The Simplest Math Problem No One Can Solve - Collatz Conjecture


12 mil. pregleda7 800

    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 -

    Datum objavljivanja: Prije mjesec


    1. Quapadople

      0:21 It's not that "Mathematics is not yet ripe enough for such questions"'s the fact that we can't measure or quantify something by lacking the cognitive capacity of doing so. On a case basis it might also be a useless and/or stupid question...asked from the point of view and/or understanding of an individual that is unable to asses its own limitations. Humans have this tendency to think they know more they actually do...and we see that throughout human history through our predictable actions. Understanding we cannot understand some's a monumental achievement. I simply know that I don't know...and I'm okay with that. Edit: 1 may most probably be the universal balance just like everything around a nucleus core will find itself at position 1(one as in the prime order(measured numerically by humans) of universal balance). As an example...let's absurdly assume you could just poor water on earth enough to cover everything. The surface of the water will be perfectly balanced around the gravitational force coming from the center of the planet...and so everything will balance to the universal order of 1. Through force/energy you can change that order...but no matter what it will always come back to 1.

    2. Saarphire TTV

      watching this video at 11PM and literally feeling my brain melting... love it.

    3. michael wilkie

      answers 42

    4. Picholas Cage

      Pff easy, he said there were rules, no restrictions, just add 103 everytime you get to one

    5. tom bash

      How come the number range doesn't start at 0. That would solve the problem, wouldn't it, as 0 would be an even number?

      1. Релёкс84

        Well, 0 is a trivial loop. But that doesn't say anything about other numbers, though.

    6. Не такой Как все

      -7 - the cycle of the number -7

    7. Не такой Как все

      Hi! Why not use negative numbers?

    8. Cezar Barreto

      I found the solution. I´ll publish in an academic paper soon. Working on it.

    9. Flapjack Mollases

      If you notice, there is a pattern where every turn, or equation change in the sequence is signaled by a prime number. It doesn't just simply reduce from an even number to a simple odd number. Each odd number encountered in any path is also a prime number. The "chain" will continue to rise in value until a prime number lower than the starting value is encountered, and then it will reduce down to 1, the lowest prime number. That was my observation doing about 20-30 samples, if anyone found anything different, I'm open to hear it, I'm not saying this is concrete, just what I have observed. The looping effect stems from the fact that 1 and 2 are both prime numbers. Since 2 is both prime and even, it creates a barrier from continuing the equation past 4, since the only multiples of 4 are, 1, 2, and 4. Taking the smallest prime number (1) and applying it to the "growing" equation of 3x+1, then the result is 4. For instance, you will never encounter the number 27, unless you start with this number. Why? there is no integer value that can satisfy the equation 3x+1 = 27. There are also a multitude of even numbers that will never be encountered, unless that number is used as a starting value. What about 54? Since 54/2 is 27? There is also no integer value that satisfies 3x+1 = 54. I haven't done enough tests to prove anything, but I observed that any even number that reduces to an odd number that is not prime is also not present. Even though this appears to be a linear equation, this is very misleading. In a linear equation, we assume any value along that line can be a solution that satisfies X. Because we can only use integers, and not all real numbers, this limits the number of possible solutions to any given situation. So you are starting the problem with the false assumption that 3x+1 is a linear equation (which it normally is), when you are limiting the values of x to integers. When you begin a problem with false assumptions, you can't possibly hope to have a real answer. Point? You can say with pretty strong confidence that any multiple of any number that satisfies the equation 3x+1= 0 is not defined by the problem. Therefore, the "solution" to this problem does not lie in the range of possible values for X. So the problem itself becomes undefined. Take 0 for example. 3(0)+1 returns 1. The smallest prime number. If we then use the rules to apply this 3(1)+1, we again get 4. If we assume x = 0 for each equation, then the first returns 1, and the second returns 0. We would assume then that the answer would lie somewhere between 1 and 0, and since we can only have integer values of x, then it becomes undefined.

    10. Azuleslight

      it makes sense to me, its basically like a tree. where the roots eventually invert to make a tap root so to speak. while the positive numbers branch out like a tree. also like a tree the tap root goes deep and slims down eventually. the divergence happens at zero.

    11. Vikram Mardi

      One two ka four Four two ka one My name is Vikram

    12. dreesee


      1. dreesee


    13. mraksoll

      This will go to infine , and you can it check :) ( try chenge first and see what hapen if give result to 2 :) def col(n): sp = [n] if n < 1: return [] while n > 1: if n % 2 == -100: n = 3 // n - 1 else: n = n * 2 sp.append(n) print(n) while True: col(int(input())) check via this def col(n): sp = [n] if n < 1: return [] while n > 1: if n % 2 == 0: n = n // 2 else: n = 3 * n + 1 sp.append(n) for i in sp: print(n) while True: col(int(input()))

    14. J J K

      We're nothing but smart apes. I hope we go extinct soon.

    15. J J K

      Some problems were never meant to be solved. Stop trying to be God, humans. Just die.

    16. Rob Adkins

      I believe the issue here is that we are trying to using decimals. There is a solution described dividing by 16, by 8, by 2....that's hex, octal, and binary. What if these numbers were translated to binary and then worked through? Just a thought.

      1. Релёкс84

        @Rob AdkinsThe representation method you use changes nothing about the properties of numbers. Also, the binary representation was explicitly used in the video 17:37, and while it's definitely interesting there's nothing that can easily be said about it that couldn't otherwise.

      2. Rob Adkins

        Interestingly, 4,2,1, are the first 3 weights of binary numbers. A decimal 4 written in binary is 100. Again, interesting that the whole number is 100, which is 3 variables plus 1 logic high..3x+1.

    17. Joe Seabert

      Not 3/2, it’s 1/6th

      1. Joe Seabert


    18. Joe Seabert

      Click and clack the tappit brothers said a long time ago, the answer is always 1.

    19. Antoine Bilbord

      The Universe is like this math problem

    20. Mrmace51

      I dont see what is signifigant about patterns set by arbitrary formulae

    21. Humbleaction

      Why is it a +1 though? Where or what is that 1 and why not any other number? Or are all the other problems solved and this is the only 1? Lol.

      1. Релёкс84

        You could also add -1 or +3 if you want, as long as it's an odd number. If it's an even number, including 0, then the answer is really simple. 3x+1 is arguably the simplest variation of the problem for which we have no definite answer.

    22. Nikola Tesla

      Well, I don't know if this means anything but the premise here is strongly related to that of a pyramid... Think of how a pyramid is laid out - 1 stone is at the top, then you have 2 stones below that, which is 3 in total, then below that you have 4 stones, which is 7 stones in total then the layer below that has 8 stones, which is 15 total stones, then below that is 16 stones, which is 31 total stones and so on - but there is always that 1 stone on the very top which is why any number of stones you chose always ends up at 1.. Of course a pyramid is a geometric object so this math riddle obviously has ties to geometry, weather or not if that means something, I don't know? Below is a simpleton "math diagram" of what I'm trying to imply er say here, lol. 1 2 ----3^ 4 ----7^ 8 ----15 16 -----31 1 1+2=3 3+4=7 8+7=15 16+15=31 32+31=63 It's the total sum of stones needed to construct a perfect pyramid.. It doesn't matter what number of stones you choose because you will always end up at the top of the pyramid... It's basically the reverse engineering of a pyramid.. Look, I'm not a math nerd, lol... It's difficult trying to explain abstract ideas but I was just watching a video where pyramids where discussed, a light went off in my head and it made me think of this video and I associated the sum of stones it takes to build a perfect pyramid with this math riddle and it matches perfectly. Now I don't know if geometry has anything to do with this or if this was a coincidence - or even an answer - but I thought it was fascinating to say the least..

    23. Bee Lynn

      This guy likes saying arbitrarily.

    24. Lalitha Janghamaiha

      An awesome equation which will be solved by future guys soon or an AI 😉

    25. Landon Cady

      imagine how you would solve this if the numbers were negitive - hmmm how's that for an math problem :D

      1. Landon Cady

        i wonder would you get the same result or diffrent outcome :D ???

    26. Christo du Plessis

      What happens if you do this with Grahams number? Wouldn't that go to infinity?

    27. Angieghost UFORick


    28. Andrey Z

      How is that sequence, if it takes random quantity of operations of dividing the number by 2 to bring it to odd number?

      1. Charles

        A even number divided by 2 can be an odd number, 6/2 = 3

    29. Wild Malibu

      It’s just a function that tapers down into a pool. That pool being a cycle that math itself created and perceived by humans of this age are not capable of breaking down and seeing further into, once math becomes evolved enough to break the cycle down, we as humans will discover that the exact same thing will take place, it is only in comparison to another intellectual age that we have evolved, in its own system function perceiving from within from a mind of its own era there will seem to be no answer, but on the brink of evolution, inspiration and awe fuels change and new perception on the very material existence of consciousness itself.

    30. mohammed ikram

      Try Pi

      1. Релёкс84


    31. Arlen Gonçalves Martins

      what about zero? 0/2 = 0

    32. tim watley

      If you look at the graph upside-down it will go up instead of down. Problem solved.

    33. Chip

      the name of this channel is based off the Latin word which means truth

    34. Gergely Roboz

      What is the most fascinating thing about this video is that I understand nothing of it, but I still find it super interesting and entertaining. You (and your team) are (an) amazing presenter(s)!

    35. Daisy Do

      Jeremy Bearimy

    36. Flapcow

      So this is calculus

    37. Guts


    38. Rohan Preis

      Wait, what’s the problem?

    39. Chandra Saud

      Really amazing. It's like organic math😅

    40. Ananya Srivastava

      Your fuel for science is so contagious, Derek! Absolutely love your work.. one of a kind creator. Hands down!!

    41. idk anymore

      answer is 21

    42. John Sessa

      Numbers are interesting on their own. But why are humans obsess with them, the numbers?

    43. Wacow

      Somewhat understanding these videos makes me feel like a mathematical genius when I'm getting loaded with my friends.

    44. nostaWrD

      If ever I believe my work is done Then I'll start back at one !

    45. mrfm1704

      Nah nah, my question is, why does 3x+1 even exist in the first place?

    46. Vansh Sharma

      I ve always hated maths and now I hate it even more.

    47. MornThunder

      "communism is the riddle of history solved, and it knows itself to be this solution. "

    48. Sunshine

      I don't need it, but it's mildly interesting

    49. Rocco Castagnola


    50. Elyes Bens

      10:36 you're welcome people

    51. Matthijs Koelewijn

      lowest number must be zero

    52. Владимир Пронин

      hello. I no every write and speak English. You give 1,57!!!

    53. •Oficxllymxli•

      10:38 is he okay??

    54. m&m _kanal

      this stuff hurts my brain

    55. Angga Hadi

      But what is the problem?

    56. Tamunotokini Pokubo


    57. TheRealLori

      The simplest way to avoid spending your life trying to solve this is by only applying it to the numbers 1 through 10 or technically 0 through 10 but zero doesn't do anything. any number that comes before 1-10 doesn't matter eg: 438 only the last digit matters while the 430 is completely meaningless because being an odd or even number is defined by the last digit.

      1. Релёкс84

        @TheRealLori What doesn't matter for what?

      2. TheRealLori

        @Релёкс84 but that doesn't really matter

      3. Релёкс84

        Sure, but the value of the other digits come into play as soon as you start dividing by 2, which starts happening no later than on the second step.

    58. Kazus

      You can't prove it true because it's not math

    59. Dank

      Now I'm wondering. What if plants are more smarter than us humans? I feel like it's somehow connected with math. Okay blame the Soviets for this question.

    60. Eriberto Acedo

      Most interesting when I feel like a challenge I will definitely try this until then I'll work on my Hot Rod, customizing is a challenge on its own because you can't buy the parts you have to create them which I've done to every inch of my hot rod. Measure two or three times then cut shape and fit all other pieces interlock with each other makes it stronger than what you can buy on the showroom floor and looks much better because it's Customized! Better than original did the same with my Harley Davison 1950 vintage totally customize it road around on it, 22 years without opening up the cases just a valve job and replace the rings which I did in the kitchen with my ex-wife helping me worked out great! some people have an Eye for customizing most people don't majority don't. that's what I do along with electrical wiring with no mistakes! From San Diego California ( Customindesign.)

    61. Sean Gray

      this seems similar to the halting problem (proven to be "undecidable" by Alan Turing) Edit: nevermind, you touched on that later lol. I need to remember to watch the whole video *before* I comment

    62. Katherine Wan

      my math teacher told me to watch this lmao-- thanks for the homework fjsfakllfsdflksjf

    63. David Roman

      Even in trying in Base 8 it still gets to 4, 2, 1 cycle But, if you ignore actual numbers and think about even distribution of stuff without numbers then if follows logic path.... The sequence of "stuff" are measured in discrete units of Z starting with nothing (0) and increasing by Z, with a repeating sequence Where Z-Z is nothing and X is cumulative of all Z, and Y is half of X it looks like: Whole numbers only X is the start value in sequence, X can be even or odd Y is always even (proof) if X is even and X=2Y then Y is even Z is always odd and a set value of 1, so Z-Z = nothing and Z+Z=2 if X = Y+Y then X is even (Xeven) If X = Y+Y +/- Z then X is odd (Xodd) If Xeven then Y becomes new X in series. if Xodd then X+X+X+Z becomes new X in series. The sum of X+X+X+Z will always be Xeven, then the disposition of Xodd's will always become even. Even numbers can be expressed as the disposition of two equal lower value Xodd's Xeven = 2Y can be expressed as (3X+Z)+(3X+Z) or 6X+2Z. If Xeven numbers can be expressed as the disposition of lower Xodd, and the disposition of lower Xodds are even, and Xevens are always halved (divided by 2) then all Whole numbers 0->infinity will eventually reduce to the 8, 4, 2, 1, 2, 1, 2, 1, 2, 1 loop unless we introduce imaginary numbers into sequence. then again I am Kinda tired and my eyes are blurry and burning....

    64. gordon b

      boycott Beijing 2022 Olympics ! Countries engaged in committing genocide don't deserve the honor of hosting an Olympic Games.

    65. sedisvacantia

      Mathematicians waste their time with this kind of nonsense, and people in Humanities are the ones everyone makes fun of.

    66. L Ruddy

      ok but whats the point, wheres the real world application of multiplying odd numbers by 3 and adding 1 and dividing even numbers by 2

      1. LordGaben

        Learning, expanding knowledge. It might lead nowhere, but the next problem solved might be the gateway to humanities future of abundance. Point being to ignore the problems of they seem pointless means sticking with the status quo and as we know that is the ultimate in destructive conservative futility.

    67. Nikola Tesla

      This is hilarious, it's like a riddle for math geeks in their language, lol.. It's like watching Rain Man trying to figure out "Who's on first?" lol

    68. Trapster

      what if i pick the initial number to be Zero? Zero is a even number, so the rule says we divide it by two. So, 0 / 2 = 0. Hence we break the loop of odd number and the end result will always be zero.

    69. Melvin Shelton

      I wonder how many students have quit mathematics in dispair over not being able to solve problems like this.

    70. the warion

      My smaet ass "its x4"

    71. Rob


    72. Joshua Banda (Jkaksb)

      In other words it's true

    73. wAdjwakdjawkidj

      I created another loophole for Collatz Conjecture. Which when you start off from 21 you end up still on 21, I also apologize for creating another loophole.

    74. yunus emre akca

      isnt infinity a loop?

    75. Markus Müller

      Great video but what happened here 13:55 ?

    76. Stefano

      Would decimals make a difference?

    77. R Dodds

      I suppose someone has tried this with decimal numbers?

      1. Релёкс84

        Since when are decimal numbers odd or even?

    78. Bool False

      19:46 - *We have no right to have solutions to all of these other problems*

    79. BananaKingdum248

      do it with 350

    80. Callibor 31

      This equation sounds like a law of infinite. And unlike the laws that humankind made, we broke trillions of them in a matter of seconds under 3x+1. Would everyone be sinners then if this equation for some reason is a law of infinite justices that even our own justice system violated?

    81. AMV FROM- Mr. Editzz

      Isn’t it 10

    82. WolfY

      So, why 7 and not other nine numbers? Why do I always see such thing when people are choosing a number from 1 to 10 and it's most of the time 7? I DON'T NEED SLEEP I NEED AN ANSWER!

    83. Moreno Franco


    84. Moreno Franco

      Amazing. My father showed me this when I was 10. That was 59 years ago. Now I instantly see patterns - in life and in peoples behavior. I see it in the traffic, and in crowd movements. Thanks, Verisatium Team.

      1. Sophistic History

        @DJ-murlock Animation ............GIGGITY...........

      2. DJ-murlock Animation

        You're 69 now? Nice

    85. Fabio Müller

      Is it possible to proof any prime number comes back to the original 4 2 1 sequence? Because of that is true we'd only need to proof any starting number eventually comes to a prime

      1. omp199

        That doesn't sound any easier.

    86. Hayate Azekura

      Sometimes I wish I wasn't such a dumbass, So I too can understand anything

      1. omp199

        Surely you can take comfort from the fact that when it comes to this notorious unsolved problem, nobody understands it. Here, even the world's greatest mathematicians are "dumbasses".

    87. ankokuraven

      Another great video

    88. Manny Norton

      The alleged sort arespectively heap because gemini psychophysically employ as a bad resolution. guttural H habitual, heavenly heavy hellish colt

    89. Tobi L

      My question is: why is that a “problem”? It is just a patrern

    90. Renato Gabriel

      Humans can live happily ever after without 3x+1

    91. Máté Kurucz

      it makes me so proud that most mathematicians mentioned from the same country were hungarians in this video. such a small country yet there are many smart people who pave the path going forward, even though i lowkey didnt understand anything in the vid :'D

    92. Ryan Betker

      Seems like someone made it up as a joke to have people spin their wheels. If you can continually divide an even number result by 2, and the setup has a propensity to produce an even number, of course it's going to get smaller. More simply, look at the last digit. Any odd number x3+1 is going to create an even number. Any even number x3+1 will make an even number in the next round anyways.

      1. Релёкс84

        It's not that simple. You could very well have a sequence of increasing odd numbers separated by even numbers. Some starting numbers reach thousands of times their initial value that way, and there's no reason to think it's impossible to grow to infinity.

    93. koyou33

      lol fractrum what a load of rubbish

    94. hassan abdeen satti

      the logo of the video

    95. hassan abdeen satti

      its 10

    96. John Westfaul

      Does it only work with whole numbers tho

    97. Oisin McCaul

      Just gonna throw out square root of 421 Billion

    98. M Najib I

      AH... My head, hurts

    99. Renu Singh

      The number q is not in this loop 😎