The Simplest Math Problem No One Can Solve - Collatz Conjecture

Dipublikasikan tanggal 29 Jul 2021
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 brilliant.org/veritasium 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.

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References:
Lagarias, J. C. (2006). The 3x+ 1 problem: An annotated bibliography, II (2000-2009). arXiv preprint math/0608208. - ve42.co/Lagarias2006

Lagarias, J. C. (2003). The 3x+ 1 problem: An annotated bibliography (1963-1999). The ultimate challenge: the 3x, 1, 267-341. - ve42.co/Lagarias2003

Tao, T (2020). The Notorious Collatz Conjecture - ve42.co/Tao2020

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. - ve42.co/Tao2019

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. - ve42.co/Conway1987

The Manim Community Developers. (2021). Manim - Mathematical Animation Framework (Version v0.13.1) [Computer software]. www.manim.community/

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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

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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 - ve42.co/3DCoral
Coral visualisation by Algoritmarte - ve42.co/Coral

Komentar

  • Your way of Explaining through Graphics is beautiful sir.

    • wanna work on that

    • TURN TO THE LORD JESUS CHRIST BEFORE ITS TOO LATE, GIVE YOUR LIFE TO HIM AND START WALKING IN OBEDIENCE, WITHSTANDING FROM ALL SIN AND WICKEDNESS, JESUS SAID THE PATH TO HEAVEN IS HARD AND NARROW, AND FEW FIND IT. MATTHEW 7:13-14, HEBREWS 5:9, JOHN 14:15, MATTHEW 7:21-26, 1ST CORINTHIANS 6:9-10, JOHN 3:16-21, JOHN 10:7-8, MATTHEW 10:26, AND LUKE 13:5. GOD BLESS YOU ALL.

    • Mathematician was correct waste of time

    • Agreed

  • As someone who goes to Syracuse university, totally caught me off guard when you mentioned it’s also known as the Syracuse problem. But I was more shocked it wasn’t a coincidence, looks like the math department here has spent a bit of time on the problem and picked up the name. I think they’ve given up on the problem because I hadn’t heard of it lol

    • @Woos Because he goes there and hadn't heard of it. No difficult.

    • @Paige Rasmussen Or a drinking school with a math problem.

    • Dogs, cats, birds use math without much in the way of a mathematical language and some are much better at it then others, some cats take forever on deciding on a jump other do it without even slowing down and always land on the target location and some house cat breeds are much better at this then others. small slender cats with long bodies and large hind legs for jumping and are very good at turning their bodies in mid air to grab prey and birds on the way down, the better hunters take their time and push their hind legs into the ground and hold still trying to remain hidden until the prey looks away and before the jump some cats eyes change color a result of the cats night vision that could be a result of the amount of concentration used by the cat. man tries to turn everything into a language. animal can even figure out what some human sounds mean without repeating such sounds, a common one is the word for food, cats can learn what certain signs mean. a petting gesture can turn into a meaning, make the gesture and some cats will stop and decide if she or he wants to be petted, you can use a gesture for the word food and some cats will walk over to their dish, some cats know the game fetch but will refuse to play when dogs , cats even when other human are around and unlike dogs always seem to place the toy right in the center of the hand. the cat version normally means knocking the object around then grabbing it and bringing it back and placing in the center of the hand for the next toss and may even end the game by mocking a dog by laying down like a dog and sticking its tongue out. Cats love to observe and the amusing thing is cats sometimes attempt to teach humans.

    • So we're not just a drinking school with a football problem but a math school with a drinking problem?

    • @Woos cuz its something important he had no idea was going on in his university

  • Hahaha the part 12:34 where he shows himself while saying "One of the worlds greatest living mathematicians..." and then showing Terry Tao, was a funny little touch :D Appreciated.

    • @Shawn A cat uses math to determine how far to jump to land on a target location or prey, this type of math is also used in sports. math without a mathematical language that may be tied into instinct, memory , subconscious mind and experience, studies on beavers lodges have suggested that beaver may be good mathematicians who work with a blue prints.

    • Had me in the first half, not gonna lie.

    • The birds and cats know math long before human added language to math to feel special.

  • Someone should try graphing the relationships between the digital roots of the target numbers. The thing I noticed right off is that in the base 10 system (0 - 9), the "+1" in the algorithm never allows the target numbers to have a digital root of 3, 6, or 9. I.E. if you start with 7 (putting the digital roots in parenthesis), it would look like this: 7(7) - 11 (2) - 17 (8) - 13 (4) - 5 (5) - 1(1) If you were to start with a 3, 6, or 9, the "+1" will force the digital roots back into the list of [1, 2, 4, 5, 7, 8] 3(3) - 5(5) - 1(1) A similar type of pattern emerges when you look at the digital roots of doubled-numbers. For example, if you start with any number other than 3, 6, or 9, and double it, then get the digital root, you'll never get 3, 6, or 9, but if you start with 3, or 6 you'll only get 3's & 6's and starting with a 9 you only get 9's. 1(1) - 2(2) - 4(4) - 8(8) - 16(7) - 32(5) - 64(1) and the pattern repeats 128(2) - 256(4) - 512(8) - 1024(7) - 2048(5) - 4096(1) etc. You'll notice in this sequence in particular, you once again have a 1, 2, 4 relationship with regard to the digital roots, in that the sequence goes: +1, +2, +4, -1, -2, -4 vs. 3(3) - 6(6) - 12(3) - 24(6) - 48(3) - 96(6) etc. vs. 9(9) - 18(9) - 36(9) - 72(9) etc. So, because the even numbers are always divided evenly and the odd numbers are always made even, it always pulls the digital root trend into [1, 2, 4, 5, 7, 8]. In cymatics, frequencies that have digital roots of 3, 6, or 9 generate standing waves. Adding or subtracting a hert will make the nodes travel forward or backward. It's hard to say how, but I think this is all related to how these numbers trend, because even if you start with multiple digit number that has a digital root of 3, 6, or 9, the next action will make its digital root a value of "1" before dividing by 2, which makes the next digital root 5, every time. In example: if we start with 15, which has a digital root of 6, then 15 x 3 + 1 = 46 (digital root of 1) / 2 = 23(5); if we start with 27, which has a digital root of 9, then 27 * 3 + 1 = 82 (digital root of 1) / 2 = 41(5); or let's try 417 which has a digital root of 3: 417 * 3 + 1 = 1252 (digital root of 1) / 2 = 626 (5) / 2 = 313(7) ...and you're back into the list of [1, 2, 4, 5, 7, 8] every time.

    • I was just gonna say this exact same thing. /s

    • This is super interesting, I also noticed something similar which is why I tried this equation using fractions. Example: 1.5 x 3 = 4.5 + 1 = 5.5 x 3 = 15.5 + 1 = 16.5 / 2 = 8.25…etc. The problem is that the .5 wasn’t changing much, so I decided that the rules apply to the .5, .7, .2, only. Example: 2.2 / 2 = 1.1 x 3 = 3.3 + 1 = 4.3 x 3 = 12.9 + 1 = 13.9 x 3 = 41.7 + 1 = 42.7 x 3…etc. This opened up a whole lot of opportunities and I did get close a few times, I also tried fractions like 13.568, but sadly even with the fractions I ended up in that same loop, it just took longer to get there.

  • Fascinating to see the correlation between the randomization of this equation and the patterns in nature and beyond, which should really be the focus in my opinion becasue the equation or problem itself doesn't seem to be a problem at all. I mean take replace 3 with 4 or with 5 and see what happens. There is always a pattern. It just so happens that the pattern of 3x+1 will always result in a sequence of dividing by 2 that will eventually overcome the multiplication of 3+1 reducing it to the 4-2-1 loop. I consider it solved, the solution which was presented in the video. Again, the real intriguing aspect is how that particular problem replicates randomness in nature, and well even other facets of this world including the stockmarket...hmmm..can anyone say matrix lol..perhaps this demonstrates that if you know the starting numerical value of anything, you can see how it will grow and perhaps when it will decline..ooops crazy conspiracy stuff leaking out now.That all being said, great job on the video!

    • @Ian Allen Appreciate that. I find that many complex problems are easily solved through simplification and visualization.

    • The problem, brian, is that an odd number can be expanded by 3/2 multiple times in succession. In fact, countably infinite number of times (that is to say, every bridge upwards is absolutely finite and eventually ends, but they grow infinitely long), and to make matters worse, you can go up a bridge, fall back some ways, and land on a second bridge that takes you even higher. And this pattern as well, repeats infinitely. But good on you, seeing that it is solved. Math is easy, when you aren't actually counting.

  • Math problem no one can solve: Exists Me: Finally I'm not the only one who is bad at math.

    • @Azlan Adil BRO I SEEN IT AT GOOGLE

    • @Lilliana Pabon It was a true or false question. What do you mean 7?

    • I solved this question with google,THE ANSWER WAS 7

    • @The Reality Gab Umm… If you pick a number multiply it by 3 then add one if it’s odd, and divide it by 2 if it’s even. Will positive integers always end up in a 4,2,1 loop. Solve that.

    • Create animations with 3x+1

  • For negative numbers, the equation should be 3x - 1, and you'll get the same results. (@ 15:04 min)

  • 15:33 i dont know if anyone else noticed this but the amount of perfect squares in each number roughly equals the sqaure root of that number. sqrt(100) = 10, sqrt(1000) = 31.6, sqrt(10000) = 100

    • @Omgdodogamer Yea you already said that. You formulated your conjecture, and I'm asking if you can prove it. I for one can, and it's a good exercice to try it yourself.

    • @Релёкс84 if you google perfect squares between 1 and 1 million it says 1000 and the square root of one million is 1000

    • @Релёкс84 15:33 "How many perfect squares are there in 100? 10." and the square root of 100 is 10. 1000? 31 and the square root is 31.6. 10000? 100 square root is 100

    • How strange... can you prove it?

  • There’s actually a simple way to prove that a divergent series can be balanced by a convergent series due to the combined functions, but it’s too much of a hassle to explain. 3x+1 always is an even number, /2 is only 1/2 odd of the time. By nature you can’t assign a limit to a non-converting series so the true answer is to ignore the givens, but apply a limit to the results using probability, and you’ll get a nice tidy converging answer.

  • The more I know about math and science, the more I realize how much we don’t know

  • I absolutely love how mathematicians always find the most random things to debate over!

    • As it is chaotic in nature, stubborn and inquisitive = mathematicians

    • @Christ Loen the solution in itself is worth it. math is art, after all.

    • this video is not about mathematics...

    • @Orezio Pancrazio There are different types of people in this world. Some may love to do theory like me. I am a mathematician turned to pharmacy. I love modeling diffusion of particles. Its fascinating. Can you imagine being a pharmacy major and talking about math? Some people spend their whole life in way of religion. Some in the way of dancing, art. Its only a dead end until someone opens it. When looking for oil in the US, everyone thought its a dead end until Drake first found oil. everyone else had left. thats why he won. Always invest time in hard and dead end things cause thats where opportunity lies

    • @Conner blalock I like to think someone somewhere on ID-tv has commented THE counter example but we will never find it amongst the trolls

  • What an amazing video. I have learned so much from just the side ways you took the explanation.

  • Im sure someone has tried this already but my first instinct to solve this would be to try to prove that all prime numbers terminate in 4,2,1. If all prime numbers do than that implies every number reduces to 4,2,1.

    • @Max Muir what about odd numbers that aren't prime? edit to pe more clear. if you prove that all odd numbers reach 124, then even numbers do as well, which is probably what you are thinking about, but why do non-prime numbers always reach one of their divisors?

    • @Joe Heitman eventually they will I could takes years for some numbers that are insanely big but yea they will and there is no other way for a loop to happen unless a prime number can be divided by 2 that isn't 2

    • @R L all prime numbers are odd and all numbers dived by two will eventually be odd so if odd number can't loop then on 1 2 4 can

    • I'm wondering how you got the idea that this is true. I know it might be true, however at first glance, I don't know why it would be true. For example, if we know that it works for 3 and for 5, how do we know that it works for 15? Is there some pattern I missed?

    • why would this lemma be true?

  • Honestly I feel like the negative numbers thing isn't that confusing It's the same as changing it from 3x+1 to 3x-1. 3x-1 has the same loops as negative numbers in 3x+1, just positive rather than negative

  • The problem "does the Collate Conjecture hold" is trivially decidable. The definition of "decidable" is (simplified) that it is possible to create a Turing machine that gives the correct answer. It is trivial to create a Turing machine that outputs "yes", as it is to create one that outputs "no". One of these machines gives the correct answer to the question. For decidability, we do not need to know which one. More generally, only problems with an infinite number of inputs can be undecidable, IF the conjecture is false. So the question "does 3x+1 terminate" may be undecidable, assuming both the answer "yes" and "no" occur for an infinite number of integers.

  • Fun fact: We are not mathematicians but we got interested by this.

    • I somehow get curious enough to open these videos but by halfway thru I’m so lost it’s a waste of time for me to keep going. 😵‍💫😵‍💫. Yes, I have a college degree, studied physics, algebra etc but never got not into math enuff to follow this stuff.

    • We want not look stupid .. but here we are 😭

    • @Amir Pakravan A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems

    • Fun fact: We are not mathematicians but youtube recommended us this video

    • @JustMonikaHerex No, the answer is the sequence 4, 2, 1 :S

  • always when i hear about problems like this - easy to explain, hard to solve - i imagine some other dimensional species looking at us thinking "oh they're so close but yet so far away. why don't they look at this problem *this way*"

  • This problem is beautiful and it touches on a vast number of fundamental concepts and outstanding conjectures. It is one of the greatest travesties of modern mathematics, that it is always introduced as "a dangerous problem that people are warned not to waste their time on." For shame.

  • You all should realize by now that 3x+1 will always point to an even number. If that number have prime number consisting of only 2, then it will get to 4-2-1 loop. So this equation will result an even number and the procedure is finding the number with all of its prime numbers consisting of only 2.

  • The best video on youtube i have ever seen and amazed that for such a trivial looking problem, no one has yet been able to prove/disprove it.

  • Whoever created all those graph animations is an absolute master in after effects expressions

  • What do you think happens to decimal numbers when followed the colatz conjecture?

  • Didn’t watch the whole video, but I consider it more of a stats problem. There are infinite possible inputs, so many possible patterns, but think of it like this; while dividing and multiplying, odd multiplication adding 1 results in evens, but many evens when divided by two can create another even. So the initial function has a likelihood of increasing it, but as we test more and more numbers, the division by two takes control. Why? Because we’ve allowed it to divide multiple times by two when it is even. You have a 50/50 chance to get an even when you multiply and you also have a chance to get that when you divide. You will always multiply once, but there are many times you will divide more than once. Not sure the actual odds, but I can imagine it’s a back of the envelope formulation. This way I can imagine it’s more likely to eventually go down than up which will lead you to 1. Or you will just hit some power of two randomly along the way anyways, which technically if we run this an infinite number of times, we will at some point do it with any number and immediately get to 1. I think it’s a combination of those two things. I think that’s actually pretty intuitive.

    • That does not mean there can't be a higher loop, or a sequence that increases indefinitely due to some currently unknown pattern. There are sequences that go far above their seeds.

  • Reading these comments, I'm seeing many people realizing that they number of steps remains consistent or close too consistent for many sizeable ranges of numbers. This is because of the tree that is the collatz conjecture in reverse. as in start at 1 and for each number you ever reach, choose to either multiply by 2 or sub one and multiply by three. This generates that cool looking tree they showed during the video. So those similar ranges are actually expected, because those values are simply at similar depths in that tree. What I've noticed that I think is more useful is the power of 0 in these sequences. If you only look at the last digit, 0 is your best case. it makes you /2 and will be either 5 or 0 again. but its that ability to recursively hit 0 that gets you some large drops in the value no matter how high you started. Also there are some "core branches" that show up more often than others. Obviously there's 2^x, but also (2^x)*10. which makes me wonder, could you prove that any number ending in 0 must converge? because if you can do that, you get 1-9 for free since that's doable even just by hand.

  • Graphics is fascinating. Thanks for all the hard work.

  • Oh my god, this poor animator. That is a serious amount of dedication. Looks fantastic!

    • @Mehtab Traveller Maybe I'll work with this someday Python fascinates me everyday

    • @Good Nintendo player how do u know it’s a “he”

    • @Lucky The Luckless Wolf I know I am

    • @Llama Man no, you're amazing

    • He needs to be paid every single day $100,000 heh

  • feel like i d never ever use this knowledge but it feel so good to learn so much!

  • So here's something interesting: if you can just prove that for all numbers >1, each number will eventually go to a number less than the starting one, you'll prove the hypothesis.

  • Although this may be of no value, it occurred to me that there is no upper limit of numbers, but there is a lower limit (the number 1). If there was no lower limit, the calculation would go in forever. The lower limit is the reason it stops.

  • If you can make every single positive integer out of a combination of 4s, 2s and 1s, then every number can be reduced to that loop. There's no number that can't be made out of 4s, 2s and 1s

  • Everyone here: "...but just a maaaaybe I'll be the one to solve it."

    • 😂😂😂😂😂😂

    • XD

    • For real

    • @Milena Vision I did, I’m right, you’re wrong lol.

    • @ℳꫀᥣꪮdᥡℳꪮꪮᥒᏟᥲkꫀ᥉~~ Sorry its not but heres the answer 0

  • I absolutely loved it you did have some divergent moments..... Don't let dark chocolate confuse you of it not being chocolate.... That's where some of your scientific arithmetics hit the wall

  • After watching this, I feel like this 3x+1 to an odd number and x/2 to an even number is just a cycle to get every starting positive integer to a number on the 2^x line, which 1-2-4 is the start of (2^0=1, 2^1=2, 2^2=4, etc.), and because 3x+1 will always make an odd number even, and x/2 will always, eventually, make an even number odd (with the end goal being the odd number 1 that only lays at the end of the 2^x line), I feel like every positive number will end at the 4-2-1 loop

  • 15:52 completely unrelated note, did anyone else notice the parabola appearing on the graph? It makes sense for one to appear since it's graphing perfect squares, but I still find that kinda cool

  • The animations that Veritasium uses makes every video more complex than what it seems.

  • This math problem is actually like my trading portfolio, I can start with any number but end at $ 1

    • you posted the solution in your post there^ - hint; single character

    • Then start at 0

    • @Luca Disgusting Light Mode

    • @Anonymous000 I love it when a plan comes together

    • Bruh I thought there was a hair on my pnor

  • According to me most of the time when we add 1 in 3x there is one time the number which we will obtain as even will be a factor of two. So i think if you want it to be infinity then start with infinity. Because if you start with any number it will eventually get to 1 until it's decimal or irrational.

    • you had it there, the answer, for a brief moment near the end of your paragraph, then poof - gone

  • It was an amazing video, thank you. I am not a mathematician at all but I do have an observation/question/thought. Maybe the answer is not within the problem but outside the problem. This might just be babbling and outrageous so pardon my ignorance. Has anyone tried to develop the same problem for numbers in different bases? Like Sexagesimal numbers? The hypothesis would be that there is a number or series of numbers that are shared among Collatz conjectures of different bases. Of course it is unlikely all bases have the same Collatz conjecture model (3N +1 for odd numbers and divided by 2 for even numbers). I guess each base has its own Collatz conjecture model and that needs to be found first. So, after finding each base model, could be possible to find those numbers that are shared among conjectures. One or more of those shared numbers could be subject of a different loop than 4-2-1 in the base 10 Collatz conjecture. Probably the interaction of those numbers could create a unique loop different from each conjecture or a different unique tree that is not constrained by the base of the numbers used, and has its own behavior. What are your thoughts?

    • Well, the problem doesn't depend on the basis used, so it doesn't matter which basis we use. The only thing that matters is whether a number is odd or even

  • I wonder if there's any correlation between the base number (x) and the number of steps it takes to get to the 4, 2, 1 loop

  • Thankyou for putting in the effort to make this video I actually found it really interesting

  • Nice work Soviets. You got me.

    • you fell into the 'trap' by yourself. Please take all the credit. What is a 'soviet'?, Propaganda is significant;y older than your avatar's face looks

    • Found the mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

    • @Ali Akram wwwwwwwwwwwwww¹wà

    • It's 4X the X doesn't mean multiply

    • Thehh uhhhh got me

  • A googolplex (10^10^100) is far bigger than a millinillion (10^3003). I also think I can prove that 1 Millinillion is clearly a Type 2 -illion and that (10^10^3003) is familiarly known as 'Maximusmillinillion' or 'Millinillionplexed'

  • Hi! Being proud as a hungarian as you mentioned Pólya, Erdős... and others who aren't mentioned here like Neumann, Teller, Kemény... just one thing about the pronounciation of hungarian names: Erdos is written with this letter: O-->Ő which sound like the french vowel "oe" in the word oeuf.

  • Maybe the problem is that everyone is calling it 3x+1 when actually the active part of the problem is the divide by 2. The operation is attempting to get to 1 and every time it hits a roadblock of an odd number the 3x+1 resets it back to even to continue the operation dividing by 2 to get to 1

  • If you use any multiple integer number of all three variables you get a different number pattern that still repeats, even if you use different multiples. Edit: The pattern can have a different length too.

  • A big shoutout ot the graphics department for making this 100% more understandable!

    • TURN TO THE LORD JESUS CHRIST BEFORE ITS TOO LATE, GIVE YOUR LIFE TO HIM AND START WALKING IN OBEDIENCE, WITHSTANDING FROM ALL SIN AND WICKEDNESS, JESUS SAID THE PATH TO HEAVEN IS HARD AND NARROW, AND FEW FIND IT. MATTHEW 7:13-14, HEBREWS 5:9, JOHN 14:15, MATTHEW 7:21-26, 1ST CORINTHIANS 6:9-10, JOHN 3:16-21, JOHN 10:7-8, MATTHEW 10:26, AND LUKE 13:5. GOD BLESS YOU ALL.

    • Yet I still can't understand

    • @Gvko A big shout-out to you who couldn't be bothered to proofread!

    • WHO DA HECK IS UNLIKING AND LIKING THIS COMMENT I KNOW UR THERE

    • Ah yes, 999 likes

  • One thing interesting when I try it is that when the num is 2^¹⁰⁰⁰ , leave it running for 30 minutes and almost all of the number took 7248 steps, some are 7249/7229, others 7428. It seems like it can't go further than that for some unknown reason

    • It seems in general that when you test numbers in some relatively range, the number of steps tends to be from a very small selection, e.g. testing the numbers M+1 through M+1000 for various M gives the following numbers of steps: M=2^1000: 7248 (97.0%), 7429 (3.0%) M=2^1001: 7249 (93.6%), 7430 (6.4%) M=2^1002: 7250 (89.7%), 7431 (10.3%) M=2^1100: 8121 (51.5%), 7428 (48.5%) M=2^1200: 9087 (100.0%) M=2^2000: 14726 (67.3%), 15243 (18.6%), 13157 (14.1%) M=3^630: 7854 (50.0%), 6652 (48.7%), 7642 (1.3%) M=3^631: 6651 (62.6%), 7853 (35.0%), 7641 (2.4%) M=3^757: 8001 (95.2%), 8973 (4.8%) M=3^1262: 14553 (99.5%), 13320 (0.5%) M=5^861: 14490 (95.9%), 14322 (4.1%) and just to make sure this also happens with "random" numbers rather than ones that are a prime power: M=round(Pi*2^2000): 13810 (82.6%), 15118 (17.4%)

    • your process is not the fallible part

  • It's a simple mathematical formula that can be explained similarly to entropy. I cant wait for humans to solve this one haha.

  • 15:05 I like to think of zero as a mirror between negative and positive numbers, and multiplication and division reflect nicely, but addition and subtraction are more lateral, you could say, so they don't reflect. So can there be a formula that results in the same 4-2-1 loop on the negative side, instead of 3x+1 and dividing by 2?

    • you posted the actual/only answer in your post 3 times. Hint; single character

    • 3x-1

  • What do you do with fractions to determine odd and even?

  • I like how you asked us what colors would represent odd and even numbers before making this video. And according to the results for most people the odd numbers would be red and even numbers would be blue just like they are in this video.

    • evens are green odd is orange of course, but three is blue

    • @Veritasium What about 0 if we take 2 cases 1st 0 is even & 2nd 0 is odd This could be the connecting link between negative and positive chains

    • @Xandor Kaine No the week starts on Sunday and every calendar has that...

    • @Veritasium Great vid as always. Now, make a vid where you and Xyla put that treadmill cart on a long conveyor belt, maybe 100ft long. But start shorter, maybe 50 ft. In the interest of science, clarity, and honesty, my friend. But mostly honesty It will be a great vid, and you will very quickly garner a million clicks, likes, subscribes, etc. so whats not to like about that? Time to get to it, my friend......

  • So Basically, you’ll eventually run into a power of 2, which will bring you down to 1. This can also work if you just do x+1 instead of doing 3x+1.

    • @Lorand Horvath I use stopwatch and obviously remember the time that I start to run it, The activity monitor in my laptop also show how long java has been using my CPU just for this loop, I couldn't really change anything in the program as it's already running, I simply make it so that it stopped and show an output when the number gets to one

    • @FOS You could implement some checkpoints to see where you're at, so you can estimate the time Is your code running blindly?

    • @Lorand Horvath Hei, if you're going to write a code about this, If possible, try 3^2378284 It has a million digit and when I put it into my program, it already takes 4 hours and it hasn't stop (hasn't hit one). I know it might be long because it's a very very big number, but in theory it only has 30 millions steps, my laptop could handle 1 millions steps in a minute just fine. So I'm really curious if I found something interesting here Edit, More than 7 hours and still going

    • @FOS Nothing in particular. Besides, there are other loops like that (seed 17 also has an unique loop, while some numbers seem to grow rapidly and then abruptly stop at around 10^16, not going higher. This might be another loop, although a very long one

    • @Aleksander Karch Do you know anything unique on why the second loop exist?

  • I looked at the thumbnail, and felt pretty confident that I can solve this. The next second I start laughing uncontrollably. Never have I ever laughed at my own stupidity like this.

    • excellent, A++ - I would hire you based on that statement alone

  • If they want to find an exception, they should be focusing on plugging in prime numbers. I think it's safe to say all non-prime numbers will eventually reduce down and fall into the ending pattern (and sometimes the whole, complete pattern) of the loops their factorials have, so only prime numbers have the potential for yielding a non 4-2-1 loop ending pattern. They should crunch the numbers on just primes and see if any of them yield an exception with which the equation could be modified / corrected as needed.

    • I don't think being prime is really relevant, since after a 3x+1, the factors of the next number will be unrelated to the prime number itself

  • If you apply the function 3n-1 to the negative integers, it would only have one loop of -4,-2 and -1. Similarly, if you apply the 3n-1 function to the positive integers, there would be many independent loops found.

    • Not "many" - three, as far as we know.

  • This sounds like a problem that we will one day show to a chaotic, but brilliant and creative child/teenager and he will just give us a counterexample in minutes and no one would know how

    • @Zain Elsayed zain the incel

    • @Stolfo Ch. No, Pags is the fool, not you.

    • @IrokoSalei that’s not what i meant. im not saying that there arent any genius women out there.

  • Now this was fascinating. Appreciate it.

  • Either it goes towards infinity or 0, to go to infinity, it always has to odd no, 3x + 1 can only be odd if x is even. Since adding 1 it makes impossible x to be odd always. Moment x is even, probability to go towards 0 increases. Lowest no is 2, so it is destined to fall in loop.

  • This is not a problem, this is an explanation to the nature.🙃

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  • A couple of days ago he had a poll on what colour would evens and odds would be if they had a colour. The poll decided blue as even and red as odd. In this video, he has the evens as blues and the odds as reds. I love how much he cares about his community and the little details.

    • Amen.

    • @Tyler Lawrence I do. My favorite number since I was a child was 7. When I learned to read I played a game in my head when I was little. I liked the words with odd letters because I would divide them in my head . Odd numbered words would have an even number on the left and right and an odd number in the middle. I liked to spell them backwards and speak backwards when I was bored. I liked it much better than the even numbered words.

    • Is a multiple of 2 : Kalm Is another number : REAL SH T

    • @Jim Balter Odd is right.

  • To continue. Since f can be rewritten in New form and can be generated as said as well as the other combos then by setting f and others in a matrix form we have the necessary requirements to map on into another. Or simply put prove x+1 and x-1 then 3x+1 is a simple spectral matrix manipulation. No trees or b's required.

  • Also makes me think of the warp drive problem where they realize when they factor in the universe is moving not just them suddenly the answer appears … what if we are missing a understanding of mathematics or life if self that would lend the poof or disproof or even better what if quantum level understanding possibly leads to a duality (granted my understand )(and I failed high math,)) is that I don’t see how it might apppy) We often find what we were so sure gets disproven sometimes totally by mistake or trying to prove or disprove something else The final answer to this I am sure won’t be the expected source or way of getting it

  • We all got trolled by Collatz Dude said "Any number except 1 or 0" and literally had people busting their brains over it for decades

  • Numerical gravity is honestly the best description for the pattern in this equation.

  • Mad respect to the animators here. That must've been a lot of work.

  • the reason there are different loops on the negative number line is because you're practically changing the function when x is a negative value, because you do the equivalent of 3x-1 instead of 3x+1

  • I think the reason why it seems impossible is because in this case, the number system is not defined as true infinity such that when a number is reduced to 4,2,1 it has nowhere to go into the spectrum of negative numbers. You can't say it's true for the number system when you only use an abstraction of the system. In fact, the equation itself puts a limit on the bounds of x. Further, we have integers that, albeit we call them even or odd, they act as both. For example, 10 is an "even" number but divided by 2, we get an odd. 2 is an even number, but divided by 2, we get an odd. 6 is even, but 6/2 is odd. That's a third of the base 10 loop that is our number system. Naturally, 4 2 1 are the least positive integers that fit the conjecture, so of course it will always amount to those 3 numbers given an unfair advantage of more even numbers available to divide. In other words, there are not truly the same number of odds and evens in the number system. This is also why 2 is even but still a prime..because it divides into an odd number and has nowhere to go by the limit defined by our number system that says it can't cross over to negative numbers. Think about all the other loops of 10 in the number system. They have somewhere to go when they're divided, until they get to the original single digits, they can't go below 0. Now, think of (ie) a base 2 number system. You don't see any other tiers of the loop, just 0,1,0,1,0,1 etc. Do you have the same problem with this conjecture in a base 2 number system? Probably not, but it would be interesting to see the results. My point is, we created the impossibility because our number system is flawed by our own definition of the actual base (0-9) versus its corresponding loops which are allowed to not follow the rules of said base. Further, if it wasn't flawed, infinity would be defined by an infinite number of positive and negative base 10 loops, not just infinite "numbers". I mean really, negative numbers, by that supposition shouldn't even exist. What we have is a 'base infinity' system that has no loop. That's my take on it anyway.

    • We can solve it the loop it impossible to happen appstore from 1 2 4 becuse all prime numbers except from 2 can't be devided by two and when two dose it gets multiplied to a multiple of its self no other number dose apart from 2 and is the only number that can be a loop

  • How the mathmatician try to solve this Conjecture gives me encouragemet to all my trivia math problems. Thanks for this amazing video

  • Interesting. I dont know if Im able to solve it, that is my wondering as well. Why say I can and afterwards the result wont be the right one.I ll watch tomorrow the rest of the video to see if my calculs fit in your response. Untill now 3x+1 shall be a linear function on xy ax...

  • I’m still trying to figure out what math problem we’re trying to solve.

    • @Jim Balter the Dunning Kruger effect is based on competence not stating facts the math equation makes it inevitable to get a bit wise number

    • @Jim Balter I writing my thoughts now because it's the only way

    • Lots of instances of the Dunning-Kruger effect in the comments here.

  • Maybe someone has made this observation (I can't go through 69034 comments), but if you keep taking log of log of log you should pretty quickly get to a number less than 1 in which case your next log is in negative territory. Then you can't take another log unless you're OK with complex numbers. So that's not a good formula to work with.

  • At first I didn't think this was very interesting because I was only thinking of the escape case (launching to infinity) as being intractable but seemingly true based on the Brownian motion argument. If we pretend the structure is stochastic, then we can argue that the probability of finding an escape case is zero. But the loop case is far more interesting because I don't see any reason why such loops wouldn't exist.

    • "Better to Remain Silent and Be Thought a Fool than to Speak and Remove All Doubt"

  • just imagine if there was a full number that was neither odd or even.

  • Brilliant as usual. Really humbled by the guest speaker when he said "nothing! we have achieved nothing!"

  • I'm not a mathematician but found this fascinating enough to watch the entire video.

  • me: thinks of a possible solution also me: *perhaps i was the chosen one*

  • This is interesting. Not something I would ever think about but all the same it is very interesting. I'm just happy to get the right change back when I'm in a store.

  • Reminds me of a joke we used to tell, where a number counted in letters back to a number, example 100, hundred, 7, seven, 5..... If you would continue this, no matter what number you use, the end will always be 4/four, we called it everything is 4. If you're drunk/high enough this will mess with your brain. Example 24, twentyfour, 10, ten, 3, three, 5, five, 4, four, 4.

  • I haven’t watched a second of this video. But the equation in the thumbnail (3x + 1), is mathematically impossible. X could equal any number able to comprehended by the human mind. So basically, X=INFINITY

    • @J Modified just to give my input. I’ll say up front that i probably have no business talking about math. I mean, hell, I failed algebra 1 this year so this isn’t my domain.

    • Why comment then? The expression in the thumbnail is just an informal name for the Collatz conjecture.

  • Pretty much every subject in school is really interesting if I’m not forced to learn it

    • facts

    • Amen.

    • You just described the main problem with the current education system on several places on the world: They don’t make you interested in learning the subject, they force it down your troath

    • Yes

  • Fantastic video, excellent visualizations, (evidence for God imo), thank you

  • to the people whom produced this video - bravo, I haven't ever had this much fun trying to help people, all in a few hours commenting on their comments and helping people is what I have done to survive for most of my life. I am 53

  • Fun fact: there is a closed loop for 5 if we subtract 1 instead of adding

  • Great fun, thank you!

  • Your "one of the world's greatest living mathematicians" joke totally killed me.

    • Found the Mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

    • @dgcfgv vgb thanks

    • @jet what? I'm so confused lol

    • 😂💯

    • Since Narak rythems with the english word ungramatmatically correct then that is correct.

  • 10:00 Oh... So the problem is to prove whether all numbers end up in the 4-1 loop or not. Got it! Guess that's what a conjecture is.

  • This is just a formula that once it hits its lowest form it loops back to itself. One of the key part to this formula is dividing by 2 this stops the number by growing infinitely. The role of the plus 1 is just so you can use the rule of dividing by 2 when the number hits uneven. The multiplication will only work on an even number like number 3 because if even number is used like number 2 the answer will always be even number which adding by 1 and dividing by 2 will not work. And also multipying by an even number like number 3 is the part that will take the equation for a ride because the answer can either be even or uneven, until it hits what I called the "looping family numbers". The looping numbers using ×3+1÷2 is 4, 2, 1 and it repeats from there, but 4 is half of 8 and is half of 16 and if you keep multiplying these numbers by 2 which is the opposite of it's dividing by 2 rule you'll end up with something like this 4, 8, 16, 32, 64, 128 and so on and this can go infinitely and this is what I call the "looping family numbers", now all the equation has to do is hit of of these numbers that when divided by 2 will take you right back to 4, 2, 1 looping numbers. The chance of hitting the "looping family numbers" using ×3+1 is not impossible thats why it always leads back to 4, 2, 1 loop, it might take a bit longer for some numbers to hit the "looping family numbers" but it's not impossible. You can have other looping equation that will have the same behaviour like ×3+3÷2 or ×5+1÷2, I've only tried these 2 looping equations as I've only known about this math problem a few hours ago.

  • I stumbled onto something with the conjecture. Waiting on someone who's working on the program to run my equations. If I'm correct, 3x+1 will be followed by an infinite number of possible equations that will repeat the same phenomenon. Wayyyyyyy too much to do by hand.

  • I should become a teacher and tell my students as homework "using this method find a number that doesn't go down into a loop"

  • Mathematicians: Dont waste your time on this problem 20.7 million people: YES

    • @Frank Chary imagine if we do figure out warp technology and they actually call it warp technology lol

    • 26million*

    • Search about Abhijeetbyte Collatz Conjecture GitHub 😎🤣🤣👍👍..... Can't share Links on ID-tv comments

    • We need to work on practical problems that solve mankind's various problems such as air water and ground pollution, developing cleaner energy, food water and resource supply and more equitable distribution. Having solved these problems then we can move on to developing Warp technology to open up the final frontier.

  • 7:57 the statistical model given is all the proof I needed.

  • I like to say that "mathematics is most often about the relationship between numbers and sets of numbers, not in computing anything." All too often we concentrate on utilitarian number crunching, which is boring, and why most students loathe math. Instead, patterns OF numbers and within sets of numbers are much more pleasing to the pattern making instincts of the human brain. Odd. Even. Positive. Negative. Fractions. Now here come the irrational numbers. Pi. Euler's Number. Phi, the Golden Ratio. Now here's a Cartesian plane. A unit circle. Sine waves, which we see and know in nature. Now here's a sphere. A hypercube. Fractals like the Mandelbrot Set. Procedures built with numbers that generate landscapes. Layers of infinity and an infinite number OF infinities, many of which we will never know. And somehow, within many of these mathematical constructs, we discover similar patterns over and over. Sometimes it feels as if there is a great unified system of patterns that revolve around each other, forming shapes, galaxies, anything we can imagine. THAT makes math beautiful. If we only taught it this way.

  • The reason 3x+1 is different using negative numbers is because when using negative numbers the + works like a - so it works like 3x-1

  • In binary: 3x+1 is left shift (2x) and then add x (2x+1x) = 3x ... and then, if the result is all ones, when you add 1 the ones will cascade and you will get a power of 2 ... and that will go to the 4 2 1 loop. But most of the time, 3x will have some zeroes to prevent that cascade from happening. Dividing by two, in binary, is right-shift ... since the number was even, this eliminates the right-most zero.

  • Imagine being a Math Teacher and you gave an entire class an activity 1. Solve Collatz Conjecture 3x+1 (10 pts.)

  • Who wants to time travel and make an old Greek philosopher's brain blow up with this?

  • If maths class was entertaining id be a 10/10 student in no time

  • just imagine like, a 12 year old, just... finding the answer in 2 seconds, and questioning how nobody could figure it out before them.

  • 3x+1 is also know as the centralized Finite Curve