1 = 3x+1 JA DIT IS WAAR

    Kamiel DraulansKamiel Draulans11 ชั่วโมงที่ผ่านมา
  • if only numbers were words...

    DevinDevin11 ชั่วโมงที่ผ่านมา
  • Obviously adding something (1) that cant be divided by 2 will end at exactly this numer (1) lol

    Georg SGeorg S12 ชั่วโมงที่ผ่านมา
  • can we use decimal

    DutchGamerXDutchGamerX12 ชั่วโมงที่ผ่านมา
  • Isn't this proof of infinity? The answer to the equation is infinity.

    Soul TheorySoul Theory12 ชั่วโมงที่ผ่านมา
  • 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 Vessel13 ชั่วโมงที่ผ่านมา
    • when you get to heaven can you ask big G what the number is

      omputer fanomputer fan10 ชั่วโมงที่ผ่านมา
  • 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 Bata13 ชั่วโมงที่ผ่านมา
    • For some other multipliers the result is easily provable.

      J ModifiedJ Modified10 ชั่วโมงที่ผ่านมา
  • great video! btw, which model/brand of the phone is he using at 6:41, anyone?

    Arun MaitiArun Maiti14 ชั่วโมงที่ผ่านมา
  • Whatever goes up must come down.

    JerzeeeDevilJerzeeeDevil14 ชั่วโมงที่ผ่านมา
  • If we let skynet solve the problem then we have solved another.

    Gamer TayhongGamer Tayhong14 ชั่วโมงที่ผ่านมา
  • So what about "3X+1" where X=PI?

    Jerry LowranceJerry Lowrance14 ชั่วโมงที่ผ่านมา
  • Binfords law can detect irregularities in ballots, except in the 2020 election that is………..

    Orange CrushOrange Crush16 ชั่วโมงที่ผ่านมา
  • 0

    Why_r_ u_gaiWhy_r_ u_gai22 ชั่วโมงที่ผ่านมา
  • Awesome

    Amin AssadiAmin Assadi23 ชั่วโมงที่ผ่านมา
  • Where's the "="?

    timothy kearnstimothy kearns23 ชั่วโมงที่ผ่านมา
  • There's nothing random about the stock market. It is highly manipulated by central banks, governments, corporations and financial whales.

    Don CooperDon Cooperวันที่ผ่านมา
  • 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?

    • 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 Modified9 ชั่วโมงที่ผ่านมา
  • 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 McKnightวันที่ผ่านมา
  • 10

    Roger GalleboRoger Galleboวันที่ผ่านมา
  • X = 1

  • There is no problem.

    Michael BerruMichael Berruวันที่ผ่านมา
  • 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...

  • 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 HickokConrad Hickokวันที่ผ่านมา
    • I meant to add the equation itself is naturally reductive so eventually it reduces itself into the smallest amount it can realize.

      Conrad HickokConrad Hickokวันที่ผ่านมา
  • 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.

    Jay RayJay Rayวันที่ผ่านมา
    • 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).

      J ModifiedJ Modified9 ชั่วโมงที่ผ่านมา
  • I have found the solution. I will release it for 100 bitcoins.

    Soorena AslaniSoorena Aslaniวันที่ผ่านมา
    • lmao

      adamadam15 ชั่วโมงที่ผ่านมา
  • This is how we go back in time

    Lion Pride MusicLion Pride Musicวันที่ผ่านมา
  • 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 RaaruAndu Raaruวันที่ผ่านมา
    • 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 RaaruAndu Raaruวันที่ผ่านมา
    • 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

      Andu RaaruAndu Raaruวันที่ผ่านมา
  • Tony Stark and Shuri probably solved this as children for funsies

    Purple KnightPurple Knightวันที่ผ่านมา
  • I’m pretty sure the answer is 1*1=2.

  • Did they test one Gazillion? I have a feeling about this one.

  • 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.

    kamran pervezkamran pervezวันที่ผ่านมา
  • I still don't get why is that a problem

  • 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)

  • 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.

    Georgia's LibraryGeorgia's Libraryวันที่ผ่านมา
  • lol, I still don't understand what they are trying to solve.

    Pranoy KumarPranoy Kumarวันที่ผ่านมา
  • Lucky I have my calculator 😁

    Fatema TasnimFatema Tasnimวันที่ผ่านมา
  • 3x(+1).

  • 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 AhuoWieh Ahuoวันที่ผ่านมา
    • 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

      Wieh AhuoWieh Ahuoวันที่ผ่านมา
  • 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 ?

    Oliver FreierOliver Freierวันที่ผ่านมา
  • What if we include decimal numbers ? Like 1.85 or like 18.092 ? Thanks.

  • This video is a gem 💎 , although the problem is mind boggling but to express us in this way it's really a gem.

    Debabrata NayakDebabrata Nayakวันที่ผ่านมา
  • 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.

    • ok lets try it 10 31 15.5 <---- this is the problem

      omputer fanomputer fan10 ชั่วโมงที่ผ่านมา
  • 4 2 1? did Jupiter copy pasted this for it's moons orbital resonance?😁

    Dr noBrainDr noBrainวันที่ผ่านมา
  • 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.

    Δημήτρης ΣταυρόπουλοςΔημήτρης Σταυρόπουλοςวันที่ผ่านมา
    • In this case infinity means that for any number you chose in a Collatz sequence, a higher number will follow.

      J ModifiedJ Modifiedวันที่ผ่านมา
  • This seems to be a problem for an algorythm, isn't it?

    Fabian WichelhausFabian Wichelhausวันที่ผ่านมา
  • think of this, not as the question, but the answer ;)

  • Humans creates numbers to understamd stuff and then doesnt understand it. Some problems might be small but are actually really hard.

    Nazeeh .vNazeeh .vวันที่ผ่านมา
  • what about imaginar number?

  • 3x+1 1x3 3

    Андрей НиколаевАндрей Николаевวันที่ผ่านมา
  • 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?

    Robel MorenoRobel Morenoวันที่ผ่านมา
  • 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.

    Alain CoetmeurAlain Coetmeurวันที่ผ่านมา
    • Possibly, but it doesn't "feel" unprovable to me.

      J ModifiedJ Modifiedวันที่ผ่านมา
The Simplest Math Problem No One Can Solve - Collatz Conjecture