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

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- Published on Jul 29, 2021 veröffentlicht
- 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 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 - ve42.co/3DCoral

Coral visualisation by Algoritmarte - ve42.co/Coral

Diego de PaulaYear ago^{+7197}Whoever created all those graph animations is an absolute master in after effects expressions

nigelft25 days ago@Official_Freehugs606090

Then you get into the realm of variable variables, which lays beyond me ...

nigelft25 days ago@Erhan Abdurrahman

An Episcopalian Christian here ...

But is there truly something called useless knowledge, especially in the field of mathematics ... or anything within STEM, for that matter ...?

Yessir5 months ago@huskai no. Your wrong. His comment was 110% accurate. Because of Jesus he get his editing skills in after effects.

Official_Freehugs6060905 months ago^{+1}@MONKE with DRIP x is a variable. Maybe in 5th grade you used x as a multiplication symbol but it's a bad habit when you start using variables.

Isaac Li6 months agotry create an expression for aah

Jack ScanlonMonth ago^{+728}I love how seemingly simple this problem would seem but how it's stumped mathematicians for so long. Makes you appreciate how complicated the mundane can be. Really makes me want to go back to school for a math degree!

D£viLiSH2 days ago@Athenri i will make sure to reach you out again whenever I got a problem ❣️ Thank u Athenri💪

Athenri3 days ago^{+1}@D£viLiSH It actually has nothing to do with our number system! We can write numbers any way we want, and it doesn't matter. The question itself is honestly not all that interesting - who cares if this function always goes to 1?

The interesting thing is that it's a question that's *so* easy to ask that it almost looks like a little toy problem - the kind you might give a clever high school student or a first year college student to puzzle over. But actually *solving it* by either proving that it always reaches 1, or that it sometimes don't? That's incredibly difficult.

I think what you want to look into in general is proofs. At heart, that's the core of math - proving that you don't just *think* that something is right, but proving that you *know* that it is right. And you can do this for infinitely many numbers simultaneously. This problem, however, seems incredibly resistant to our proof techniques, so we'll probably have to develop some new and fancy math to some day prove this.

It's probably not leading to new technology or anything - sometimes the fun and beauty is in the hunt itself, as with the Collatz conjecture.

D£viLiSH3 days ago@Athenri hey Thanks for replying

Oh now i get it but still I have a question.

It doesn't matter what number you pick if you put it through this algorithm it'll end up in a 4-2-1 loop and because there're infinity numbers it's almost impossible to find if this is 100% correct.

So what does it mean to maths in general is it a new path of maths or something that we haven't found yet or is it just a normal thing!?... For me this loop thing shows off the beauty of mathematics and I really want to believe that there's something more than what we know. For example a new way or a new kind of pattern humans aren't very familiar with... I know it sounds weird but this is a big fat universe and I believe there should be a lot of theories. Humans have a number system and they built their civilization around it but just imagine if we found a different path at the beginning it might lead us to a whole new level of technology.

Yo I'm sorry 😐 These are just things in my mind and I was so happy to see you replied to me. I would love to hear from you again 😍 btw if you know about any good content on Clip-Share around these kinds of ideas please let me know. I'm obsessed with mathematics🎭✨

Athenri3 days ago^{+1}@D£viLiSH

When you do math at the college level, sometimes also high school level, math is very often not going to be "calculate this and that". Instead, it might be something like "prove that there are infinitely many prime numbers" or "prove that these two expressions are equivalent" or "prove that there are no choices of integers p/q that equal the square root of 2".

That kind of stuff. The collatz conjecture then says "I think that if you have a function that for all odd numbers multiplies by 3 and adds 1, and for all even numbers divides by 2, it will always reach 1, no matter where you start."

But it's a conjecture. That, in the math world, means "educated guess" - and usually one we'd like to turn into a theorem (which is basically something we *know* to be true.) We don't *know* if this is true. And it turns out that answering this question in a way that's certain, the way math people want, even though it only really involves really simple expressions, is *incredibly* difficult.

Feel free to ask follow-up questions.

D£viLiSH3 days ago@Athenri hey can you explain about it with more simplified version please 🥺

Phillip MeeceMonth ago^{+141}I'm sure someone already knows this, but if you do 3x-1 on negative numbers it seems to have the same affect, and if you put positive numbers into 3x-1 it comes out with the same loops as the negative numbers do in 3x+1. I find that fascinating.

Тут Был Я9 days ago^{+2}@J Modified And with the same logic 3*n-1 for positives should behave as 3*n+1 for negatives. Sorry, but no mystery or fascination here.

J ModifiedMonth ago^{+33}That's because multiplication, odd/even, and division are all mirrored about zero. The only thing in Collatz that isn't is the "+1", so if you change that to -1 you get the exact mirror image of the problem.

Mauricio IturrietaDay ago^{+3}I cant even imagine the amount of work put into the visuals of this video, and of course the research and everything. Thank you so much for this!

FourKKMonth ago^{+92}Would be interesting to see this problem expanded into the complex numbers. They are beautiful to work with and it might lead to bigger insights into the original problem. Though I am quite sure this has already been done

bored person26 days ago^{+1}Only integers have parity.

Aerxis29 days ago^{+2}@Athenri gaussian integers (complex integers) have unique prime factorization, hence the terms even and odd ARE properly defined

FourKKMonth ago^{+1}@Athenri True, but you could i.e. look at the absolute value of the complex number, or look at the real and imaginary parts seperately and see if there might be an interesting corrolation

AthenriMonth ago^{+10}They don't really have even or odd numbers though, so you'd have to think hard about what your extension would work like.

TheOriginalFayariMonth ago^{+127}I feel like Mersenne Primes are the key to solving 3n+1. 2^82,589,933 − 1 would be an interesting one. It's the largest known prime number, and it ends in 1. Meaning that for as long as the number lasts, its ending number will always be 1, 4, 2, and back to 1. Whenever it loops back to having 1 as its ending digit, it shoots way up again, and it has over 24 million digits to run through before it can even think to drop to 1.

lol5 days ago@E Stolee because the cpu would run forever and still not find the number, so it goes back to the original problem right?

E Stolee5 days ago@lol What you proposed is a program that would solve the halting problem, which is obviously not possible. Good thought, but it's not enough

lol14 days agoi dont understand, the whole problem is set up so you will eventually reach 4,2,1 loop right? its like zeno's paradox? and if we just run a super CPU that checks for any number that breaks the rule, it will never find one. so what is the big thing here?

MEHDI OMAR20 days ago^{+1}I discovered the pattern of prime numbers, how can i make it public?? (Im outside the US)

Cosmic NomadYear ago^{+18569}I absolutely love how mathematicians always find the most random things to debate over!

You16 days agoMost people who are saying we have to test all numbers up to infinity is false. However, in this scenario, casework would be awful to do. The reason we cannot explain itmis because we dont have a reasonable amount of axioms to describe why or how this avalanche occurs. Thus, we can not prove or deny this conjecture. For those who say this is a waste of time, many conjectures like this prove to show usefulness in a real life context. We also cannot assume this loop goes on forever because we dont have a reasonable axiom to show this holds for all positive real numbers.

Kristian Dior29 days agoword

Dipa TariganMonth agoYup, just like my wife

bill stockMonth ago@H B mathematics is a human invented tool to help map out measurements, analysis,and predictions. This tool does not account for the more complex calculations taking place at all times for all things. A ruler has measurements in increments of inches,centimeters and millimeters, does that mean a measurement other than what is displayed on a ruler not exist?

When I speak of random I am referring to calculations on a vast scale for all things and not limited to a man made construct called math.

H BMonth ago@bill stock Random means that a number in a sequence cannot be predicted from what went before. Unless you can prove that ALL sequences MUST have a generating algorithm, you are very wrong.

Timofey Gerasimov2 months ago^{+55}It's really interesting how a completely deterministic problem like this is best characterized in terms of geometric Brownian motion and probability distributions, which describe random systems. It makes me think that probabilistic modeling captures something deep and profound about many phenomena in the world that at first glance might not appear to have any randomness whatsoever.

bored person26 days ago@H B something tells me you've never heard of quantum mechanics.

H BMonth ago^{+1}@Timofey Gerasimov If science is everything we believe it to be, there is NO intrinsic randomness anywhere, just outcomes whose input conditions we cannot fully characterise.

Timofey GerasimovMonth ago^{+1}@Nickers I guess it's hard to think of a real-world example with no randomness at all, since most real-world phenomena have some amount of intrinsic randomness. But chaos theory (sensitive dependence on initial conditions) is one example where deterministic behavior can give rise to seemingly-random behavior.

NickersMonth agoWhich phenomena you are referring to?

СМЕШУСИК3 months ago^{+54}I love simple problems and the way someone can explain their use in real life. Most of Veritassium videos have clever names, so clever that your hand just itches to press play. But I noticed one repeating pattern too. I am not a mathematician. I loved statistics in College and I understand how everything we have would not be possible without countless people solving problems for millennia.

EVERY TIME video begins simple enough and easy to follow. But about 1/3 of it's starting to get so complicated that a regular person like me has no idea how to catch up and understand. Maybe I'm just slow. But I know I'm not that dumb. Does anyone else have the same problem as me? So I actually almost never ever watched a full video without falling asleep or just getting too bogged down by facts, words, numbers, and other fancy stuff

Please, if anyone experienced the same I would love to hear your side of it.

One other thing that bothers me. So many likes, views, and smart comments. Obviously, lots of people understand this stuff. And I'm happy for them. But it makes me feel a little too far behind. Ohhh. Boo Hoo. Poor me.

chance bonnett22 days ago^{+1}Honestly, the largest issue is overcoming the pain of not knowing and continuing to expend time(energy) into something that has no assurance will actually bear fuit in the mind.

Most intelligent people I've seen are only really intelligent in fields they are specifically interested in. If you meet someone who finishes these vids regularly there's a very strong chance they also don't understand but still finish the video for other reasons (wormholes look dope)

If you end up on a subject that is officially declared to be above human thought (quantum mechanics) then the people in that field will constantly reassure you that they too do not really understand it.

nonamecha0sMonth agoХахатунчики в чате

Nancy FancyMonth ago@Artists Aliens 💀

Nancy FancyMonth ago@aneyesky agreed! ifyt.

Jacob SimsMonth ago^{+7}I would be interested to see what the data set looks like without prime numbers. I have a feeling there will be a predictable symmetry between the factors of ascending numbers

H BMonth ago^{+4}Some part of the answer to this problem may well involve how rapidly and how completely we fill number space using this algorithm. The requirement that just a single number in any sequence is very small tells us something about the distribution of numbers in each sequence. The rest is, as they say, left as an exercise for the reader.

John Xina20 days ago^{+3}This was truly a great video to see! Keep up the great work!

The Musical Stylings of Brent BunnYear ago^{+2881}Mad respect to the animators here. That must've been a lot of work.

Bobby Smith11 days agoThis is not a problem to solve. It's the simple result of using this procedure. Using the number 3 to multiply a number most likely will be a odd number then add 1 will make it a even number. The constant dividing by two will keep the number from becoming to high. Remember if the sum is a even number you divide by 2 again. This can lead to another even number wich will be again divided by 2. That reduces a number by 50% then 50% and again 50% basically a 95% reduction. while only multiplying a sum by 3. Then add 1 will ensure the number will be a even number enough times to be reduced to the lowest single even or odd digit.

aashna27 days agoFinally some recognition for animators...otherwise it's just 'editors'

Mesonaldo Editz2 months ago^{+2}Bro you could do the add first and then times by 3 3 ez

S M5 months ago^{+7}Thank you for saying animators instead of just “the editor”

Devang agarwal5 months ago^{+1}Its pretty basic so im pretty sure anyone will be able do it

Dan LeidalMonth ago^{+3}I think what's so interesting thing about this idea, not including the paradox or its principles, is the pattern. What if instead of choosing this as an integer problem, but instead look at as a model for 2^x where x is countable iterations? Take instead of 3x + 1 or x / 2 and do the set {x sub n, rx sub (n - 1) + y, x sub (n - 1) / 2, ... } then we could have some models for calculating persistence in quantum states

Skybird Projects3 months ago^{+153}Ok, I'd like to see how this problem holds up to non integer numbers, or even if it can be modified to include complex numbers.

cawayy23Month agocheck out Michael Penn's video about the Collatz conjecture. He goes on about both these extensions at the end.

Donald StinnettMonth agoInclude imaginary numbers too.

x lorrix -Month ago@breadsticck you could make it so if the nunber is closer to odd/even then it does the thing

GonPlays3 months ago^{+2}@breadsticck correct

breadsticck3 months ago^{+52}Well the whole even/odd thing wouldn't work for non-integers

RådiøåctiveMonth ago^{+2}Can decimal numbers be used? Or fractions? Just wondering, it may be possible that way.

Dru Oswald2 months ago^{+5}I think this is a great exercise. The use of it is in learning to enjoy thinking, not calculating. If you approach this as, "I am going to calculate all of the numbers to..." then you still enjoy calculation. But if you approach it as "I am going to develop an algorithm to hunt for the number that avoids the loop," then you've gotten much use from it.

Rearrange Your Guts10 days agoThis really is such an interesting problem, and I can't help but wonder how much of the complexity results from the piecewise nature of this function (which really is more of an algorithm than a function). Effectively, 3x+1 is a stitching of the function 3x+1 with the function x/2 in an even/odd piecewise nature. It's the complex piecewise nature of this function that gives it a historicity, and effectively makes it a computational machine (in logic form). So for that reason, I'm almost certain this function, and many other piecewise functions of this flavor, fall into the halting problem. They aren't really functions, but amalgamations of functions that switch between each other, resulting in chaotic motion and unpredictability.

ParkieLYear ago^{+4345}Oh my god, this poor animator. That is a serious amount of dedication. Looks fantastic!

KalliboyYear ago@Mehtab Ghumakkad Maybe I'll work with this someday

Python fascinates me everyday

NotYayaNguyenYear ago^{+1}@Timmyrbx how do u know it’s a “he”

Llama ManYear ago@Lucky The Luckless Wolf I know I am

Lucky The Luckless WolfYear ago@Llama Man no, you're amazing

TimmyrbxYear agoHe needs to be paid every single day $100,000 heh

LinkGuy082 months ago^{+5}The negative numbers dont have the same effect because the +1 is kinda like -1 to natural numbers, thus the number will shrink or rise (-1+1=0 -10-1=-11) (whatever you prefer to say)

If you were to change it to 3x-1 Natural numbers would have the same cycles as negative have with 3x+1 and vise versa

Matt Bruce2 months ago^{+16}That was a fascinating video Derek. However I don't think this should be called the "3X + 1" problem because this is really an iteration problem. This is just what happens when you make rules that apply "3x + 1" with some logic based on opposites such as even/odd, hot/cold, positive/negative etc.. It seems intuitive that the rules of the Collatz Conjecture will always cause the answer to get smaller until it bounces between 4, 2, and 1 infinitely. In this case once you hit an even number that is part of the Geometric sequence with a factor of 2 THEN down you go into the 4-2-1 trap.

H BMonth ago@Mahalalel Christ taught us mortals many things, but he didn't teach us a while lot about mathematics. Even deities have their limits.

MahalalelMonth agoThe only thing that matters in this life is whether or not you know the Lord Jesus Christ who is the love of your life, and you His.

All we like sheep have gone astray; we have turned every one to his own way; and the Lord hath laid on him the iniquity of us all. Isaiah 53:6 ✝🌅

Even from the days of your fathers ye are gone away from mine ordinances, and have not kept them. Return unto me, and I will return unto you, saith the Lord of hosts. But ye said, Wherein shall we return? Malachi 3:7 ✝🌅

“Repent ye therefore, and be converted, that your sins may be blotted out, when the times of refreshing shall come from the presence of the Lord. Acts 3:19 ✝🌅

Shane TolfMonth ago^{+3}Since 1952 this thing has had the brightest minds thrown at it. Thank you B. Thwaites, your legend is being cared for by the ever so geniuses of Clip-Share!

Norman Graham2 months ago^{+2}Also note: 2**a at the closest point to 3**b, is an increasing sequence. (They are getting further apart).

Axer101Month ago^{+6}My gifted teacher from last year is teaching kids this year about the collatz conjecture, and it looks amazing.

Demens Clay10 months ago^{+2726}A big shoutout ot the graphics department for making this 100% more understandable!

Jim Greene6 months ago@Anndy Arguedo lol

Jim Greene6 months ago@icebreaker900 Broad is the path that leads to 1. Narrow the counter examples to this conjecture.

Josiah Brady8 months agono problem

gerkey8 months ago@myUserName thanks

icebreaker9009 months ago^{+3}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.

bruce okeloMonth ago^{+3}I ran the numbers one through nine and the series closed off with nine containing all the units appearing in the rest of the numbers. 9 (28) 14 7 (22) 11 (34) 17 (52) 26 13 (40) 20 (10) 5 (16) 8 (4) 2 1 4. I also run it backwards from one trying to see what generated the number I had a 0 1 4 2 possibility which could only be generated with a zero prior to one that was divided by three and one added. But beyond that all other paths got disqualified and the chain from the end going up was 1 2 (4) 8 (16) 5 (10) and then from here 3 or 20. With divergent points indicated within the brackets as also in the example of nine above. And they all had a root number of 1 4 or 7 when the digits were added. And they all were divisible by 3 when one is deducted. So the paths in my analysis of one to ten was easy for some cause they naturally fell into the path. For others it was harder because they had to find that point that was root one, four or seven and was divisible and then work it's way down safely from there. But the divergent points is where they gain entry into being able to work their way down to one. While passing several on the way down. And my guess are those points are the ones in sequence and not repeating

Sarthak Bhatia19 days agoI used to wonder for a similar situation like this where if its an odd, add 1, and if its even divide by 2.

even this will eventually lead to 1

and i think it would be true for each odd no.(x+1,3x+1,5x+1,7x+1...)

Bintang NaufalMonth ago^{+2}aside from the problem, these animations are really cool. what tool did you use to make them?

G LakersMonth ago^{+1}Awesome video I love to find out about examples like this. Makes me want to reinvest my time into mathematics. Thank you!

Edit: Just grammar

Mary FurrMonth ago^{+1}I wonder if the reason they always end up at 1 is because that's the only number every single number is divisible by

shadyceddyYear ago^{+20817}Fun fact: We are not mathematicians but we got interested by this.

Aleksandar Jovanovic2 months agoyes haha

BloxxingDinosaurus2 months ago^{+1}I am the 20202nd like and the 222nd reply to this comment.

bi nq2 months ago_i’m addicted to the Collatz Conjecture pls send help._

no like really when i’m in class I end up Collatz-ing instead of actually focusing on my studies lmao

QuinticGhost3 months agosame lol

quantum flucs3 months ago@Subrina Campbell this is cuz u look math the most wrong way.. its not about solving.. its about where do nature use it-what can i see trough it .. what does it want to teach me ? ; have fun

Mile Šakić2 months agoIf there are no other loops it only needs to be proven that all numbers eventually end up as a potention of number two. I think proving there aren't any other loops is the harder task.

Michael McGuire3 months ago^{+1}Some of the graphs, such as the one at 22:08, indicate less of a decrease than a value being halved.

I believe the analysis should focus on the occurrence and variability of even numbers.

Tokas1981Month agoRegarding the negative side of the numbers, of course its differnet because you are adding 1 instead of subtracting 1. In the additive form, the stream is going upwards, so the equation is 3x+1. The Negative side should be a total mirror, meaning -3x-1, and there the -4-2-1 loop will appear 100%.

J ModifiedMonth agoThat explains why it's different, but not why there are multiple loops in one case and not in the other.

SwangoMonth ago^{+1}15:55 kinda fascinating how there's like a "wave" made out of perfect squares going from right to left on the upper part

ClownpierceMonth agoShowed this to my math teacher on test day and he spent the whole hour working on it so we didn't have to do the test.

WalkastrayYear ago^{+803}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.

H BMonth ago@Jim Balter Absolutely not - it is a purely social construct.

H BMonth ago@Art Thingies No Spinal Tap for you then.

Pratana KangsadalYear agoAmen.

Maria MariaYear ago^{+1}@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.

Dawid Markiewka3 months ago^{+2}The Collatz conjecture is a mathematical problem that involves iteratively applying a certain function to a positive integer, and then repeating this process with the resulting value. The conjecture is that no matter what positive integer you start with, you will always eventually reach the number 1 through this process.

Here's how the process works:

Take a positive integer n.

If n is even, divide it by 2 to get n/2. If n is odd, multiply it by 3 and add 1 to get 3n + 1.

Repeat step 2 with the resulting value of n.

For example, if we start with the number 6, the sequence of values would be 6, 3, 10, 5, 16, 8, 4, 2, 1. If we start with the number 7, the sequence of values would be 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

The Collatz conjecture has been tested extensively and has not yet been proven or disproven, so it is still considered a conjecture. It is also known as the "3n + 1 conjecture" or the "Ulam conjecture," after the mathematician Stanislaw Ulam, who is credited with introducing it.

Valiebro3 months ago^{+26}Basically any number that eventually reaches a power of two lands back on 1, so theoretically any number that doesn't land up on the power of 2 when multiplied by 3 could be the solution. (my way of thinking, no idea if this is correct).

H BMonth ago@Dan Treadwell Powers of 2 are an extremely small subset of the positive integers and therefore make very little difference to calculation times.

Dan TreadwellMonth ago@ExtravagantPanda it would make the job "easier" by removing a subset of numbers outright, and allows for faster computation by allowing you to stop if you reach a power of 2 number. Even though it's kinda irrelevant, as removing an infinite subset from an infinite set still leaves an infinite set to deal with. But hey, maybe it can give some solace from the idea that you have fewer numbers to check. . . 🙃

ExtravagantPanda3 months ago^{+6}You're correct in identifying that a power of 2 always lands back on 1. However just finding a number that, when multiplied by 3 (and I think you meant to say when multiplied by 3 then added to 1) is not a power of 2 is not sufficient, because that does not guarantee that the rest of the sequence does not eventually land on a power of 2. Take 3 as a counterexample: 3*3 + 1 = 10 which is not a power of 2, however the sequence for 3 goes 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1.

Parinita Gowda DVMonth ago^{+3}Very truly said "Mathematics is not yet ripe for such questions"😁

Arath Alatorre Muniz2 months agoI only recently begun thinking about the conjecture with my programming assignment. I completed it it returned sequences from 1 to 100. My professor would then go on and ask how we could prove 3x+1 with so much technology and still be wrong. After a lot of thought, no wonder a lot of math mathematicians say it's useless or pointless.

The Golden AtlasMonth agoWhat rules do you use to calculate it?

For me if youre brute forcing through everynumber you don't actually need to calculate all of them.

1: all starting even numbers go to a number already tested.

2: Any operation that makes the current number lower than the starting number sends it to the 421 loop since you've already tested below the starting number you know this for fact.

3: If the current number becomes the starting numbed a loop has been found.

With just these 3 rules you can basically brute force for as long as you want. Since you can't test for going to infinity with certainty you have to test for a loop.

bluejamMonth agoi like how mathematicians treat numbers like they're little critters observed in their natural habitat

toe gangYear ago^{+2236}I’m still trying to figure out what math problem we’re trying to solve.

Kevin Peralta3 days ago@Richard Farmbrough I can answer that for yoy, yes

brenda williamsMonth ago@quiz All things work on math problems. I feel most things do counting. Like humans breath but something is counting. That kind of thing.

quizMonth agoSomeone working on a math problem is a person working on mathematics as a whole.

Mathematics as a whole is one of the best tools that humans have ever developed (and are developing!!). Therefore it is usefull.

The only reason why there are still debates over the usefullness of mathematics is because no one actually does it.

brenda williams3 months ago@Andreas Philippou deep

Diederik Huys2 months ago^{+1}I love how the quote at 13:56 ("Maybe we should spend more energy looking for counterexamples") coincides nicely with Karl Poppers' demarcation criterium of falsifiability :-)

H BMonth ago^{+1}If you test a number, you are simply testing the conjecture. You are neither looking for a counterexample, nor seeking to find further support. The only relevance that Popper has is that one single counter example is sufficient to render the conjecture invalid. Looking for counter-examples involves trying to establish what properties they might have rather than carrying out a brute force test on a random number.

Tom McKernan2 months agoApplying 3X+1 for odd numbers and dividing all even numbers by 2, to the first ten numbers, [being 0 1 2 3 4 5 6 7 8 9], produces two distinctive results. These results are 0 divided by 2 = 0, with the remaining 9 numbers all = 1. In the negative state, converting 3X+1 to -3X-1, simply equalizes the equation. So, -0 divided by -2 = -0. The remaining nine numbers all result in -1. For example, applying -3X-1 to -3 produces -3 X 3 - 1, resulting in -10, -10 divided by -2 = -5, -5 x-3-1 = -16, and the following -8 -4 -2 -1 results. I expect the matter of 0 being a number will be problematic and the translation into minus format for all minus numbers to be somewhat challenging. However, the traditional base line for single numbers is plain wrong. Here is the value of 0. One, zero, minus one, or 1, 0, -1. Ann owns one dollar, Bill owns no dollars and Jill owes one dollar. Ann is two dollars richer than Jill and one dollar richer than Bill. Bill is one dollar richer than Jill.

Regarding the assumed conjecture status of 3X+1, has anybody sequentially proven the existence of number infinity?

Ich bin TupiniquimMonth agoWunderbar!!! Vielen herzlichen Dank.

Benguira DavidMonth ago^{+1}4-2-1 has to be the only loop because 1 is the only number for which 3x+1is the equivalent of x4 so after having done 3x+1 dividing by 4 will bring you back to your original number or 1

J ModifiedMonth ago^{+3}That means it is the only loop of length 3 of the form even-even-odd, not that it is the only loop. It doesn't even mean there aren't other loops of length 3 (though it is trivial to prove that there are not).

Lightning First20 days agoIf they find just one number that provably blows up to infinity, that will mean there are infinitely many numbers that do so.

That's kinda freaky to think about.

ChristopherYear ago^{+1280}I'm not a mathematician but found this fascinating enough to watch the entire video.

Anshu NeupaneYear agoSame

Artificial intelligence plus lotteryYear agoFound the mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

MatthewYear agosame lo

kpink gamerYear agoSameee

ThaKidKaneYear agoWhat was the answer I don’t go time to watch the video but it should be 3 if it’s not it’s wrong

Lukepuke311Month ago^{+1}Lets say that an odd number * an odd number always = an odd number (which is true I think): all of them will become even and even if that becomes odd the number will go even again. Every time you do the calculation it starts another calculation and the more calculations you do, the more likely it is it won't be going to infinity. Every one we've checked has not disproven it and i'd say due to the way this works it always will go down to one of those half chains

Peter ALHachem2 months ago^{+1}I haven't looked deeply into this problem and although I think it would be interesting to see all the covering aspects...But in my opinion it is fundamental to go back to the concept of mathematics and its representation in reality..Just like this video has protrayed the Collatz Conjecture as a height representation and also the frequency of other sequences such as the Fibonacci and golden ratio in nature, taking into account that reality and nature is made of positive integers I feel that through this conjecture we are not able to find 2 consecutive odd numbers that follow each others (I haven't profoundly performed it on large numbers) and thus we are not able to perform two consecutive 3x+1 consecutively...I do not claim anything but as they spoke in the video trying to find a counter argument will be the solution to this problem and not continue to feed its correctness.

PoroMonth agoYou cannot perform 3x+1 two times because by definition, 3x+1 will ALWAYS give an even number

Robert Brunson16 days ago^{+1}I'm not educated, but I've gone thru this problem for 45 plus years at one point it almost drove me mentally unstable.

Boredom Buster20 days ago^{+1}Hello, my name is Rohit. I really enjoyed your video, and I hope you continue to make them. So I took an irrational number (root 2=1.41) and applied the 3x+1 rule, and the result is that irrational numbers also end with a 4, 2, 1 loop, indicating that irrational numbers, not all but a few, do follow the 3x+1 rule. I asked you to forward this message to everyone at every institute and mathematician.

Esage19 days agoHow? You can never get an odd or even number

Ray Gaskell6 days agoI wonder if the answer might come from looking at other conjectures like 5x + 1, or 7x +1...as opposed to just 3x + 1, 3, 5 and 7 all being primes too, but I would assume these have been analysed too.

Mr ScientificYear ago^{+31098}Nice work Soviets. You got me.

runciter naki10 months agoyou 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

Artificial intelligence plus lotteryYear agoFound the mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

Surya PrakashYear ago@Ali Akram wwwwwwwwwwwwww¹wà

Martina OmearaYear agoIt's 4X the X doesn't mean multiply

SameeknowsbestYear agoThehh uhhhh got me

Aiganym ToleukhankyzyMonth agoThe most fascinating video I have seen in my life😍 Can you tell please which program did you use to make this video?

Stan Meer2 months agoOk so I think I may have solved this equation or at least question this. How about adding the positive number to negative numbers in the minus zero and then combining the equation together so instead of seven times three plus one it would be seven times negative three plus minus negative one and then divide my negative that number and then add the positive numbers in the negative sequence and go back and forth that way it opens up a whole new set of numbers and can even go the same way in the negative and comes back around unless you keep adding the minus number to another minus number then divide by a positive number

Alexander WeckMonth agoPerhaps it is usefull, to use the reverse funktion and look for completeness of all integer numbers produceable by this function. Multiplying by two is easy, because its always an integer. Minus 1 and then divided by 3 needs the boundary condition, that the result is only true, when the number minus one is a multiple of 3, and therefore the result an integer.

[ZenkaiSoul]2 months agoThis is why I love mathematics

Sometimes the simplest equations make the unsolved problems

For example 3N+1 /2

We can make complicate it even more

Like if N is a real value

Using N² or Square root , negative values , and the homie 0

JulianMonth agoI clicked on this video for the sole reason that I thought i'd be able to solve this problem super easily and mathematicians were simply missing my amazing abilities the whole time they've been working on it and now I don't really know what to do

nwogamesalertMonth ago^{+1}Just wait it out until some of the other Clip-Share geniusses have solved it. In the mean time enjoy yourself with wine, women and song.

Levy ChevyYear ago^{+1225}My calculus professor just introduced this conjecture to us last week, and ever since then I've been shamelessly addicted to just bringing up a random number generator for a starting point and wasting away the hours.

H BMonth agoWhy are you repeating tests which have been done many times before ? What do you hope to discover by doing so ?

bi nq2 months agoglad to see i’m not alone lol I do be Collatz-ing instead of actually focusing on homework

Artificial intelligence plus lotteryYear agoFound the mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

DHRUV SHARMAYear ago^{+1}Same brooo

James ProngerYear agoBlah blah blah more replys. "Think differently and simply"

Hope RockMonth agoHere are 2 things I realised since 421 loops, the conjecture can be summarised as all numbers lead to 16 (which leads 8 and then to the 421 loop). Also there are probably many problems exactly like this we just haven't searched for them, like, how do we know x/2 if even and 5x+3 otherwise doesn't create the same problems with a different loop?

awebmateMonth agoWho are "we" and who told you they didn't search for them lol sorry, i can guarantee you that all variations of this problem have been heavily studied.

There even is a proof for 1093x+1, that almost NO numbers go to 1. Also, surprisingly, 181x+1 has a loop beginning from 27.

If you are interested, John Conway came up with some exciting rules creating similar kinds of unpredictability.

Andrew WernerMonth agoI would be interested in what happens if you apply this with some adjusted rules to complex numbers. Perhaps all numbers must be of the form r*e^it where r is a whole number. Then apply 3x+1 if r is odd and x/2 if r is even... well, actually there will have to be a different rule for 3x+1. I suppose that would have to be applied only if a+bi uses integers for a and b or else it may not have integer results.

Perhaps it would be based on the first digit or the whole number portion of something.

H BMonth agoOff you go. Please let us know when you have something of interest to tell us.

LixuskzsMonth agoThe key is +1. Whatever you do, you are creating an even number, doesn't matter how big is the number, you will get to the end.

Calvin Williamson3 months ago^{+14}I would just like to say that obviously it doesn’t work the same for negative numbers. It’s not asking the same question, you’d have to mirror the equation making it 3x-1 to achieve the same result with negative numbers.

J Modified3 months ago^{+1}They're not identical, but there's no known reason that 3x-1 would have multiple loops while 3x+1 would not.

zbGiovannihe SpiaagMonth agoPlaying around with this problem makes me strongly believe that all positive integers will reach 1 and that the 4-2-1 loop is the only loop

TheNordicNormieYear ago^{+2612}Pretty much every subject in school is really interesting if I’m not forced to learn it

GuyYear agofacts

Pratana KangsadalYear agoAmen.

Victor KappelYear ago^{+6}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

How the turn tablesYear agoYes

Masons2 months agoAnd I think this is the point where all math teachers gets their thinking caps on and try to solve it

FKA Gaming & CraftsMonth ago4:44 I wonder if you could program a random number generator with using similar equations and methods.

RangerMonth agoIts not that this math problem is hard, it is just that it is so time consuming.

De ViceCrimsinMonth agoAs a regular how, It sounds like a problem tied to the mechanism of entropy. At least on the most basic level I can imagine and am aware of... Edit: forgot my time stamps. 0:00 - 3:45

KOMonth agoWOW! I understand this video's subject matter and I very poor at math, but I like it.

SpookwormYear ago^{+1651}I have never been someone who liked math during school, but for some reason I find it so completely interesting to learn about on my own time.

nowherenearbyYear ago@Spookworm nope

SpookwormYear ago@nowherenearby You're entitled to your wrong opinion :)

nowherenearbyYear ago@Tza16 i dont think so, watching a surgeon do his work doesnt mean youre learning it (only if you already know a lot about medicine but whtvr)

Tza16Year ago@nowherenearby it is for some people

nowherenearbyYear agowatching these types of vids isnt learning math but ok

B. Germocen2 months ago^{+1}Question, Do they only use whole numbers for this exercise? Does it work the same way with fractions? Also, who made up the rule we had to multiply by 3 and add 1, then divide by 2? Or, is that the issue of the problem itself? Determining IF we use this rule, it will always give us 1 as a result? I'm confused on to why Clip-Share recommended this at 6am, but now I am engaged and intrigued.

PoroMonth ago^{+1}They only use whole numbers because fractions dont work with odd/even distinctions

Also it is 3x+1 because it just is part of the problem

Messenger 12212 months agoI wonder what loop you would get if after dividing by 2 add 1 which would turn it odd again each time.

Tobias Urs Marti MüllerMonth agoI love how you can prof in mathematics, that you'll never be able to prof your problem correct or wrong.

J ModifiedMonth agoYes, there are relatively simple true/false questions for which we have proven we can never know the answer, yet they must be either true or false. Collatz may or may not be one of them.

Asher WilkinsMonth ago^{+1}Okay so it's kinda like life, it can take insane twists and turns but ultimately it always ends the same way, with death

FoxiesboiMonth ago^{+1}I wonder what happens if you put in decimals. I don't think the video talked about when you add something like 1.5

emayarYear ago^{+761}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.

H BMonth ago@Veritasium For information. The international aviation community defines Monday as day 1 of the week. Microsoft defines Sunday as day 1 of the week.

And Therefore6 months agoWait, if you do this 3x+1 for 1.5, you will always find an odd number, and will eventually lead to infinity

promessa_exYear agoevens are green odd is orange of course, but three is blue

Tejasvin KansalYear ago@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

An Earthian12 days ago^{+2}The even simpler question that no one can solve is

"How is that question simplest when no one can solve it!"

Aditya Mishra8 days agothats actually quite deep, you know before everybody thought they know why mercury's orbit differs from predictions ( possible undiscovered planet ), then we found we didnt understood gravity itslef...

i think same goes here, we think we understand the problem , but all or predictions/proof methods fail to prove it... maybe we need a radical new insight which no one has seen yet, thus in that way it is actually a not so simple problem and not many people may even understand it

J Modified11 days agoThe question is simple - easy to understand. The solution may be impossible, difficult, or simple but very hard to find.

irakaram Laurent21 day agodoes this apply to decimal numbers also?

anton4o_protoMonth agoi'm honestly more impresed by the fact that you guessed that i chose 7 at the start of the video than the actual video

apersonMonth agoyou may have notice the pattern inevitably leads to powers of 2, which is why they keep dividing until 1

GawędziarzMonth agoI think that it's kinda pointless... because it's all just bouncing around within rules we create.

A set of instructions that for any input produces a logic loop. That +1 is key there, in my mind at least, since it both adjusts the number of times we divide instead of using formula, and ensures that entire thing will settle at 421 eventually. To be honest input doesn't matter at some point this arrangement will produce a number that will bring it down to 421 directly. So it's a sort of unstable system, that fluctuates until it can settle at sort of low energy state.

Franz PattisonYear ago^{+496}Can I just take a moment to applaud the animations created for the visual representation of the concepts?

Artificial intelligence plus lotteryYear agoFound the Mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".

Sincical PVPYear ago^{+2}yes you can.

putrycydesYear ago^{+2}Thought exactly the same while watching

Mark ShimanYear ago^{+5}3blue1brown animations

FinneasAF2 months ago^{+2}I love numbers, but when I found out that at the most basic axiomatic level, the rules often contradict themselves or prove absurdities. Our math is not yet advanced enough indeed. I can't help but love it.

H BMonth agoAny "rule" that contradicts itself is by definition not a rule. Any rule that leads to absurdity is not a rule. In the Collatz conjecture there are no contradictions, nor are there any absurdities.

UnPrankAble 6662 months agoIf we go the other way, keep doubling from four up, we get an infinite number of seeds that always go straight to the 4-2-1 loop, without rising up. Quadrupling can work as well, along with any even multiplication.

H BMonth agoTrue but irrelevant. You are allowed to start from 3 if you wish.

Decorum Pan Troglodytes 🌈🦧8 days agoI’m pretty sure this problem would not have any predictable outcome as its components makes it extremely randoms; specifically how the concepts of odd and even don’t really mean much in terms of values considering it only works with whole numbers.

BODOQUE2 months ago^{+1}I'm just surprised at how mathematicians are very closed minded... if they just sit back for a second, they would, could see that the loop created in the end IS the infinite.

Max LebowMonth agoIt seems to me you proved it when you showed that the actual multiple was 3/4, less than 1, therefore will always tend to lower until finally it rests at 1.

WissleYear ago^{+7053}I don't care about math whatsoever but this was incredibly interesting.

Amazingly presented, dude!

Shreeya Sharma10 months agoyess

Ms Harris About God10 months ago@walt dutchak appreciate you, thanks. i never liked too much visibility. my body have 26 years old.

numai bine walt, you're a good person.

walt dutchak10 months ago@Ms Harris About God _ You are doing pretty good 12E (It would be nice if you used your real name like I do. Are you afraid of something?)

_ I am in Canada - native language Ukrainian, born in Germany - age 75 and many years of experience in many fields.

_ It has been nice sharing with you.

_ bucurie

Katie2 months ago^{+1}If you map the digital roots results of 3x+1 they will always either be 1, 4, or 7, which means the equation can only map to 1/3 of possible even numbers that can exist. This seems pretty significant.

David TircavuMonth ago^{+1}by watching this video i figured that mathematicians have a lot of free time

ButterRadish2 months ago^{+2}i'm so happy i found this video again, i could'n find it anywhere. idk if anyone has said it before (someone probably has), and it's not really news, but i haven't seen this yet. we know that once we get to a power of 2, we will go straight to 1. but how do we get to 2? well it just so happens that 2's powers, except for 1 and itself, work in a system. 4=3x+1, then 8=a prime number (pb) +1, then 16=3x+1, then 32=pb+1, then 64=3x+1, then 128=pb+1 and so on. basically, if the exponent is even, the number is equal to 3x+1 (x varies from one power to another), and if the exponent is odd, then the power is equal to (prime number)+1.

example: if the graph reaches 682, it will divide by two, giving us 341. that's odd, so we multiply by 3 and add one, getting 1023, and then 1024, which is 2¹⁰. we are then forced to go all the way to 1(obviously, because it's a power of 2, so it'll be divided by 2 until it's odd, so until 1).

example: prime number 2047 (look it up or do the maths). add one, boom: 2048=2¹¹.

the thing is that 3x+1 will never be a prime number, because it will always be even. my current knowlege in maths doesn't really help me prove that 3x+1 wil always, eventually, become a power of 2 and i'm running on 1% brain juice because it's late, but maybe someone else can do something with what little i've said

Hackall360 Gaming25 days agoDon't remember if this was mentioned in the video, but I thought of this video a year later, and attempted to feed the equation into chat-gpt, and after a very long conversation about complex mathematics, imaginary and real numbers, apparently the equation is solvable with x = -1/2

Athenri25 days ago^{+1}Excellent example of chatGPT being absolutely terrible at math. If you permit numbers like x=-1/2, how are you even determining evenness or oddness?

Jokes on YouYear ago^{+3863}Teacher: Why did you not answer the questions on your test.

Me: Because the Math is not ripe enough for me to answer these questions

UTU49Year ago@Delaraam

I love WOTH.

My other favorites are MNT and FUOPH.

DelaraamYear ago@UTU49 YESSSSS 😂😩😩😩im so happy you know her

UTU49Year ago@Delaraam

Shizuku? Is that you?

DelaraamYear agolmaoooo

Faris Akhal3 months ago^{+15}My gut feeling says that there must be a way to logically rule out different possibilities other than 4, 2, 1 loop

MonkeBrainMonth ago^{+3}Yeah make it 7

AJ'sMonth agoThe visual presentations are the best. I wonder who makes them!

HussainMonth agoMultiplying by three only scales the number. Adding one turns it into an even number. Dividing by two breaks down the construction of the number until it reaches one. It’s like a pro player playing Tetris and clearing all the lines on screen

Harllem Alves3 months ago"One is all, all is one." FMAB proved it years ago 😎