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

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. —

The Manim Community Developers. (2021). Manim – Mathematical Animation Framework (Version v0.13.1) [Computer software].

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 —


  1. Given these specific laws of composition and decomposition, the whole system should inevitably grind to the smallest possible decomposition since decomposition is stronger

  2. The answer for the question in the thumbnail is 3

  3. You can never find a counterexample that is climbing. I mean when do you stop. It will go on to infinity.The only way is to find a loop.

  4. I think that the answer to this solution lies in the nature as it is the manifestation of all knowledge

  5. Eventually we'll get caught in a power of two which crashes us down to 1.

  6. if it always reaches 4 and loop, why dont they start backwards from 4?

  7. Wenn Du die gleiche Regel im negativen Bereich anwendest hast Du eine ganz andere Perspektive. 3x+1 verhält sich negativ wie 3x-1 im positiven Bereich wäre. Du spiegelst die Achse.

    Entsprechend hat 3x-1 die GLEICHEN Loops im Positiven wie die 3x+1 im Negativen.

    Diesen Teil fand ich jetzt doch offensichtlich unsauber. Die Dimensionen stimmten überhaupt nicht überein.


    The same rule of the positive dimension applied to the mirror of the negative dimension is not the same. The equivalent to 3x+1 in the negative realm (to "grow" the axis by 1) is 3x-1. Thus using 3x+1 in the negative realm creating those 3 loops is like using 3x-1 in the positive realm creating the same.

    That part of the video was not well researched. Comparing different functions for their respective dimensions is not clean.

  8. [3+1=4][(4/2)/2=1] The X's cancel out. There is your answer.

  9. İngilizce bilmeyerek bir süre dinleyerek anlamaya çalıştıktan sonra ben: level up

  10. Ah yes, the first result of me having a 🦆ing seziure and slamming my keyboard for searching “ù< ùñ% ~Nõ%⊫ ñ% ù öõq'yy `< ø⊫6'y't'6-“.

  11. “But have you tried putting it into wolfram alpha?”

  12. This kid one time went up to me and asked “are you good at math?” I said “yeah kinda” and he said “then what is 3x+1?” As a veritasium subscriber I knew he had no idea what he was talking about and just looked at the thimbnail 🤣

  13. This video does a poor job of explaining what the goal is? What are they trying to solve. It’s a chaotic pattern. So what? Are they trying to discover a pattern? Trying to find a number that it won’t work with? What’s the point of this video? This channel sucks.

  14. hmm could the organic looking structures must be a result of the relatively arbitrary choice to use base 10? I wonder how those sorts of structures would change under different bases. Fascinating.

  15. In one dimension, the pattern is simply unsolvable. In two dimensions: that holds given that it after all boils back to one. However, in reality; space is complex. That is, multi-dimensional. Such that no object is immune to transformation. Brute force? NIL.

  16. I am sorry you should start with why do I want to solve it not how I can solve it 10 min in and still I don’t care about the problem cuz I don’t know why I want to solve it in the first place

  17. im confused on what the actual problem is, there is nothing to solve?

  18. You go into fractions and you stop the loop.

  19. Also interesting to do a *Collatzception*: start with whatever number, count the steps and enter the number of steps as a new number. All the tests I did came to 5. And 5 needs 5 steps to get to 1. 16 leads to 4.

  20. What is the "problem" you're trying to "solve" that there might be one number that doesn't conform to the principle of 3x+1? What would the relevance of that be of you proved it either right or wrong?

  21. Nothing is perfect! Like in nature, square root does not exist to infinity!

  22. I'm pretty sure it's always possible to get to one, because eventually, if you get an infinite number of chances for each number to get to one, than it's always possible.

  23. In this game, there’s more chance of hitting even than odd. Odd +50% even -50%. Even always follows odd. Odd or even can follow even.

    There’s also a chance of hitting a geometric sequence of evens (64, 32, 16, 8, 4, 2).

    I probably missed something important but that’s how I think about this.

    Rules of the game are rigged in favour of evens always winning (eventually) and then getting trapped in 4-1 loop. The loop is just because of the +1 I think.

  24. If I do solve it, what do I get? And how is this applied?

  25. The amount of intelligent people on the comment is staggering. While I'm here trying to light my wet zippo with a candle.

  26. The narrator suggests that this is something that can't be brute-forced, but I'm struggling to understand why a supercomputer couldn't simply churn through numbers, checking them one by one? Given modern computing power, I'd think such a setup could surpass the current known limit relatively quickly.

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