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 Ivy 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. I’ve been messing around with some stuff on sheets and I noticed a pattern where 3x+1 is always power of 2 starts when the sum of 4^n from n=0.
    Anyone able to explain this, and if this does stop being true can you let me know when?
    Edit: So is it 3*sum(4^n) + 1 is 3 + 1 + multiples of 4? So sum(3*4^n) + 4

  2. I think its like counting
    when u start counting, it starts with 1 so it always end with 1. To be simple you start something it has an Beginning and ending. A start (counting) ends with the star.

  3. Covered the entire internet and you've manifested downfall of participation

  4. This seems… stupid? So all odds get 3x+1 and all evens get divided by 2. Why? What purpose does this serve? Who came up with these absolutely arbitrary rules that could honestly be anything? Of course it'll always reach 1, it has the potential to be halved over and over, but never to be 3x+1'd over and over as any odd number that goes through 3x+1 will become even but not every even number divided by half will become odd. It is ultimately a reductive formula.

  5. i just began the video so i don't know if this is what we're doing but 3x(+1) = 3??

  6. What’s the point of solving it I wonder. Furthermore what is there to “solve”.

  7. Guys, is it solved or not: lim x to infinity f(x) = infinity. The neatest equation ever invented by Mankind. Congratulations Mr. Tao! My physical observations show, that you're right and that's it! There is nothing more there in Math, but I may be wrong 👾

  8. What a massive waste of time and resources. The very first question should be, "How would it actually matter whether it is true or not?" If you can't come up with a positive outcome either way, then it's a fanciful waste.

  9. I solved this:

    Find the X and Y intercepts for: 3x + 4y = 12
    Y-Int Sub x = 0
    3x + 4y = 12
    3(0) + 4y = 12
    4y = 12
    4, 4
    Y = 3 or (0.3)
    X-Int sub y = 0
    3x + 4(0) = 12
    X = 4 or (4,0)

  10. What of treat as a logical problem? Consider x = 0, than you have 3+1=4, divide and you have your first and unic loop. Than use logic, if no number (zero) leads to a loop, so every number leads to the same loop. No?

  11. Después de la tormenta, siempre viene la calma…

  12. I got 7.5.. tf is the problem here-

  13. I found the loop one
    1 is an infinite loop
    I need my Nobel prize

  14. What happens if you do it with a decimal?

  15. I spent too long at the beginning like "but 1 times 3 is 1… add 1 and its 2…?" Then I realized I shouldn't be doing anything with mathematics when I somehow confused × 3 + 1 with x^3 + 1

  16. i still dont understand the problem so what if it ends with 1-4 loop 😂

  17. its not impossible i put it on my caulcuator and its 3 thats the awnser

  18. Commenting a few minutes into this… isn't it just that the +1 ends up taking away the oddness of the factors of the initial number? And the dividing by 2 means any evening doesn't get put back to being odd.
    Let's say you just did "+1 or /2" instead… 27 -> 28 -> 14 -> 7… Your 3x3x3 became 7x2x2. You still have an odd factor in the 7, but you removed 2 of the initial 3s. Then 7 becomes 8 and it's now all 2s.
    With the x3 included, you're adding some 3s to the new factors, but also evening some of those out with the +1. It slows down the evening out (compared to just doing +1), but ultimately doesn't stop it.
    9 -> 28 -> 17
    edit: And after watching more, that still seems the case. The worst examples (that grow the most) are those with only uneven factors. 9663 seems to only be divisible by 3(?) So 3221 and 3 are its only factors. You need to play with those a lot for them to get closer to something that's all evens.
    It's wonky in the negatives, because you aren't turning a "remainder of 1 into a 4" (kind of) – so it doesn't divide properly afterwards. It's similar to trying it out with "5x+1" – your 1 becomes a 6, not a 4, which won't divide as easily.
    edit2: And I understand that proving that logic applies to all numbers is the difficulty. I'm no mathematician, but seems to me looking into the factors would be the way to go about this. The 3x+1 seems to change the factors in a way that would always break it down into different factors, which eventually end up just being many smaller factors.

  19. two odd numbers in your function usually won't give you patterns. math nerds have at it

  20. I know it’s not real but what about i

  21. It's the "divide by 2" part when it's an even number that this entire equation is about. . The real question is when will it hit 3 or more even numbers in a row so you can divide the total in half multiple times to get down to 1

  22. A mathematical waste of time but mathematicians love numerical puzzles. The Mandel bro set is much more interesting and chaotic. A nice video and very nice graphics well put together

  23. This conjecture is like logarithms in everyday life (except for sound engineers, volcanologists, geologists and electrical engineers). It is useless in my opinion

  24. I can't be the only one who thought the answer was just 1.

  25. Like your videos but this is pointless and a waste of time. There is no benefit to any of this.

  26. Is it odd that I find the conjecture graph to be eerily similar with how the Laniakea Galactic Supercluster looks?

  27. I'm clearly not a mathematician, but it seems like a lot of the video is the statistical analysis of numbers. Please pardon my ignorance, but why not try some different higher dimensional shapes to model the problem? It was my understanding that the introduction of a datum in the number line with "0" created issues we needed to account for with approaches like complex numbers. Perhaps there's some 2, 3, or N dimensional model that can match the 1D number line behaviour; like a projection of a higher order shape of the problem?

  28. 0:08: 💡 The Collatz Conjecture is a simple yet unsolved problem in mathematics that involves applying two rules to a given number.
    3:53: 📚 Mathematicians struggled to make progress on the 3x+1 problem, with some believing it was a Soviet invention to hinder U.S. science.
    6:53: 🔍 Benford's Law is a mathematical principle that can be used to detect fraud and spot irregularities in elections.
    9:36: 🔢 The Collatz Conjecture states that for any positive integer, you can create a sequence by repeatedly dividing even numbers by 2 and multiplying odd numbers by 3 and adding 1, and eventually you will reach the number 1.
    12:57: 🔍 The Collatz conjecture is still an unsolved problem in mathematics, and efforts to prove it true may be hindering progress.
    16:36: 🔢 The Polya conjecture proposed in 1919 by George Polya asserted that the majority of natural numbers up to any given number have an odd number of prime factors, but it was proven false in 1958.
    20:04: 🧮 Numbers are peculiar and mysterious, connecting in intricate ways that we still struggle to understand.
    Recap by Tammy AI

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