The Simplest Math Problem No One Can Solve - Collatz Conjecture

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The Collatz Conjecture is the simplest math problem no one can solve - it is easy enough for almost anyone to understand but notoriously difficult to solve. This video is sponsored by Brilliant. The first 200 people to sign up via 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.

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/

Special thanks to Patreon supporters: Alvaro Naranjo, Burt Humburg, Blake Byers, Dumky, Mike Tung, Evgeny Skvortsov, Meekay, Ismail Öncü Usta, Paul Peijzel, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Juan Benet, Ruslan Khroma, Robert Blum, Richard Sundvall, Lee Redden, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson, Sam Lutfi, Ron Neal

Written by Derek Muller, Alex Kontorovich and Petr Lebedev
Animation by Ivy Tello, Jonny Hyman, Jesús Enrique Rascón and Mike Radjabov
Filmed by Derek Muller and Emily Zhang
Edited by Derek Muller
SFX by Shaun Clifford
Additional video supplied by Getty Images
Produced by Derek Muller, Petr Lebedev and Emily Zhang

3d Coral by Vasilis Triantafyllou and Niklas Rosenstein - ve42.co/3DCoral
Coral visualisation by Algoritmarte - ve42.co/Coral



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Cosmic Nomad
Cosmic Nomad Yıl önce
I absolutely love how mathematicians always find the most random things to debate over!
You 13 gün önce
Most 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 Dior
Kristian Dior 25 gün önce
Dipa Tarigan
Dipa Tarigan Aylar önce
Yup, just like my wife
bill stock
bill stock Aylar önce
@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 B Aylar önce
@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.
Jack Scanlon
Jack Scanlon Aylar önce
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!
me me
me me 3 gün önce
you want but might not fit....
Athenri 4 gün önce
@steven chen if it were as simple as just saying WeLl ClEaRlY this happens, there would be no problem - but how do you know it never gets stuck in a loop elsewhere? If you do 3x-1 and /2, you get stuck in a loop with anything that hits 5, for example. So it's either: Prove that no such alternate loop exists Prove that such an alternate loop exists Show such an alternate loop.
steven chen
steven chen 4 gün önce
I still don’t understand what they are trying to solve here. Is it pick a number that doesn’t end in the 4 to 2 to 1 sequence? Because I don’t think that’s possible since they made it divide it by 2. Just like how infinity works no matter how long a number go since there is a divide by two there will be a point to where goes down to 8 4 2 1. I don’t get what we are trying to solve
Athenri 8 gün önce
@Tanner Muhle Does it? How are you so sure that it does?
Tanner Muhle
Tanner Muhle 8 gün önce
​@Athenri what question? I'm super confused what the actual "problem" is? Yea, it's randomness, liked millions of things in math. And it makes sense, you chose to set RANDOM rules. Odd is 3x +1 and even is ÷2. What is there to solve other than the fact that it always ends in a loop.
Phillip Meece
Phillip Meece Aylar önce
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 Я
Тут Был Я 6 gün önce
@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 Modified
J Modified Aylar önce
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.
FourKK Aylar önce
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 person
bored person 23 gün önce
Only integers have parity.
Aerxis 25 gün önce
​@Athenri gaussian integers (complex integers) have unique prime factorization, hence the terms even and odd ARE properly defined
FourKK Aylar önce
@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
Athenri Aylar önce
They don't really have even or odd numbers though, so you'd have to think hard about what your extension would work like.
Timofey Gerasimov
Timofey Gerasimov 2 aylar önce
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 person
bored person 23 gün önce
​@H B something tells me you've never heard of quantum mechanics.
H B Aylar önce
@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 Gerasimov
Timofey Gerasimov Aylar önce
@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.
Nickers Aylar önce
Which phenomena you are referring to?
TheOriginalFayari Aylar önce
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.
lol Gün önce
@E Stolee because the cpu would run forever and still not find the number, so it goes back to the original problem right?
E Stolee
E Stolee Gün önce
@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
lol 10 gün önce
i 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 OMAR 17 gün önce
I discovered the pattern of prime numbers, how can i make it public?? (Im outside the US)
a nobody's nobody
a nobody's nobody 20 gün önce
Yeah. I feel like that too. 🙄
K. PacificNW
K. PacificNW Yıl önce
Everyone here: "...but just a maaaaybe I'll be the one to solve it."
systim30 Aylar önce
@John D I understand clearly. You are explaining this because of what? What are you trying to prove ... and who are you trying to get into a debate with... 🤔
John D
John D Aylar önce
​@systim30You don't understand second step 22/2 is not equal 11, the 11 is 11 pieces 2,2...., this is main problem of mathematics 22/2 can't be equal simple 11 because in left side we have 22 and do split it by 2 = result. result can't be in real world 22/2 = {2,2,2,2,2,2,2,2,2,2,2} = 22 = 11*2 X/2=half of X? where is second half? The true is X/part size=Y parts Finally question what are you trying to do when you multiply Y^ 3+1? what's the point of that?
John D
John D Aylar önce
@JR Bros I Solve You don't understand second step 22/2 is not equal 11, the 11 is 11 pieces 2,2...., this is main problem of mathematics 22/2 can't be equal simple 11 because in left side we have 22 and do split it by 2 = result. result can't be in real world 22/2 = {2,2,2,2,2,2,2,2,2,2,2} = 22 = 11*2 X/2=half of X? where is second half? The true is X/part size=Y parts Finally question what are you trying to do when you multiply Y^ 3+1? what's the point of that?
Traveller Aylar önce
Chestradamus Teutonic
Chatgdp will
СМЕШУСИК 3 aylar önce
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 bonnett
chance bonnett 18 gün önce
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.
nonamecha0s Aylar önce
Хахатунчики в чате
Nancy Fancy
Nancy Fancy Aylar önce
@Artists Aliens 💀
Nancy Fancy
Nancy Fancy Aylar önce
@aneyesky agreed! ifyt.
Skybird Projects
Skybird Projects 3 aylar önce
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.
cawayy23 Aylar önce
check out Michael Penn's video about the Collatz conjecture. He goes on about both these extensions at the end.
Donald Stinnett
Donald Stinnett Aylar önce
Include imaginary numbers too.
x lorrix -
x lorrix - Aylar önce
@breadsticck you could make it so if the nunber is closer to odd/even then it does the thing
GonPlays 3 aylar önce
@breadsticck correct
breadsticck 3 aylar önce
Well the whole even/odd thing wouldn't work for non-integers
Jacob Sims
Jacob Sims Aylar önce
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 B Aylar önce
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 Xina
John Xina 16 gün önce
This was truly a great video to see! Keep up the great work!
The Musical Stylings of Brent Bunn
Mad respect to the animators here. That must've been a lot of work.
Bobby Smith
Bobby Smith 8 gün önce
This 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.
aashna 23 gün önce
Finally some recognition for animators...otherwise it's just 'editors'
Mesonaldo Editz
Mesonaldo Editz 2 aylar önce
Bro you could do the add first and then times by 3 3 ez
S M 4 aylar önce
Thank you for saying animators instead of just “the editor”
Devang agarwal
Devang agarwal 5 aylar önce
Its pretty basic so im pretty sure anyone will be able do it
Dan Leidal
Dan Leidal Aylar önce
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
Dru Oswald
Dru Oswald 2 aylar önce
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.
Rådiøåctive 26 gün önce
Can decimal numbers be used? Or fractions? Just wondering, it may be possible that way.
LinkGuy08 Aylar önce
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
Sarthak Bhatia
Sarthak Bhatia 15 gün önce
I 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...)
Diego de Paula
Diego de Paula Yıl önce
Whoever created all those graph animations is an absolute master in after effects expressions
nigelft 22 gün önce
​@Official_Freehugs606090 Then you get into the realm of variable variables, which lays beyond me ...
nigelft 22 gün önce
​@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 ...?
Yessir 5 aylar önce
@huskai no. Your wrong. His comment was 110% accurate. Because of Jesus he get his editing skills in after effects.
@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 Li
Isaac Li 5 aylar önce
try create an expression for aah
Matt Bruce
Matt Bruce 2 aylar önce
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 B Aylar önce
@Mahalalel Christ taught us mortals many things, but he didn't teach us a while lot about mathematics. Even deities have their limits.
Mahalalel Aylar önce
The 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 ✝🌅
remy njeu
remy njeu 3 aylar önce
Isn't it easier to work only with odd numbers ? Odd multiples of 3 have no odd antecedent. Also, if an odd number N can be written N = 8k+5 (k integer), then it exists a smaller N'=(N-1)/4 which gives you the same next odd number.
Gaurav Mitra
Gaurav Mitra 22 gün önce
Also what I thought. So I looked into it and formulated a general proof for all odd numbers. I have written it in the latest comment to the video.
H B Aylar önce
@John D In all numerical bases other than 2, 22/2 is 11. You will find *nobody* that disputes this assertion.
Poro Aylar önce
@John D What are you even saying 22/2 is 11
John D
John D Aylar önce
I Solve You don't understand second step 22/2 is not equal 11, the 11 is 11 pieces 2, this is main problem of mathematics 22/2 can't be equal simple 11 because in left side we have 22 and do split it by 2 = result. result can't be in real world 22/2 = {2,2,2,2,2,2,2,2,2,2,2} = 22 = 11*2 X/2=half of X? where is second half? The true is X/N=Y parts Finally question what are you trying to do when you multiply Y^ 3+1? what's the point of that?
Twitchy Tyrant
Twitchy Tyrant 3 aylar önce
presumably their "brute force" strategy takes these and other shortcuts, but the skipped numbers can still be considered "tested"
Rearrange Your Guts
Rearrange Your Guts 6 gün önce
This 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.
bruce okelo
bruce okelo Aylar önce
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
Axer101 Aylar önce
My gifted teacher from last year is teaching kids this year about the collatz conjecture, and it looks amazing.
Mr Scientific
Mr Scientific Yıl önce
Nice work Soviets. You got me.
runciter naki
runciter naki 10 aylar önce
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
Artificial intelligence plus lottery
Found the mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".
Surya Prakash
Surya Prakash Yıl önce
@Ali Akram wwwwwwwwwwwwww¹wà
Martina Omeara
Martina Omeara Yıl önce
It's 4X the X doesn't mean multiply
Sameeknowsbest Yıl önce
Thehh uhhhh got me
Norman Graham
Norman Graham 2 aylar önce
Also note: 2**a at the closest point to 3**b, is an increasing sequence. (They are getting further apart).
Ray Gaskell
Ray Gaskell 2 gün önce
I 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.
Shane Tolf
Shane Tolf Aylar önce
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 TRvid!
Mile Šakić
Mile Šakić 2 aylar önce
If 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.
Bintang Naufal
Bintang Naufal Aylar önce
aside from the problem, these animations are really cool. what tool did you use to make them?
shadyceddy Yıl önce
Fun fact: We are not mathematicians but we got interested by this.
Aleksandar Jovanovic
yes haha
BloxxingDinosaurus 2 aylar önce
I am the 20202nd like and the 222nd reply to this comment.
bi nq
bi nq 2 aylar önce
_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
QuinticGhost 3 aylar önce
same lol
quantum flucs
quantum flucs 3 aylar önce
@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
Michael McGuire
Michael McGuire 3 aylar önce
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.
G Lakers
G Lakers Aylar önce
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
Andrew Werner
Andrew Werner Aylar önce
I 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 B Aylar önce
Off you go. Please let us know when you have something of interest to tell us.
Valiebro 3 aylar önce
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 B Aylar önce
@Dan Treadwell Powers of 2 are an extremely small subset of the positive integers and therefore make very little difference to calculation times.
Dan Treadwell
Dan Treadwell Aylar önce
@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. . . 🙃
ExtravagantPanda 3 aylar önce
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.
Mary Furr
Mary Furr Aylar önce
I wonder if the reason they always end up at 1 is because that's the only number every single number is divisible by
ParkieL Yıl önce
Oh my god, this poor animator. That is a serious amount of dedication. Looks fantastic!
Kalliboy Yıl önce
@Mehtab Ghumakkad Maybe I'll work with this someday Python fascinates me everyday
NotYayaNguyen Yıl önce
@Timmyrbx how do u know it’s a “he”
Llama Man
Llama Man Yıl önce
@Lucky The Luckless Wolf I know I am
Lucky The Luckless Wolf
@Llama Man no, you're amazing
Timmyrbx Yıl önce
He needs to be paid every single day $100,000 heh
Tokas1981 Aylar önce
Regarding 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 Modified
J Modified Aylar önce
That explains why it's different, but not why there are multiple loops in one case and not in the other.
Swango Aylar önce
15:55 kinda fascinating how there's like a "wave" made out of perfect squares going from right to left on the upper part
Lightning First
Lightning First 17 gün önce
If 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.
Hope Rock
Hope Rock Aylar önce
Here 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?
awebmate Aylar önce
Who 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.
zbGiovannihe Spiaag
Playing 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
Levy Chevy
Levy Chevy Yıl önce
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 B Aylar önce
Why are you repeating tests which have been done many times before ? What do you hope to discover by doing so ?
bi nq
bi nq 2 aylar önce
glad to see i’m not alone lol I do be Collatz-ing instead of actually focusing on homework
Artificial intelligence plus lottery
Found the mathematical phenomenon A very interesting channel - " Artificial Intelligence plus lottery".
Same brooo
James Pronger
James Pronger Yıl önce
Blah blah blah more replys. "Think differently and simply"
Arath Alatorre Muniz
I 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 Atlas
The Golden Atlas Aylar önce
What 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.
Boredom Buster
Boredom Buster 16 gün önce
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.
Esage 16 gün önce
How? You can never get an odd or even number
Clownpierce Aylar önce
Showed 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.
Tom McKernan
Tom McKernan 2 aylar önce
Applying 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?
aperson Aylar önce
you may have notice the pattern inevitably leads to powers of 2, which is why they keep dividing until 1
Walkastray Yıl önce
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 B Aylar önce
@Jim Balter Absolutely not - it is a purely social construct.
H B Aylar önce
@Art Thingies No Spinal Tap for you then.
Pratana Kangsadal
Maria Maria
Maria Maria Yıl önce
@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.
Calvin Williamson
Calvin Williamson 3 aylar önce
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 Modified
J Modified 3 aylar önce
They're not identical, but there's no known reason that 3x-1 would have multiple loops while 3x+1 would not.
Diederik Huys
Diederik Huys 2 aylar önce
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 B Aylar önce
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.
bluejam Aylar önce
i like how mathematicians treat numbers like they're little critters observed in their natural habitat
Parinita Gowda DV
Parinita Gowda DV Aylar önce
Very truly said "Mathematics is not yet ripe for such questions"😁
Alexander Weck
Alexander Weck Aylar önce
Perhaps 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.
Demens Clay
Demens Clay 10 aylar önce
A big shoutout ot the graphics department for making this 100% more understandable!
Jim Greene
Jim Greene 6 aylar önce
@Anndy Arguedo lol
Jim Greene
Jim Greene 6 aylar önce
@icebreaker900 Broad is the path that leads to 1. Narrow the counter examples to this conjecture.
Josiah Brady
Josiah Brady 8 aylar önce
no problem
gerkey 8 aylar önce
@myUserName thanks
icebreaker900 9 aylar önce
Lukepuke311 Aylar önce
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
Stan Meer
Stan Meer 2 aylar önce
Ok 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
Lixuskzs Aylar önce
The 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.
Peter ALHachem
Peter ALHachem 2 aylar önce
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.
Poro Aylar önce
You cannot perform 3x+1 two times because by definition, 3x+1 will ALWAYS give an even number
FKA Gaming & Crafts
4:44 I wonder if you could program a random number generator with using similar equations and methods.
TheNordicNormie Yıl önce
Pretty much every subject in school is really interesting if I’m not forced to learn it
Guy Yıl önce
Pratana Kangsadal
Victor Kappel
Victor Kappel Yıl önce
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 tables
Benguira David
Benguira David Aylar önce
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 Modified
J Modified Aylar önce
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).
Timothy Gunter
Timothy Gunter 2 aylar önce
Seems that this could address multiple issues with RNG that is developed by game creators. This has natural, mathematical randomness built in.
Ranger Aylar önce
Its not that this math problem is hard, it is just that it is so time consuming.
B. Germocen
B. Germocen 2 aylar önce
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 TRvid recommended this at 6am, but now I am engaged and intrigued.
Poro Aylar önce
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
Max Lebow
Max Lebow Aylar önce
It 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.
Agentkp Yıl önce
Mathematicians: Dont waste your time on this problem 20.7 million people: YES
Liam Wilson
Liam Wilson 10 aylar önce
@Frank Chary imagine if we do figure out warp technology and they actually call it warp technology lol
soobin doll
soobin doll 11 aylar önce
Abhijeet Yıl önce
Search about Abhijeetbyte Collatz Conjecture GitHub 😎🤣🤣👍👍..... Can't share Links on TRvid comments
Frank Chary
Frank Chary Yıl önce
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.
Dawid Markiewka
Dawid Markiewka 3 aylar önce
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.
[ZenkaiSoul] 2 aylar önce
This 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
De ViceCrimsin
De ViceCrimsin Aylar önce
As 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
Helou20 2 aylar önce
The equation 3x+1 is known as the "Collatz Problem" or the "Ulam Conjecture" and is famous because no one has been able to prove its result. The problem requires you to take an integer and, if it is even, divide it by 2 and, if it is odd, multiply it by 3 and add 1. You can repeat this process as many times as you want, and the conjecture states that any integer starting with an integer will eventually lead you to the number 1. Although this has been verified for all integers up to a certain limit, it has yet to be proven for all integers. This makes the problem intriguing for mathematicians. Example with 100: 100 is Even, so divide by 2. 100/2 = 50 (50 = Even, so divide it by 2.) 50/2 = 25 (25 = Odd, so do 3x+1) 3x25+1 = 76 (76 = Even, so divide it by 2.) 76/2 = 38 (38 = Even, so divide it by 2.) 38/2 = 19 (19 = Odd, so do 3x+1) 3x19+1 = 58 (58 = Even, so divide it by 2.) 58/2 = 29 (29 = Odd, so do 3x+1) 3x29+1 = 88 (88 = Even, so divide it by 2.) 88/2 = 44 (44 = Even, so divide it by 2.) 44/2 = 22 (22 = Even, so divide it by 2.) 22/2 = 11 (11 = Odd, so do 3x+1) 3x11+1 = 34 (34 = Even, so divide it by 2.) 34/2 = 17 (17 = Odd, so do 3x+1) 3x17+1 = 52 (52 = Even, so divide it by 2.) 52/2 = 26 (26 = Even, so divide it by 2.) 26/2 = 13 (13 = Odd, so do 3x+1) 3x13+1 = 40 (40 = Even, so divide it by 2.) 40/2 = 20 (20 = Even, so divide it by 2.) 20/2 = 10 (10 = Even, so divide it by 2.) 10/2 = 5 (5 = Odd, so do 3x+1) 3x5+1 = 16 (16 = Even, so divide it by 2.) 16/2 = 8 (8 = Even, so divide it by 2.) 8/2 = 4 (4 = Even, so divide it by 2.) 4/2 = 2 (2 = Even, so divide it by 2.) 2/2 = 1 (When it reaches 1 and still continues, it will be on a Loop of 1 -> 4 -> 2 -> 1) Tradução para Brasileiros (Translation for Brazilians): 3x+1 é conhecido como o "Problema de Collatz" ou o "Conjectura de Ulam" e é famoso porque ninguém conseguiu provar seu resultado. O problema exige que você tome um número inteiro e, se for par, divida-o por 2 e, se for ímpar, multiplique-o por 3 e some 1. Você pode repetir este processo quantas vezes quiser, e a conjectura afirma que qualquer número inteiro começando com um número inteiro levará você ao número 1. Embora isso tenha sido verificado para todos os números inteiros até um certo limite, ainda não foi provado para todos os números inteiros. Isso torna o problema intrigante para matemáticos. Exemplo com 100: 100 é Par, então divida por 2. 100/2 = 50 (50 = Par, ou seja, divida-o por 2.) 50/2 = 25 (25 = Impar, ou seja, faça 3x+1) 3x25+1 = 76 (76 = Par, ou seja, divida-o por 2.) 76/2 = 38 (38 = Par, ou seja, divida-o por 2.) 38/2 = 19 (19 = Impar, ou seja, faça 3x+1) 3x19+1 = 58 (58 = Par, ou seja, divida-o por 2.) 58/2 = 29 (29 = Impar, ou seja, faça 3x+1) 3x29+1 = 88 (88 = Par, ou seja, divida-o por 2.) 88/2 = 44 (44 = Par, ou seja, divida-o por 2.) 44/2 = 22 (22 = Par, ou seja, divida-o por 2.) 22/2 = 11 (11 = Impar, ou seja, faça 3x+1) 3x11+1 = 34 (34 = Par, ou seja, divida-o por 2.) 34/2 = 17 (17 = Impar, ou seja, faça 3x+1) 3x17+1 = 52 (52 = Par, ou seja, divida-o por 2.) 52/2 = 26 (26 = Par, ou seja, divida-o por 2.) 26/2 = 13 (13 = Impar, ou seja, faça 3x+1) 3x13+1 = 40 (40 = Par, ou seja, divida-o por 2.) 40/2 = 20 (20 = Par, ou seja, divida-o por 2.) 20/2 = 10 (10 = Par, ou seja, divida-o por 2.) 10/2 = 5 (5 = Impar, ou seja, faça 3x+1) 3x5+1 = 16 (16 = Par, ou seja, divida-o por 2.) 16/2 = 8 (8 = Par, ou seja, divida-o por 2.) 8/2 = 4 (4 = Par, ou seja, divida-o por 2.) 4/2 = 2 (2 = Par, ou seja, divida-o por 2.) 2/2 = 1 (Quando chega em 1 e mesmo assim continuar, ficará em um Loop de 1 -> 4 -> 2 ->1)
Foxiesboi Aylar önce
I wonder what happens if you put in decimals. I don't think the video talked about when you add something like 1.5
Hanya manusia biasa.
Me : "That's interesting puzzle, maybe I can solve it" Me 22 minutes later : "oh."
The Rayven
The Rayven Yıl önce
@Miso_Soup fair enough...
Grzegorz Hordejuk
1 instead of 0 on the scale
The Rayven
The Rayven Yıl önce
@Miso_Soup which means you *HAVEN'T* solved it... If you had solved it, you would be more than willing to publish your findings, to prove you solved the equation... However, your lack of willingness to prove you solved the equation, only proves that the equation has not been solved because there is no proof that the equation has been solved... So in reality, the equation remains unsolved...
The music King
The music King Yıl önce
@Miso_Soup what is it
FinneasAF 2 aylar önce
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 B Aylar önce
Any "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.
Aiganym Toleukhankyzy
The most fascinating video I have seen in my life😍 Can you tell please which program did you use to make this video?
Gawędziarz Aylar önce
I 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.
Robert Lukac
Robert Lukac 2 aylar önce
For a counter example one would have to prove for a number that it never ends up at 2^x because from there it's always a straight way to 4-2-1. And is it possible to be creating even numbers that never end anywhere in the 2^x?
Ich bin Tupiniquim
Ich bin Tupiniquim 28 gün önce
Wunderbar!!! Vielen herzlichen Dank.
Spookworm Yıl önce
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.
nowherenearby Yıl önce
@Spookworm nope
Spookworm Yıl önce
@nowherenearby You're entitled to your wrong opinion :)
nowherenearby Yıl önce
@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)
Tza16 Yıl önce
@nowherenearby it is for some people
nowherenearby Yıl önce
watching these types of vids isnt learning math but ok
ItzVoid 3 aylar önce
I've noticed something, I've put in extremely big numbers into a 3x+1 grapher and they always seem to go 5, 16, 8, 4, 2, 1, I'm just confused.
ItzVoid Aylar önce
@MonkeBrain what
ItzVoid Aylar önce
@Maddox Monteza no
Maddox Monteza
Maddox Monteza Aylar önce
have you tried multiples of 64
Nate Regan
Nate Regan Aylar önce
yes! they do! however, just because they end in the same pattern does not make it a loop. 4-2-1 is the only loop here because from 1 you get 4, then 2, then 1. you'll never have it back up to 5, 16, or 8.
Messenger 1221
Messenger 1221 2 aylar önce
I wonder what loop you would get if after dividing by 2 add 1 which would turn it odd again each time.
Decorum Pan Troglodytes 🌈🦧
I’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.
irakaram Laurent
irakaram Laurent 17 gün önce
does this apply to decimal numbers also?
Marco Kapusta
Marco Kapusta Yıl önce
This math problem is actually like my trading portfolio, I can start with any number but end at $ 1
Martin Garixx
Martin Garixx 6 aylar önce
This is like whar casino algorithm looks like
Shepherd Marima
Shepherd Marima 6 aylar önce
S.R. 6 aylar önce
This seems to be the most devastating global pandemic.
Aman Rajput
Aman Rajput 6 aylar önce
god damn 😂
Not Sure
Not Sure 7 aylar önce
This is life, no matter how often you do something odd, you end up as one…
Masons Aylar önce
And I think this is the point where all math teachers gets their thinking caps on and try to solve it
Katie 2 aylar önce
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.
Victor Serrano
Victor Serrano Aylar önce
This might be a silly question but the functions are calculated on base 10 (as we humans determined the base based on our digits). I am not a matematician but, might there be another numeric system that can help simplify the pattern behaviour? (my thought is that a new counting system, or a re-thinking of the counting system (e.g. real numbers) could help simplify the problem.
J Modified
J Modified Aylar önce
Not really. You can change the x / 2 to a binary right shift, and 3x + 1 to left shift with carry and add to original. That doesn't appear to provide any useful mathematical insight, but it does show you how you could efficiently implement Collatz testing on fully custom hardware.
Gargamel Aylar önce
Theoretically, don't you need to find just 1 out if the number in loop for it to loop?
Miguel Prytoluk
Miguel Prytoluk Aylar önce
I've tested that both 2^420420 - 420 and 2^420420 - 69 still abide to the Collatz Conjecture returning to 1 in 5654853 steps.
emayar Yıl önce
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 B Aylar önce
@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 Therefore
And Therefore 6 aylar önce
Wait, if you do this 3x+1 for 1.5, you will always find an odd number, and will eventually lead to infinity
promessa_ex Yıl önce
evens are green odd is orange of course, but three is blue
Tejasvin Kansal
Tejasvin Kansal Yıl önce
@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
Julian Aylar önce
I 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
nwogamesalert Aylar önce
Just wait it out until some of the other TRvid geniusses have solved it. In the mean time enjoy yourself with wine, women and song.
Hussain Aylar önce
Multiplying 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
Hackall360 Gaming
Hackall360 Gaming 21 gün önce
Don'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
Athenri 21 gün önce
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?
Tobias Urs Marti Müller
I love how you can prof in mathematics, that you'll never be able to prof your problem correct or wrong.
J Modified
J Modified Aylar önce
Yes, 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.
KO Aylar önce
WOW! I understand this video's subject matter and I very poor at math, but I like it.
Pansoti Yıl önce
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
Lily Liao
Lily Liao Yıl önce
@Zain Elsayed zain the incel
Jim Balter
Jim Balter Yıl önce
@Stolfo Ch. No, Pags is the fool, not you.
Zain Elsayed
Zain Elsayed Yıl önce
@IrokoSalei that’s not what i meant. im not saying that there arent any genius women out there.
UnPrankAble 666
UnPrankAble 666 2 aylar önce
If 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 B Aylar önce
True but irrelevant. You are allowed to start from 3 if you wish.
WoofWizard Aylar önce
We'll see in many years, when we finally have enough resources to create a crazy computer that can process near infinite bits
H B Aylar önce
How can we expect to understand the behaviour of one single algorithm unless we look at other similar algorithms. How do x+1, 5x+1, (2n+1)x +1 behave ? How do x-1, 3x-1, (2n+1)x-1 behave ? Is there anything that we can say about a counter example other than it does not satisfy the conjecture ?
Tomato 2 aylar önce
When you apply 3x + 1 to a number like 1 then all the numbers it generates will end in the 4 loop so that could improve computing time
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