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

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

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

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Written by Derek Muller, Alex Kontorovich and Petr Lebedev

Animation by Iván Tello, Jonny Hyman, Jesús Enrique Rascón and Mike Radjabov

Filmed by Derek Muller and Emily Zhang

Edited by Derek Muller

SFX by Shaun Clifford

Additional video supplied by Getty Images

Produced by Derek Muller, Petr Lebedev and Emily Zhang

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

Coral visualisation by Algoritmarte - ve42.co/Coral

## ความคิดเห็น: 56 104

I don't care about math whatsoever but this was incredibly interesting. Amazingly presented, dude!

## Davy Bloggs

## 8 วันที่ผ่านมา

@Baz Snell Yeah, the Americans bastardise (with an 'S' btw, not a 'Z'!) the English language to an unbelievable degree. They even make up complete words which don't exist! eg, they continually say Aloominum, when the material they're trying to talk about is Aluminium! There is NO SUCH THING as Aloominum - Yanks, it doesn't exist! Same with Solder, used for Soldering wires etc - they say Sodder!! No such thing guys. There's an 'L' in there - don't you see it? Rant over:-)

## Rowena Braddock

## 12 วันที่ผ่านมา

I care a lot about maths, but also didn't understand a word. Some people have clever brains, I want some.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Frankie freitas

## หลายเดือนก่อน

yup

## Tracy D

## หลายเดือนก่อน

cringe

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.

## Math Simplified

## 3 วันที่ผ่านมา

@Kelly Swenson Thank you !!!

## Kelly Swenson

## 3 วันที่ผ่านมา

@Math Simplified you joined two weeks ago and have 500 subscribers! Nice job!

## Math Simplified

## 3 วันที่ผ่านมา

I feel, when you like your teacher, you will learn the subject well !! I completely agree with your statement. I was the same.

## Kelly Swenson

## 3 วันที่ผ่านมา

Scholars in higher academia that is the teachers of the rest, just refuse to acknowledge they aren’t special.. unique, sure, but come on. And. This IS the real math. Rule followers are great but creative minds are the true leaders we need to be raising by giving them creative freedom

## Kelly Swenson

## 3 วันที่ผ่านมา

If bath was this.. we could have all been mathematicians

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

This for a mathematician seems like the equivalent of just jumping on Fifa when you're bored 😂

## Harrison Achunche

## 10 วันที่ผ่านมา

wagwarn bro. didn’t expect you here can’t lie 😅

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## ZabookaHD

## หลายเดือนก่อน

Lol

## Extense

## หลายเดือนก่อน

Fax

The reality is that super-computing this into infinity would run for decades, and that's the only way you're likely to have a chance to find THE number that is the counter. This is a level of calculation that we will still be trying to solve when we reach Star Trek - level physics in our space programs. And that's what makes it great: all of our advances in math and science and technology have been built on the supposition that anything can be solved, if we are smart enough, dedicated enough, patient enough, creative enough, even wealthy enough to finance it all. Just look at the recent helical drive material that supposes to push us right to the edge of the speed of light. It all takes passionate people to make all this happen. And videos like this are amazing, because it breaks simple-yet-not-so-simple mathematical dogma into digestible pieces that the average person has a chance to grasp - maybe not entirely, but enough to be intrigued. Kudos to Derek Muller and Alex Kontrovich for executing this lesson perfectly. Not everyone has the skills and personality to pull it off this well. And a special kudos to the animators who visualized these concepts so beautifully, too allow the average viewer that chance to grasp them. I learned the Collatz Conjecture briefly as a thought exercise in college, and it never went very far, for obvious reasons. Thanks to this team for reminding me how interesting it is, and how unlikely we are to ever solve it anytime before aliens colonize Earth and bring us more advance math.....

Your way of Explaining through Graphics is beautiful sir.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Ian Horler

## หลายเดือนก่อน

Agreed

## Pengii

## หลายเดือนก่อน

@OnideusMadHatter the fact that you’re so wrong is making me crack up 💀

## Kragboks Gaming

## หลายเดือนก่อน

Thats the beutifull thing of math,it is a world that perfectly evolves around rules and formulas that we can turn into a equation or a formula or a gridd position

## Jim Balter

## หลายเดือนก่อน

@OnideusMadHatter Nothing you said there is right.

Math is the Queen of all sciences. Even if one thinks something is easy you'll get humbled very fast. Great video

## Muhammad Faran Aiki

## 13 วันที่ผ่านมา

@Ravi Dodeman Some math can be proven, how can you prove something in philosophy? If you meant the axiom of math is a philosophy, well, only if there is a question.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## IrokoSalei

## 26 วันที่ผ่านมา

@Ravi Dodeman Maths doesn't exist, it's all based on axioms and definitions , if you change them you get other maths. People who say maths "exist "are mostly ignorant about its plurality.

## Michał Włodarczyk

## หลายเดือนก่อน

@Ravi Dodeman hard to say. It's possible in a strange realm of quantum physics that we create reality by our actions and observations. However it's also possible that the reality is totally independent of us. I guess it's up to one's philosophical standing.

## Ravi Dodeman

## หลายเดือนก่อน

@Michał Włodarczyk but that’s the part where people dissagree, some say math exists and we are discovering it and others presume we are bending reality to fit in the confines of our rules

Lol i remember hearing about this in grade 11 but had enough of solving for x definitely sounds like an energy or scienece issue. Math has always been a language for me. Cool to know we tried negatives. Wondering if we used another odd prime and not 3 but 3s never involved how is 6 divided by 2 not an oçcurence. Either way i spent alot of time doing other patterns. And if you can relate it to energy and 1 being state of zero energy just gotta find vrelatinships between numbers and atomic weights possible. Math is definitely universal obviously enjoyed the video. Wish i hadtime i mean it is a quarantine but... sorry just remembered if you got 3 youd be increasing again for 4 more

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## starlight blues

## 29 วันที่ผ่านมา

the prime 6765 54-566=2 anne is bben to make begttr

Amazing video! The amazing thing is how something so simple can't be solved. Love it!

If only I knew about this for my senior seminar paper! Math has always left me curious and intrigued. This video reminds me why I chose to study math in college. The subject can be concrete and so mysterious at the same time…..I love it!

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## Häherfeder

## 2 หลายเดือนก่อน

How?

## cisse de clercq

## 5 หลายเดือนก่อน

we do or best.

## Tố Quyên

## 5 หลายเดือนก่อน

Someone's back is hurtt

## some idiot

## 5 หลายเดือนก่อน

i agree, but there are other people that have animators do even more like haminations (he's a story time animator)

This is a great video, I feel like maybe we're not supposed to solve it mathematically, test it then learn from and then learn from it what it has to offer us about our perceived reality

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Question. Isn't the problem itself flawed in that the rules around odd/even force depreciation and therefore it will decrease to the smallest positive integer?

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## Raccoonious 🦝

## 21 วันที่ผ่านมา

I was thinking similar - but probably not. I'm guessing there could be a cycle where it doesn't have 2^n in its cycle (in which case it would decrease to the smallest positive integer 1). Is one way to think about it - that there will be sufficiently large 2^n that ever odd number will reach in its life cycle? (7:58) We could try expressing the fraction of odd numbers that end up with 2^n somewhere in its path and add them together? I'm guessing the guys who've been researching this probably are in a better position to answer this though. Perhaps that's why it's a conjecture not a proof. Also, if this is as simple as that - I guess there won't be a department mathematicians who haven't solved it for 20 years.

i was wondering, could one possibly solve this conjecture by proving that all sequences will have a power of 2 (8,16,32,64 etc etc) in it? because the moment we hit a number of the form 2^n the only thing we can do is divide by 2 till we hit 1, because a power of 2 will only have 2 as a factor and thus be even until we reach 1, where we will hit the loop. I'm pretty sure epople have already thought and tried this, if so, what went wrong?

## Andrej Novák

## 9 วันที่ผ่านมา

nothing went wrong. it does not matter if you hit 2 ^n number or not, you will always end up with 4 2 1 loop anyway

Just thank you for these videos. The quality in there, the fun. Everything is perfect. Thank you

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## Brent Lucas

## 5 หลายเดือนก่อน

Yeaaaaaaaaaaa boyyyyyyy 🕶🕶🙈💼🤘🤘👖🩳🩳😄😃🤣🙃🙃😍😚😋🙃😔😣😢😖😫😟🤩🤨

## Lee Holmes

## 5 หลายเดือนก่อน

@Mehtab Edhan I like your thinking 👍

## Lee Holmes

## 5 หลายเดือนก่อน

Not really its the same to apply a passenger in a car thinks it's really hard to drive that car RIGHT when we all know it's pretty simple to drive

## ParkieL

## 5 หลายเดือนก่อน

@idk a name for this that’s an annoying way to try and get more viewers

great Job Derek, as always extremely interesting. you are such an awakened mind.

If there is an exception, there are infinite exceptions because every number that results from the seed to infinity is also an exception.

The way you explained was amazing.. loved this!

Great video. Thank-you for giving plenty of visual analysis.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## Only Gaming

## หลายเดือนก่อน

I was like, im not watching a 20 minute video about that, here i am

## Heather Thomas

## หลายเดือนก่อน

I am 😳

## Purpledstan123

## หลายเดือนก่อน

I find this by being hella bored in science

## Sparvin

## หลายเดือนก่อน

True

Feel like the answer ultimately has something to do with powers of 2. Like the counterexample will inevitably be a loop that just barely misses a power of two, or it turns out that every possible power of two is reachable via 3x+1.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Every other exponent of 2, when you subtract 1, results in a multiple of 3. 4, 16, 64, 256, 1024, 4096, 16384, etc ad infinitum. The probability that an even number, when reduced by a factor of 2, will result in an even number is 50%. But the probability that an odd number multiplied by 3 and then plus 1 will result in an even number is 100%. A downward trend is to be expected because the probability of reducing by 2 is twice as high as the probability of multiplying by 3. As the number trends downward by a factor of 2, the probability of the eventual 3x+1 operation resulting in an exponent of 2 increases exponentially. Once the result of 3x+1 is an exponent of 2, the game is over.

While it might not be of any importance, I was playing around with the Collatz Conjecture's equation after watching this. By messing around with the value that is added to 3x I think I've determined that you can add any of the first five odd numbers and it will seemingly always create a loop, although I only tried a few numbers for x so I'm not entirely sure. Anyways, let's say I do 3x+3 instead of 1. No matter what number x seems to be, the equation will eventually create a loop that ends with 3 instead of 1. This seems to be true for all the other prime odd numbers up until 9, with larger numbers for x simply taking longer to get to the loop like with 3x+1. +3 ends in a loop of 3, +7 ends with 7, and +9 ends with 9. However, for some odd reason, while 1, 3, 7, and 9 all create a loop ending in the numbers added, 5 doesn't always loop back to 5. Sometimes it loops to it's added number, but other times it instead creates a loop that ends in 19. I don't have enough knowledge with mathematics to understand why 5 is such an exception to the other numbers, but it doesn't want to loop back to it's added number as consistently as the other numbers. Now, with all that said, I don't know how true all of this is. I am not an expert in this field, and I was just messing about using the information given in this video. If anyone would like to correct me then by all means feel free.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Zestastic

## หลายเดือนก่อน

I think everything you said is accurate, but unfortunately it doesn't help a whole bunch. To solve the problem you either need to find a way to completely prove that every number will go to 1 or that there is at least one number that doesn't go to 1. The likelyhood that there is a number that won't go to 1 is very unlikely since numbers are infinite, so they can just keep going up until they inevitably come back down. But that doesn't matter if we can't actually prove that every number doesn't work, and we can't just try every number.

Maybe the fact that perfect squares are also 3.1 percent might mean something or why are the negative and positive looked at as polar opposites when it might be both negative and positive. Lastly when you look at the charts not numbers but all those dots, it looks just exactly the same in patterns that it could be that with all the numbers checked thus far it could be accurately assumed the it won't stop the same patten

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

## _RealNoob

## วันที่ผ่านมา

Not being able to solve a problem too advanced for any current math theories doesn't make you bad at it. It just makes the problem too good.

## Arturo Ordonez-Hernandez

## 6 วันที่ผ่านมา

@Kicky my sister believes she has dyscalculia, although she was never evaluated and diagnosed for it. Honestly, since I have some learning disabilities myself: Autism, Amblyopia, and Mild Cognitive Disorder, which slightly affects my brain's processing speed, have all hindered my ability to learn, so at this point, I don't doubt that's a real thing. However, even someone with dyscalculia should be able to make improvements after enough practice; it won't be easy of course but I don't think it's impossible. I'm fortunate to have an Autism spectrum disorder in the sense that I gained an early interest in math; otherwise, I'd be pretty terrible at it too. Despite this, I also had to work really hard to learn it at times; It didn't always make sense right away with little to no effort. Has it helped me? Perhaps it has indirectly. I'm not making money solving math problems all day, but software development involves quite a bit of critical thinking so I think math certainly helped my brain develop those skills.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## Teemu Leppä

## 23 วันที่ผ่านมา

@Monkey D. Luffy killing a patient doesnt make you a bad doctor

## Bunnix

## หลายเดือนก่อน

But verrrry bad at plus

The only possible demonstration is by induction, so we initially tried to figure out how to partition the problem. Of course, sequences starting with even numbers can be ignored, and demonstrating that all the numbers in from 3n+2 are always present in lower sequences is relatively easy: example 5 appears in the sequence 3;10;5. 3n are always "primitives", so they never appear in previous sequences, but we can demonstrate that all the 3n obey to the conjecture. 3n+1 are more tricky. But what happens in functions with more than two choices? We found another unresolved problem that was analyzed by Collatz, called g(n). This function divides by 3 instead of by 2 and rounds the result up or down to the next or previous multiple of 3. So we started wondering how functions with an arbitrary number of choices behave and we have found the reason why the Collatz conjecture must be true. It's all detailed in a PDF, but before finding somebody interested in reviewing it, my friend passed away for a bone cancer. I would like to have it reviewed in his memory, even if it's found wrong. A hint: collapse the two operations on even numbers to a single (3x+1)/2. Comparing its behaviour with that of the functions with > 2 choices is easier.

Great example of concept learning amazing teacher

11:27 "If you can show that in every 3x+1 sequence there is a number that is smaller than is smaller than the original seed then you have proven the Collatz Conjecture." Wouldn't that mean that a loop other than 421 is impossible? That the only way for it to be false is if it would approach infinity?

Not only the content of this channel is amazing but also the graphics and animations.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Victor Kappel

## 2 หลายเดือนก่อน

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

## OwO

## 2 หลายเดือนก่อน

Yes

## Richplaya Hi-Top 901

## 2 หลายเดือนก่อน

.

## Hydra head Clan

## 2 หลายเดือนก่อน

I was force to watch it so I would say this is bad

I am by no means a mathematician, but having spent the past year studying statistics for psychology, I instantly recognized the shape at 6:01. My thoughts were, “GTFOH! That looks like the positive half of a normal distribution!” Then by 6:36, I had to pause and reevaluate what I know and understand about the world and reality.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Stephen DeVore sadsongsishere

## หลายเดือนก่อน

What did you realize after reevaluating?

Are there other similar problems? I think the operation looks rather random. So why spend time on a particular instance of this problem or is 3x+1 known to be unique?

Every integer number above 9 can be expressed as a sum of the preceding decade number +10. e.g. 11= 1+10 ; 21= 11+10, 12= 2+10, 47= 37+10, 1345 = 1335 +10 etc. If a mathematician can prove that by adding 10 to any number the sequence must be a always eventually return to a value less than the preceding number and therefore result in the sequence 4,2,1 . Run the sequence on the number nine. Run the sequence on the number 10. Show that the addition of 10 adds a defined specific value to the sequence. Then for any integer number you can demonstrate that the sequence outcome is the previous decade plus the sequence for the number 10. Why can't they do that?

OMG! I think my brain just blew a circuit breaker! What an awesome video, thanks so much!!

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## IrokoSalei

## 26 วันที่ผ่านมา

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

## Uloola

## หลายเดือนก่อน

@Jim Balter Odd is right.

## Uloola

## หลายเดือนก่อน

@Jim Balter No it's an odd statement, doesn't make it a wrong statement.

## Jim Balter

## หลายเดือนก่อน

@notahotshot 4 leaf clovers are considered lucky in our culture.

Math exercices: 1+1=? Math mini-tests: 7*9=? Math exams:

## prakriti bhatt

## 22 วันที่ผ่านมา

Yes🤣🤣🤣🤣🤣

I tried with negative numbers and I ended up on a whole new loop 😂 instead of 1 and 4 loop, it was 5 and 7

Infinity does not work with this. Take infinity as an odd number, you multiply infinity by three and then add one. It’s still infinity. Take infinity as an even number, so infinity divided by two is half of the infinity which is still infinity.

The number of numbers to try between 1 and 2^60 is the same as that between 2^60 and 2^61, and the time taken to search must be a lot longer since the numbers are larger. Still, I hope they keep the machine running.

i wrote this comment to appreciate that those graphs were not just random. There were exact and to the scale.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Rich bob

## 5 หลายเดือนก่อน

exact and to scale? The 'coral' ones were all textured, coloured and shaped in editing (not to scale at all) and the others are just basic graphs anyone could make in ten minutes on powerpoint ffs. If it was exact and to scale we would be looking at a black and white mess of straight lines. The graphs are mostly to look pretty.

## T H A N A T O S

## 5 หลายเดือนก่อน

@Anastasia O I don't think anyone is wrong, we're all correct and absolute. We are all Human and have our flaws and talents... yet we will never come together to realize our true purpose once we step inside the classroom of thank you for reading this entire comment and not getting anything out of it. Have a nice day.

## Anastasia O

## 5 หลายเดือนก่อน

@EpicVideos Okay, but the original comment was not on the animations. It was just on the actual graphs and making those graphs. You just don't want to be wrong here lol.

## Chris Dawson

## 5 หลายเดือนก่อน

@AxxL Ah yes the dude with two """girl"""friends

I always love when you have problems like this which are near-unanimously agreed upon by experts to be unsolvable, and yet people in the comments still think they’ve solved it in a couple of minutes

They tested up to the number 295,147,905,179,352,825,856 and they all ended back at 1. It’d be hilarious if the number 295,147,905,179,352,825,857 is the one that shoots off to infinity

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## J Modified

## 27 วันที่ผ่านมา

I'm fairly sure they just reported when they hit a power of two, and kept going.

thank you for making this video this math question was in my university test and i knew a lil about it

All I thought about while watching was that the decimal system of measurement was invented by people. If we came up with another, how would our mathematics change and what all these problems would look like.

## J Modified

## หลายเดือนก่อน

This problem does not depend on number representation, as most don't. You can do it in base 87, Roman numerals, or handfuls of marbles - it's all the same. For the number base to matter, there would need to be some dependency on a particular digit value or digit count.

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

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Karan

## 5 หลายเดือนก่อน

@dgcfgv vgb thanks

## dgcfgv vgb

## 5 หลายเดือนก่อน

@momo what? I'm so confused lol

## Sarath Sreedhar

## 5 หลายเดือนก่อน

😂💯

## momo

## 5 หลายเดือนก่อน

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

The reason the strings behave differently on the negative number line is because +1 takes the string in a different direction on the negative side of things. They will still end up in one of three possible known loops, as far as we know, but they move through thought-time with a different motion. Do that seaweed-looking rendition with the negative numbers. The paths move differently.

When I look on the histogram generated by this equation it resembles the model of tree of life. Most complex organism can be reduced to single cell thus the single cell is at the base of tree of the life stem. The beginning of life is still a mystery despite many theories on the subject. Math is not ripe enough to resolve equation of life.

This is the same patterns you'd find when you map primes. Could it be related? Because the factors that impact 3x+1 is whether 2 is a prime in the number within the original pick, or when a prime of 2 would come back up in the sequence (which makes me want to do math), only large unique prime combinations could increase significantly (e.g. 27 which is 3^3), so then the proof lies in the nature of if there are any large prime combinations that would become unbound by the dampening of the 3/4 reduction to introduce a factor of 2.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Have they looked at the pattern creation in binary? If the least significant bit(lsb) is 0 then shuffle all bits down one place, if the lsb is 1 the place a 1 in the column ahead of all existing 1s & set lsb to 1. Seeing an infinite pattern looks easier to me like that but I’m not a mathematician so maybe I’m severely missing something…

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.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## promessa_ex

## 5 หลายเดือนก่อน

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

## Tejasvin Kansal

## 5 หลายเดือนก่อน

@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

## Masters Rubin

## 5 หลายเดือนก่อน

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

## 37rainman

## 5 หลายเดือนก่อน

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

With `c3` being the first lower value in the trajectory of `c2`, a stopping (AKA dropping) pattern can be derived: ``` 2^p2*i+c2 -> 3^p3*i+c3 (i >= 0) ``` `(p2 >= 2, c2 odd, c2 For example `[7] -> (22,11) -> (34,17) -> (52,26,13) -> (40,20,10,[5])`, and 5 is less than 7, which involved 7 divisions-by-2 and 4 multiplications-by-3. The "+1 part" remains inside the `c2`/`c3` values. This means that the first lower value of any `(2^7)*i+7` is `(3^4)*i+5`, so `128*i+7 -> 81*i+5 (i >= 0)`. Example for `i=1`: [135] -> (406,203) -> (610,305) -> (916,458,229) -> (688,344,172,[86])`.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

I learned more from this than I did in 4 years of high school

I only understood nearly a quarter of this video but I'm still, wholeheartedly, amazed

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## Ima Ok

## 24 วันที่ผ่านมา

Don't waste any more time on it, everything goes to 1.

I also add that the 3x+1 sequences end in a loop 1-2-4-1. There are only 2 possible endings for a sequence: loop or grow to infinity. The only possible loop is 1-2-4-1 (or 1-2 if we collapse the two consecutive operations on even numbers to (3x+1)/2. Now let's make a function with 3 choices: if N = 0 (MOD 3) then N=N/3; If N = 1 (MOD 3) then N = (N-1)/3 IF N = 2 (MOD 3) Then N=(N+1) MOD 3 Then let's vary the coefficient of the decrement: so that N = 0 (MOD 3) then N=N*2/3. Those two functions behave very differently: In the first, there are no crossings: Each number belongs to one and only one sequence. Some sequences end in a loop, other grow to infinity. There are only 4 possible loops: 1-1-1... ; 2-3-2-3-2-3 ...; 4,5,7,9,6.. and one longer starting with 44. The second function behaves very similarily to the 3x+1. Let's call "base" the number of choices. If the coefficient is >=base/2, the sequences have crossing and all end in a loop otherwise some grow to infinity and there are no crossings (crossing="numbers can appear in more than one sequence").

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

## IrokoSalei

## 26 วันที่ผ่านมา

Trading is leeching 🤮

## THE CAR

## 28 วันที่ผ่านมา

Same 😐😕

## BABA YAGA

## หลายเดือนก่อน

Its my love life:(

## The Black Cat

## หลายเดือนก่อน

Jesus, I wanted to write exactly the same comment))

## Г.Анхбаяр

## หลายเดือนก่อน

@o GENIUS

So, here's the thing about these rules. Multiplying the odd number by an odd number is not a big issue. The division of whatever you get by 2 will obviously result in decay. The result of opposing multiplication of living cells is the opposite of what you get when life proceeds uninhibited. The ending sequence looks like 421. It isn't. It is, but it is incomplete. The digital root opposite of 421 is 578. These are opposites in terms of wedding up to 9, the highest single digit number in base 10. 4 and 5, 7 and 2. 1 and 8. The master is 124, the mirror is 875, going forward. 124875 comes out of 1 2 4 8 16 32 and so on. If you know how to use that. Cell multiplication. Check out the last six numbers in the longer decays. Read backwards, they are 124875. The rest of the effects are due to the interference of increasing the numbers by the highest possible number 3. There's the first bit of gobbledygook. It works for all of those numbers. 295 quintillion numbers.

I do not consider myself any good at math, but my first question after listening to this was: If they have already tried a big set on natural numbers and negative numbers, why didn't they also try the rational numbers? It would be interesting to test the fractions, specifically such fractions where 0

## MarineIntel89

## 11 ชั่วโมงที่ผ่านมา

In this particular case they specified x is an element of the positive integers.

This is actually interesting and informative.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

What happens if you change either the multiplication factor (I.e. 4x, 5x, 6x....) or add more than 1 and does an odd number or an even number make a difference.. as I only have a pen and paper I'm not going to try but I suspect regardless of the multiple or the addition the sequence will always end up in a loop...

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

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Zain Elsayed

## 26 วันที่ผ่านมา

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

## IrokoSalei

## 26 วันที่ผ่านมา

@Zain Elsayed I really hope you are being sarcastic. You couldn't ever come close to the genius of women like Emmy Noether buddy.

## Sandra Gagnon

## หลายเดือนก่อน

I am only watched the video to see if my 9 year old daughter was correct. She looked at the problem and gave me an answer.bStill don't if she is right and I still don't have an answe

## stolfo

## หลายเดือนก่อน

@Pags the claim isn't that 3(x+1)/2 = 1 though, the collatz conjecture is a recursive algorithm. also you just checked your work to an algebra problem then called it a proof either that or it's a joke and i just made a fool of myself lol

All you need to know is 3, 6 & 9. another hint: what do the numbers between the twin primes have in common as well as in scaling?

I never studied math beyond Algebra II and Geometry in HS and 1...maybe 2? required Math class in College. I've stumbled across several of the Ve videos and while the mathematical symbols and functions are beyond me, I can mostly understand the logic. However, the main awe I feel is the genuine excitement and passion that I see in the videos--and see when reading through the comments. I appreciate and respect the enthusiasm of all of the professionals, scientists, and students of math. It makes me think I'm missing out on experiencing the source-energy of math. I'm 55yrs old, but I'm still going to head over to Brilliant and see if I can understand it's basic packages or if I need to start somewhere else.

10:01 things escalated really quickly from .. "This is a strange numeric pattern .. " to "They can define organic shapes " ..

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

I think the question is "Why does this happen?" Well, any number which this rule had been applied, will eventually become a full frame number of two. Since by adding 1, you make the number even then by dividing it you eliminate an option and start off with another one. By eliminating all the options you eventually get a full frame number of 2, which will get you to 1 - 4 loop. That's why this situation occures.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Found the Mathematical Phenomenon A VERY interesting channel - "Artificial Intelligence plus lottery"

Mathway: “Am i a joke to you?” Photomath: “Answer the question.”

## West Explains Best

## 3 หลายเดือนก่อน

Ugh those programs are virtual math teachers worst nightmare.

## Dino_Tokic

## 3 หลายเดือนก่อน

Lmaoo

## Lacrimosa Knight

## 3 หลายเดือนก่อน

Hahaa

## CryptedMage

## 3 หลายเดือนก่อน

Lol wassup homie

## ColtonTheElytrian

## 3 หลายเดือนก่อน

Lol

@Varitasium, I think, it has the simple explanation of its solution that it will always fall back to the loop. Just because we are looking for one series only and just ignore the other series hidden in it. Whenever the power of 2 numbers meets with 3N+1 numbers, it will fall back to the loop. The 3N+1 has to meet with the power of 2 numbers due to its nature. So we don't need to look for any bigger number as explained in this video. It's just a pattern number involved with randomness. This is the final solution.👍

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

There is always a counter reaction to everything. There is a heads to the tails to the coin. The only Smartest thing a man has said I don't know.

This seems less like a math problem and more like a programming problem. It's like using a a while loop with 2 if conditions. Honestly if you can solve this problem, you instantly make programming insanely efficient. Less loops means less operations required to complete a task, which means less cost (time, money and energy) to complete a task. Also if you want the negative side to have the same loop as the positive side and vice versa, just use these 2 rules. 3x-1 (for odd) /2 (for even) If you use +1 in the negative side you will practically have a different effect.

So we are trying to prove this impossible versus improbable right? We know it is improbable, but can we say with certanty it is impossible? Similar to the problem of prime numbers, or creating a perfect circle without having the finite circumference?

I'm proud of myself for voluntarily watching a video about Math

## Griffin

## 5 หลายเดือนก่อน

Every journey to the couch begins with a single step.

## Ultimate monster

## 5 หลายเดือนก่อน

This is an epic gamer moment

## Ultimate monster

## 5 หลายเดือนก่อน

This is an epic gamer moment

## Andrew & Jessica Hersh

## 5 หลายเดือนก่อน

Haha my sentiments exactly!

## President Fresh

## 5 หลายเดือนก่อน

big brain moment

You just solved the problem here, you're applying an equation that will definitely produce an even number every time, and it's 2 times more likely to produce another even number when devided by 2, so naturally the curve will always tend to fall into a multiple of 4 and from there decend and fall in the 4-2-1 loop

## Grizzly01

## 21 ชั่วโมงที่ผ่านมา

@Mohamed Salah El-Din Mostafa No, it didn't. Not at all.

## Mohamed Salah El-Din Mostafa

## 22 ชั่วโมงที่ผ่านมา

@Grizzly01 as I said, the video proved it when he explained the even and odd probabilities

## Grizzly01

## 22 ชั่วโมงที่ผ่านมา

But can you prove that in a rigorous mathematical way? Many have tried...

Doesn’t the negatives having more loops have to do with the fact that when you add the number one you are making the absolute value of the number smaller?

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## DangerMaster19 Gaming

## 28 วันที่ผ่านมา

@J Modified idk bro I’m not a math genius, just had a thought and I noticed that when you add positive 1 to a negative it makes the absolute value smaller which changes the number that comes out of the equation.

## J Modified

## 29 วันที่ผ่านมา

@DangerMaster19 Gaming Those are specific examples, but I don't see that more loops follows directly from "when you add the number one you are making the absolute value of the number smaller".

## DangerMaster19 Gaming

## 29 วันที่ผ่านมา

@J Modified because, like 5 for example, if you do 3(5)+1 you get 16 which goes to 8 then 4 then 2 then 1, but if you do 3(-5)+1 you get -14 which then goes to -7 then to 3(-7)+1 which is -20 then -10 then -5 and thus, the loop repeats.

## J Modified

## หลายเดือนก่อน

Why would that mean there must be more loops?

Ooofff... the animations are always interesting. Even if its just showing you simple graphs of mathematical equations.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Often times the simplest answer is the best answer. 1) the high end can scale infinitely 2) the low number hits a low at one (can never go lower that happens to put it in a loop) Conclusion mater how high the number goes. Infinity it can always go higher but when it hits its low it can never go lower and placed in a loop Also the loop is more likely to happen at the low end far less variables as for the high end far many more variables (infinite) I couldn't see how this would stump many.

"What do you do for a living?" Mathematician: "I am studying 3x+1."

## Rob Peters

## 11 วันที่ผ่านมา

😂😂😂

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

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

## Tom Petitdidier

## 3 หลายเดือนก่อน

@UTU49 😂😂😂

## UTU49

## 4 หลายเดือนก่อน

"What do you do for a living?" Mathematician: "Sometimes I multiply by 3... and then I add 1. After that I always divide by 2. If I can, I just divide by 2 right away until I can't anymore. Then I multiply by 3 and add 1. Some days I do a lot of the multiplying by 3 and adding 1. Other days I do more dividing by 2." "Oh."

## UTU49

## 4 หลายเดือนก่อน

@Sam N. This confusion would not have happened if Mathematicians had habitually called this "The 3n + 1 Problem" or "The 3a + 1 Problem". It's kind of too bad.

I respect the fact that every number ends up looping onto the 4-2-1 cycle,but.......im just asking, why dont you also try it with the decimal numbers and the fractions? And what about négative numbers too? Cuz i mean...they exist

4:20 Imagine, this is the cell reproduction. So the curve is the lifespan with the numbers of cells.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

i tryed it several times and found you CAN get other loops... if you go into the negativ numbers ^^ one is at (-17) he didn't say x as a part of the natural numbers (i also tryed it with something like 13.123, i think there i should multiply by 3+0.001) is >0 (i hope i dusted of my highschool math ... and english well enough ;) ) but yeah, greate! i enjoyed it very much :)

What's also interesting is what happens when you change the rules slightly. When picking the number, instead of applying 3x+1 for odd numbers and x÷2 for even numbers; apply 3x+1 for both odd and even numbers. If x=1, then 3(1)+1 =4. So now the rule for even numbers applies and eventually end up with one. 4÷2=2 2÷2=1 If x=2, then 3(2)+1=7. So now the rule for odd numbers applies and eventually end up with one. 7(3)+1=22, 11(3)+1, 34÷2, 17(3)+1, 52÷2, 26÷2, 13(3)+1, 40÷2, 20÷2, 10÷2, 5(3)+1, 16÷2, 8÷2, 4÷2, 2÷2, 1. When x=10 and x=18 it takes a lot longer to reduce to one, but eventually does. I've only went as high as x=20 by hand. What was also interesting was that the pattern of reduction for odd and even numbers eventually became the same.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Me: “tries to do it in negative” “Gets in a loop anyways”

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Found the Mathematical Phenomenon A VERY interesting channel - " Artificial Intelligence plus lottery"

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Found the Mathematical Phenomenon A VERY interesting channel - " Artificial Intelligence plus lottery"

## Merp

## 3 หลายเดือนก่อน

@Shweta Tiwari yeah you're right, i was thinking of the wrong thing

## Shweta Tiwari

## 3 หลายเดือนก่อน

@Merp hey, how 1? Wouldn't it become 0?

## Jesus has given you all. Repent or die.

## 3 หลายเดือนก่อน

Repent to Jesus Christ!! “Your word is a lamp for my feet, a light on my path.” Psalms 119:105 NIV O

I like the nr 9. It has so many strange constant patterns. Pick a nr between 1 and 9 and keep adding 9 to it. The outcome will always return to the nr you started with. 4+9=13 1+3=4 13+9=22 2+2=4 22+9=31 3+1=4 this goes on. Another one with nr9 is, pick any nr and do it x 9 like 6473x9=58257 now add all the numbers from the outcome to each other 5+8+2+5+7=27 and this makes 2+7=9 and you can do this on and on but will always return to 9. If you start playing with this then you will see more strange constants in the outcomes. Sorry for my chaotic comment.. I find it hard to express this without the numbers.. :P

The last equation can be solved with the method of two unknown with an infinite amount of results if continued with that method wich is not needed but like it can be used

This is very nicely animated!

Cant you prove it by looking at the pattern at which either 3x+1 is used for odd numbers and x/2 for even numbers? After all, the prime reason for such directed graph phenomena to occur is due to the varying time the rules for odd and even number is used. lets use 341 in your example: the number 341 has used the rule for odd number (3x+1) once, while it used the rule for even numbers (x/2) 10 times Then use the number 26, it has used the rule for odd numbers (3x+1) twice, while it used the rule for even numbers (x/2) 8 times Of course the magnitude of the number should be taken into consideration when proving this conjecture.

"Is 2^68 a lot?" "Depends on the context. Dollars? Yes. Numbers to check for a simple math problem? No."

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Found the Mathematical Phenomenon A VERY interesting channel - "Artificial Intelligence plus lottery"

## Celia YAHCCS

## 5 หลายเดือนก่อน

I can't work out 2^68 accurately on excel - all power of 2 over 2^50 have 0s at the end as it can't cope with more than 15 significant figures.

## 3 RandomTrippy Colors

## 5 หลายเดือนก่อน

@columbus8myhw I am confused by your comment, enlighten me lol

## SiMe

## 5 หลายเดือนก่อน

In theory the context of dollars might not be enough either due to ultra hyper inflation that results in the dollar being so worthless that 2^68 dollars is an insignificant amount. Granted, at that point we will probably abandon the dollar as a currency, but theoretically it is still possible. So what I would change it to is specifically "the current dollar" just so it avoids this theoretical pedantic rhetoric :]

## Brandon Luján

## 5 หลายเดือนก่อน

@Matthew Talbot-Paine they would be a lot of digits of pi to be fair, in any application you will never need that many. For the sake of being able to generate them? Yeah, any number of digits is not even close to the size of the true amount of digits there is.

I can actually solve it... Bro it's very simple. It equals 7 😁 Very easy and simple... It's not a Times symbol it's a X symbol in math So if you use 3X+1 the strategy is 3×1 =3 +1=4 4+3=7

Maybe a quantum computer coupled with AI might be able to solve the Collatz Conjecture problem.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Can we use fractions instead of whole numbers? Because if we can, i.e. ½, then (½x3)+1=1½. So we continue to divide by 2 because we pay attention only to the denominator, and basically we never get number 1 as a result, but a +infinity. Of course, that is if we are allowed to use fractions...

Question: If 3x+1 is turing complete, can't we find the program wich halts and then find the corresponding number to see if it loops?

## Dirac Sea

## 6 วันที่ผ่านมา

Yes but doing that is just automated brute force. We need something else; I think maybe a quantum computer; if nothing else it would be an interesting test of the a different computational matrix then the more common 0 and 1. Maybe this line of thinking is just my curiosity thou, basically throw everything we have at a problem, haha.

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.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## DHRUV SHARMA

## 3 หลายเดือนก่อน

Same brooo

## James Pronger

## 3 หลายเดือนก่อน

Blah blah blah more replys. "Think differently and simply"

## Diego Castro

## 3 หลายเดือนก่อน

Nerd

## Anime Fan Club

## 3 หลายเดือนก่อน

You have a great teacher if they motivated you to spend hours on this!

Here is a question: Has the Xn + y space been explored much? Basically asking which sets of X and Y behave like 3 and 1, and which behave differently What if you change the Evens rule to be something other than divide by 2? What I think might be achieved by that is that if we find sets of numbers for which the patterns are wildly different, with multiple loops, or diverging sequences... but then if there are many which behave just like 3n + 1, but maybe they have 1 or two diverging strands, it might give us a clue where in the number sequence to check for a diverging sequence in 3n+1

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Found the Mathematical Phenomenon A VERY interesting channel - " Artificial Intelligence plus lottery"

thanks for making this video that is because I don't have to do real math now I can just skip extra work at home.:)

There is a stucture which includes every positive integer and shows that all converge to 1. It is of infinitel size but, throughout, any integer can be shown to be part of a regularly repeating pattern of formats.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

If you do 8x+1 and you choos 7 as your picked number, every last 2 digits on your progression up are 57. I'm still in a straight up trajectory

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

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

## the tank

## 2 หลายเดือนก่อน

same lo

## kpink gamer

## 2 หลายเดือนก่อน

Sameee

## ThaKid Kane

## 2 หลายเดือนก่อน

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

## im bored

## 2 หลายเดือนก่อน

Same

Cool video, but I think Goldbach's conjecture even simpler and to quote wiki "one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even whole number greater than 2 is the sum of two prime numbers."

## J Modified

## 10 วันที่ผ่านมา

I think Collatz "looks" easier to prove.

3x+1=0 Subtract the 1 on both sides which will give you 3x=-1. Divide the 3 on both sides which then will give you the answer x=-1/3.

## Artificial intelligence plus lottery

## 16 วันที่ผ่านมา

Once it hits 2^n then it will be down into the loop there is no limit to n so eventually you'll run into one and then the number would go into the loop.

The seed doesn't follow the pattern. All numbers in the sequence, sans 1, follow the pattern. There's actually 2 patterns. 3x + 1 and halving. Applying the pattern to even numbers is inconsistent. I would first break this up to only analyze odd numbers as seeds. Then perhaps individually apply the 3x + 1 formula to all odd numbers within range. These microcosms will give you the ability to predict the combo. This is just what a programmer would do