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
The Manim Community Developers. (2021). Manim - Mathematical Animation Framework (Version v0.13.1) [Computer software]. www.manim.community/
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Special thanks to Patreon supporters: Alvaro Naranjo, Burt Humburg, Blake Byers, Dumky, Mike Tung, Evgeny Skvortsov, Meekay, Ismail Öncü Usta, Paul Peijzel, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Juan Benet, Ruslan Khroma, Robert Blum, Richard Sundvall, Lee Redden, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson, Sam Lutfi, Ron Neal
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Written by Derek Muller, Alex Kontorovich and Petr Lebedev
Animation by Ivy Tello, Jonny Hyman, Jesús Enrique Rascón and Mike Radjabov
Filmed by Derek Muller and Emily Zhang
Edited by Derek Muller
SFX by Shaun Clifford
Additional video supplied by Getty Images
Produced by Derek Muller, Petr Lebedev and Emily Zhang
3d Coral by Vasilis Triantafyllou and Niklas Rosenstein - ve42.co/3DCoral
Coral visualisation by Algoritmarte - ve42.co/Coral
Everyone here: "...but just a maaaaybe I'll be the one to solve it."
"I could write a computer program to try and solve it". Because I'm sure nobody has tried that before 😪
You can actually instantly solve for half of all numbers. If all numbers up to an odd N works, (n+1)/2
I too thought i could solve it :D
What is there to solve? There is nothing to solve
@@rabiebabies7812 0 is not positive but it forms a loop. Its also not negative but no number ends up at zero so it is independent loop of itself
“Pick a number” Me:Fou- “Seven? Good choice” Me:but I-
I said 4, I usually say 3 but I said 4 😂
He said seven because seven is more likely to be chosen lmao
I think Im the only one who chose 7
Only reason I'm not liking is bc tbe lile count is at 69
I didn't choose a number at all because no one can make me do math.
3:17 I like how the 4-2-1 sequence animation is accompanied with a melody whose notes match the distance between numbers.
But it doesn't. To my ears the interval is the same between 4 and 2, as it is between 2 and 1. It should drop a whole step and then a half-step. But maybe my ears aren't as good as they used to be. Anyone here with perfect pitch or really good relative pitch that can either confirm OP or confirm me?
@@jackelewish1568I actually have a perfect pitch. The notes are B, A and G#.
@@joemyk cool! Thanks!
@@joemyk The interval between B and A is a full step, with a half step between A and G#. Not 4-2-1 but 2-1-.5.
They don’t match the distance, but they do match what it’s divided by,, (4/2=2, 2/2=1), 2 each time,
I love how he makes us think that he is the world's greatest mathematician by showing us his picture when saying that, but then shows the other half of the picture.
Well, that isn't even a person who solve the task. It's a computer programme which tries to explain something what isn't actually relevant.
lol
😂 lol
This math problem is actually like my trading portfolio, I can start with any number but end at $ 1
you too? :)
I tried to remove the eyelash on the display 🤭
There's that damn eyelash on my screen again!
@@luca6819 .same lol
@@luca6819 You're using youtube in lightmode? ;o
Teacher: Why did you not answer the questions on your test. Me: Because the Math is not ripe enough for me to answer these questions
facts
Imma use this
@@lordsiomai be honest, no you won't
@@anyaburke6636 its 6
@Human Kind its already a 1000 We can make it 2000??
I was once talking to my math teacher and i asked her "why was Bhaskara so annoying" and she told me "he wasn't, he simplified a lot of things into a single calculus", then my friend said "so mathematicians have to simplify math because humans find it too hard when humans themselves created it" and my teacher said "yes, the point of math is simplifying something we made dificult in the first place". That chat was all i thought about throughout this video.
Love this anecdote. Thank you. For the video, it only exist because somebody said “no, you need to divide by 2 if it’s an even number” why? “I don’t have a reason why but it makes it complicated so do it”
As someone who is basically math illiterate, this makes almost no sense… makes me ask why we even made math, how it works, how do things even get figured out like the mathematical theorems and stuff… my confusion in difficult math related thingies makes me want to actually understand it more than my school education 😂
@@fletch4231 My thoughts exactly. Invent problems and then try to solve them. As if we didn't have enough problems!
@@fletch4231I'm not very good with math myself, but its origins are very simple. It started as a system of tallies for counting things (typically livestock). That's likely why most counting systems have changes at 5 and/or 10-those are the number of digits on a hand and both hands, the simplest way to tally (cf. Roman V & X, Indo-Arabic 9 to 10, crossing four tally marks, etc.). You probably tally with your fingers subconciously. It evolved into counting more complex things, and you may even have needed to create units of measuring, for parcelling land, weighing grain, and measuring beer. On the note of beer, written language may have developed in the Near East as a medium of accounting, as evidenced by some very early cuneiform tablets mathematically tracking grains and even beer. It slowly got more and more complex as people realized that math is better at describing the universe than it has any right being. Ultimately, it was serendipity that it happened to be our best way of describing the universe, which is absurdly complex. TL;DR: Math started as a way of tallying and became more complex as its applications were discovered. Math is never purposefully obtuse-it's just our best medium of describing the universe. As the universe is absurdly complex, any system describing it must necessarily also be absurdly complex.
@@jasondelay2931I mean exploring things like that might lead to good discoveries
Certainly one of the finest mathematical videos on KZhead.
Me: Where should we eat? Girlfriend: Mathematics is not yet ripe enough for such question
Noo
I love your girlfriend. Wait, no, it's not what you think it is!!!
LMAO🤣🤣🤣
😝
Lol
I absolutely love how mathematicians always find the most random things to debate over!
ridiculous too
I totally agree. What a waste of time
Not a waste of time. If you can find this solution, probably there is something you can achieve and get.
@@christloen4077 No way
In your mind
Here because I have the Collatz Conjecture in a Brilliant coding question. I have found Brilliant the easiest to jump back into when I have time. Great loop back Veritasium.
"coding" im always bewildered when people say that. coding is just another notation, nothing more.
@@armin3057 dork
@@armin3057 coding is a verb. It's an action. You code something which is to say you program it to behave in a certain way. What you're referring to is code itself, which is another form of notation. Regardless of the programming language, any mathematical function can be represented in code. What you're saying is out of place in response to this comment as it was not referring to the programming language, but the problem-solving aspect of writing code which in itself has nothing to do with notation.
When negative numbers have 3 different loops, it makes me wonder if I change not only the seed sign but the operand. And turned out that if I apply 3n-1 for negatives (likewise changing the sign of the objectvie function on the concavity on linear programming), the graphical representation is a mirrored one of the positives. This way the whole set can apply the same rules for positives on 3n+1 including a single loop (-4, -2, -1)
If we try the polynomial 3x + 1 for negatives then we will be stuck in a loop of -7 or -1 everytime for any negative integer. Try this too!
Yes, there are 4 different loops if negatives are used.
The +1 is the key tbh. If you go into non whole numbers any .25 .5 or .75 will loop as an odd number infinitely. Because 1 doesn't make it positive and non of these numbers rise a above 1 ever. they keep a 5 at the end. So it only works on numbers because we have no define of 0 as even OR odd, or both, and dividing it is weird, but also because adding 1 is what DEFINES changing from odd to even.
@@eon2330 That’s what I was thinking if you start the equation from 0 3x0=0 which is still nothing until u add 1 now we’re positive but u can’t get 2 from 1 so now the loop starts u will never get 0 again
@@DaPoloJaywhy did you multiply and add to zero, it's not an odd number. Why do you not divide by 2? 0 divided by 2 is 0. . .and if a previous comment is correct that zero is neither even nor odd, you can't even start at all.
Fun fact: We are not mathematicians but we got interested by this.
People that know math are are mathematicians and also if thay do math they are mathematicians
ikr
I am
Hmmm
@@amirpakravan4389 shut up u ruin the vibe
Whoever created all those graph animations is an absolute master in after effects expressions
Amen.
BR?
The thumbnail equals 1 cause 3x_ is 3x nothing so if I did that it would be 0 and if I plus the 1 it = 1
Math
BY "NO ONE" : He meant about Americans cause he himself is a american who dont knows anything about the outside world .
You can accelerate the conversion by allowing division by 3 beside 2. I noticed that in my own limited search. Fascinating stuff.
I’m glad I found this channel. Amazing quality content ❤️🙌
Mathematicians: Dont waste your time on this problem 20.7 million people: YES
Just cuz you said that I'm going to code a program that runs through all posible combinations on scratch
3 years year old me : what is maltiplikaton?
Ok
13 Million*
more
Me : "That's interesting puzzle, maybe I can solve it" Me 22 minutes later : "oh."
Lol
Same I was like I'm gonnna guess a random number and try to do it..but 2⁶⁰ is really a big numbers they tried
@@theultimatetime8029 well, Derek (the narrator in the video) did say that 2 to the 60 is nothing compared to the other numbers tried in Polya's conjecture. The counterexample which disproved Polya's conjecture was 1.845 × 10^361, an immense number. Still, 2 to the 60 is BIG.
@@mjzudba5268 yes ofcourse
@@theultimatetime8029 try 70!,it's bigger or even 2^70!
So, here's my observations. 2 to the power of any natural number will fit the conjecture. In addition, if we undo the opperation, we take (2^n)-1 and divide by 3, any output that is an odd, whole number will also fit the conjecture. My idea for solving it would be to see if there was a whole number that *didn't* fit the conjecture but that's just me.
I wanna know who edit his videos.... The hard work ❤ ! We appreciate you bro !!
-showing his own face “One of the greatest mathematicians” Dudes pops out of nowhere “Mr. Tao” Lol you had me at the first half not gonna lie
He had me too! Brilliant!
He had me too
@samridh sood infinity is a number, any number, or all numbers should i say, and no, this is not the problem with this conjecture.
LOL
@samridh sood I think you´re on to something. The Fields Medal is in reach!
Mathway: “Am i a joke to you?” Photomath: “Answer the question.”
Lol wassup homie
Lmaoo
Ugh those programs are virtual math teachers worst nightmare.
❤ A matemática é realmente encantadora 😮😮🎉 estou apaixonada!!
I’ve always wondered why mathematicians only look at the patterns of the ’hailstone integers’ with this problem. Maybe the pattern is found in the numbers that are skipped after doing the equation either with 1 number or after a million numbers
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.
nerd
Atleast find better ways of procrastination
@Hence Forgot bricks bit to though to bite on man ill rather have alloyed steel
Ez Answer Is 9 I was Doing my math Homework Bruh
You have a great teacher if they motivated you to spend hours on this!
i wrote this comment to appreciate that those graphs were not just random. There were exact and to the scale.
Ikr
Ikr I wonder how many days or months it took to build all of those. Unless he wrote a program for it then maybe a day or two
@@Sintinium of course he wrote a program for it but I expect the developer probably spent at least 2 weeks on making it.
You dislike the stuff that gets uploaded by my fingers clicking upload? Are you just a h8er boi? I say see you l8er boi. Don't watch the stuff that gets uploaded by my fingers clicking upload anymore. Your dislikes are damaging my good good GOOD reputation. I am a superstar, dear kd
@@Sintinium I think he paid some small company to do that, a single person is unlikely to do that
Found a very cool pattern, I know it doesn't prove and solve anything, which is useless but here, any odd number(represented by Y) multiplied by 3 then added by 1 and divided by 2 will always equal to Y+[(Y-1)/2+1)]. For example "5" since its odd then do 3n+1 which equals 16 and divide it by 2 which gives us 8, and 8 is equals to 5+[(5-1)/2+1)].
I found another pattern of odd numbers. 1 takes a single 3x + 1 to get to a multiple of 4. 3 takes two to get to 4x (3, 10, 5, 8), 5 takes 1, 7 takes 3 (7, 22, 11, 34, 17, 52). Basically, an odd number (Y) will take log2(Y + 1) repetitions of 3x + 1, x/2 to get to an even number.
Actually, this solves it. Only infinity can go of to infinity. Even with that, (2^31) - 1, for example, will take 61 runs through the function to decrease.
5 + ((5 - 1) / 2 + 1) does not equal 8. I'm guessing you meant 5 + (((5 - 1) / 2) + 1). But yeah, that's kind of a neat way of compounding the two steps into one.
@@soulsand4287impressive! Do you gave any idea how to prove this though?
@aadichinmaysharma4000 I don't think it's possible to prove. I definitely am not going to be the one to prove it.
The change of 3x+1 from branches pattern to 3x+1 on the -y axis where the branch pattern changes to a bar chart is reminiscent of the probability scenario on your video about the stock market and demonstrated with your ballbearing prop, also on -y axis 4,3,1 is viewed eventually as a singular number commencing the bar graph pattern.
Pretty much every subject in school is really interesting if I’m not forced to learn it
History of the entire world, I guess convinces me.
Pretty much every subject in university is really interesting if I"m not forced to learn it
School in a form of forced education kills interests and produces stupider people. Coersion always makes things worse.
English, grammar
Sorry to hear that you did not have good teachers. I was fortunate to go to a great school that had many good teachers that were able to teach stuff like this in interesting and engaging way. It was the teachers that failed you not the environment where you are 'forced to learn'.
Oh my god, this poor animator. That is a serious amount of dedication. Looks fantastic!
i was just gonna say that! Amazing work by the editor.
It seems like it is made by the same software that 3b1b uses.
Amajing
@@remenyo what is it??
i generated these graphs with python matplotlib, and then save the changing graphs for value of x, in an image sequence, then played them in premiere pro, voila..no animation needed for graphs and bar graphs 😁you can generate graphs with python
Nice video as far 2:00 i see I have these numbers thoughts since school,goes back to when i learned mathematics to do in mind calculations & once learned formulae,started applied,looked for patterns Though i never formally write those down (maybe in school times,but i lost them) often feel about these, Maybe,maybe i gotta write them down,like some pattern in maths,numbers,look how far they can go,how it can be constant at certain point,arises new etc,,. I often think it may goes huge for calculations(let alone brain thinking) so i doesn't write them down Now i feel like write them down (though anyone sees it in future,see its absurdity) Maybe it helps me do "logical thinking,calculations easy" before i go wild thinking constantly (keep thinking)😅 Instead of avoiding,running over,gotta take care of it,grow myself,train enough to over power it (life hurdles,goals be like) Thank you so far (no time to see fully) Wish all be well 😃🌟✨🙌 21.02.2024 01:31am ist (77k+ comments,871k+ likes,38,625,062+1views)
I've created two simple C++ programs that solve the multiplication of binomials such as (3x + 1), and am looking to do more. Thank you for this unique & interesting concept. College Calculus major.
Isn't 3x+1 equal to 3x=-1 so X=-1/3 ? Idk what's the problem here btw 😂
@@adamrozek5782Lol this shows the whole video went over your head What they are trying to solve is , They want to find a number which does not go in 4 2 1 loop
@@adamrozek5782Also it's not a 3x+1=0 😂
A big shoutout ot the graphics department for making this 100% more understandable!
a big shout down to yoy that you were'nt able to get such a simple equal...
I really hope this is satire 🤣🤣
@@josiahjray baited :D
@@gniewko123456 Hope so lol
Ah yes, 999 likes
Mathematicians: *_cries in proofs_* Scientists: *_laughs in null hypotheses_*
@@Sinaloabricks hypotheses is the plural 🙄
@@Sinaloabricks Who says that we have only the one hypothesis?
Statistician: *does both in bipolarity*
Is not mathematics merely just a part of science anyway?
@@andrewcramer9200 Bipolar Person: "Finally, someone DOES understand me"!
Consider the following. 1. there are an infinite number of 2^n numbers. 2. Consider an algorithm that selects a random number and tests it against whether it is a 2^n number. And if so applies the second half of the 3n+1 problem, in which case it will always collapse to the 4 -2-1 sequence. 3. What is the probability that the random number generator will never hit on one of the 2^n numbers. 4. I contend that the probability is zero.
It's not a random process. Consider that 3x-1 has identical statistics but multiple loops.
“Pick a number, any number” Uhh… 7? “7? Good choice!”
"What do you do for a living?" Mathematician: "I am studying 3x+1."
havent watch the whole video but 3x+1 is impossible to solve bc it has infinite solutions??
Big maffs
no one not even no one me: 3x+1 equals 1 because 3x nothing is 0 amd + 1 is 1
When I saw that picture I was like it’s obviously 4x I disagree
@@fanaticjay3825 bruh what
Math problem no one can solve: Exists Me: Finally I'm not the only one who is bad at math.
Not able to do a math problem, doesn't make you bad at math.
@@risav202 please explain. i dont agree
@@risav202 Nah.
I literally just saw you on Nas Daily...
@@risav202 I assume that you're not referring to math in general, just a specific math problem. Those of us with dyscalculia find even basic math challenging, to say the least.
It always and ultimately comes down to one or shall I say come up to one... the one and only... wherever you'd dig you'll find the one. Just need to open your heart following your eyes opening.
The Collatz Conjecture, also known as the 3n + 1 problem, is a mathematical problem that has remained unsolved since its proposal in the 1930s. It involves a simple sequence of operations on a positive integer. Here's how it works: 1. Start with any positive integer, n. 2. If n is even, divide it by 2 (n/2). 3. If n is odd, multiply it by 3 and add 1 (3n + 1). 4. Repeat steps 2 and 3 until the sequence reaches 1. The Collatz Conjecture states that no matter which positive integer you start with, the sequence will always end at 1. Despite extensive research and numerous attempts, the conjecture has not been proven or disproven, making it one of the most intriguing unsolved problems in mathematics.
yes... yes, we watched the video?
I think a lot of us clicked on this video thinking: *”oh it can’t be that hard”* edit: Jesus I didn’t post this comment so ppl could just argue in the replies. It was supposed to be a joke
Its only hard to find if you only work with whole numbers, at least assuming thats how mixed numbers would work Never mind a simple search says decimals cant be odd or even only integers, so yes it is that hard
@@mlpfanboy1701 i just solved this lol
@@Auromaxis what is it?
It’s easy, 0.
@@Auromaxis ?
The fact that this is the basis to making an organic shaped coral mesmerized me.
wait really? lmao
Wow 4th
Can we not use decimals?
it grows, makes an unpredictable, chaotic but somewhat beautiful image, and then inevitably falls back down to 1. like life and death cycle.
Found the Mathematical Phenomenon A very interesting channel - " Artificial Intelligence plus lottery"
This reminds me of the Philosophy Wikipedia page, where if you click on the first link in any Wikipedia entry you'll eventually get to Philosophy.
I looked up a train. I ended up at philosophy...
Well, that makes quite sense: any even number divided by 2, eventually - keeping adding 1 to any odd result and dividing it again by two - becomes 1. Despite the 3x or not. Multiplying it by an odd number would just makes the process longer, but eventually it goes to 1. No way out. I would say it is quite intuitive.
The most obvious issue for your intuition is that it stops being true if you, for example, ask about 3x-1 or 5x+1 instead. Like, we can all agree that it seems pretty clear that almost all numbers reach 1. But it's *very* hard to show that it's all numbers.
@@XCC23 - you may be correct, or you may be not. It would takes hours and hours of logic and calculations to know it. And I won't. For the time being, I am happy with the stimulation of this video. Thanks anyway for the comment.
@RicThai63 I am correct, because your reasoning doesn't predict that for 3x-1, the sequence 5, 14, 7, 20, 10, 5 exists. The hard part is ruling out that sequences like these can exist.
The class: 3+5 The homework: 3 times the square root of 4 The exam:
Exactly we go over short division then the exam is like (2a+1b)/10 the times by 10
For real the homework and class work are like 3 x 2 and the test is like calculate the diameter of the sun and multiply it by the amount of water molecules are in a single bottle of water.
Homework equals 6🕺🏾
@@Smdday._ Dang it! I was gonna say that!😂
The answer is six
The amount of graphic work that had to be done for this video is insane.
Exactly what i was thinking, i was like man props to whoever worked on this video
Try a Captain Disillusion video ... And he does those all himself
Really.. wow. Entropy maybe
@@peterh222 *disillusion
Listen ...don't look
My first thought is, since all numbers which are a power of 2, so 2^n, end up as 1, wouldn't it be easier to rephrase the problem and try to prove (or disprove) that by applying 3n+1 if odd and n/2 if even, all primes eventually give a result that is 2^n?
Choosing random numbers and applying the rules until you get caught in the loop is a great way to kill time in class, I do it all the time
Mad respect to the animators here. That must've been a lot of work.
And how much work on calculator.
Looks like 3blue1brown's framework manim at work
i agree, but there are other people that have animators do even more like haminations (he's a story time animator)
Someone's back is hurtt
we do or best.
"This math is weird because of math. We can't do enough math to solve the math - there's just too much math!"
You could start by calling it MATHS
my dumbass brain is quaking
Pretty much lol
Weapons of Math Instruction?
@@holdontoyourwig Unless he's British, why should he?
For those wondering, Alex K. is the narrator and voice behind the Quanta Magazine’s stunning video on the Riemann Hypothesis. This is like a collab of dreams!
I feel that the answer lies in a different representation of the natural numbers that kind of represent 'power of 2'-ness, and showing that that can never increase under the operations described.
"Pick a number" -"Uh seven?" "Seven? Good choise!" -"WHAT THE-"
BRO😂😂
Everyone chooses 7 cause 7 wins everytime lol
SAME
Lol same
I think it’s 3
Imagine being a Math Teacher and you gave an entire class an activity 1. Solve Collatz Conjecture 3x+1 (10 pts.)
The issue isn't solving it but proving it. :)
It's not a problem, it is a pattern. There is no solution. This is literally the formula for how all life grows, 124875 this sequence repeats infinitely, with alternating "branches" of 36363636 also repeating infinitely.
I once had a professor set the proof of the Boltzmann equation on a midterm. That proof exists but for a nonmathematician/nonphysicist (I was studying materials science) it was a beast. The equation is s = k * ln (m). Looks simple doesn't it? That was twenty years ago and I'm still traumatized. Mad props to mathematicians.
Smart in class: *Gets 10pts*
He/She would be barred from further teaching due to academic cruelty beyond comprehension.
I would place my bet on if there IS an exception, it's a new loop not a number that shoots off to infinity (in which case that number in itself would have to be proven to shoot off to infinity and not get into some other insane loop at numbers too high to comprehend)
I have noticed that the numbers that have the most difficulty in going down to 1 are those that precede an even number with the characteristic of repeatedly decaying into an even number many times in a row (i.e. those that get to 1 more easily). Example, the even number below decays several times repeatedly into an even number, easily arriving at the number 1 (as do all the numbers belonging to the group 2x2x2x2x2x2x2): 64 ---> 32, 16, 8, 4, 2, 1 Both the number 63 (the previous one) and 62 (the even number before 64) both have extreme difficulty going down to 1, so the numbers preceding 64 are in the opposite condition to that of 64 (which instead decays very easily to 1, without ever rising upwards). So, if I choose the number 2048 ( = 2x2x2x2x2x2x2x2x2x2), I assume, based on the above, that 2047 and 2046 decay to 1 very slowly (having many ups and downs).
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.
Good pickup!
Wow I did the poll a few seconds before scrolling to the video and this comment, I was wondering what the poll was for
Good catch. I like the social experiment that is in itself. That is such an arbitrary question that it should be close to 50/50. But it seems something is tilting us one way. Is it nature or society?
@@NandR I was also thinking the same. Maybe people who prefer the color blue also prefer even numbers, or people who prefer the color red also prefer odds? Just a thought
What about color blind people, there choices may be just a valid, pick any of the two, for maybe they are different shades of the same color??
The animation is everything here.
@DON'T stfu
@DON'T stfu bi-
You both just fell for his trap lmfao
@@WillCrewMusic i didnt even read the pfp the text is too small to see LMAO
@DON'T IM GONNA SAY THE N WORD
I 've been waiting to get struck by a lightening and receive the answers in my singed head. I will let yo u know. Great video!
If we try the polynomial 3x + 1 for negatives then we will be stuck in a loop of -7 or -1 everytime for any negative integer. Try this too!
Your "one of the world's greatest living mathematicians" joke totally killed me.
I was looking for this comment lol
Same here. I thought it was very clever.
* You're , btw I am better than you
@@cortnetisjustbetter not you’re but ok
I immediately knew this would be in the comments as well lol.
Looks like a good formula for generating Mountains in a virtual environment.
Ye
that's how they make roller coaster rides
Perlin noise: am I a joke to you?
not really, mountains aren't created by random processes.
If you use a decimal the number will go for ever as eg: 1.23 you would x3+1 =4.69 4.69x3+1 = 15.7 the decimal number will always be multiplied by 3 leavening you with a always odd decimal. If you start with an even decimal the decimal will keep getting divided by 2 until the decimal meets 1 then it’s will continue to rise. Adding a decimal is a way to bypass the number having to turn even every time you times the number by 3 and add 1. You are welcome for me solving it.
Schools need more material like this to inspire kids
I would say that 3n+1 is trying to reach the number to the sequence of 2^n and the division of 2 is used to say "You are coming close to join my family ,but try with someone else". what do you think about. maybe binary representations could make it easier to understand.
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.
cause you don’t have an exam and your future on it while watching this video, but at school, yes
@@ultraslanmc4619 That's a very good point! No stress to learn it 😂
Actually i liked it at school. But it annoys me at school
Same. Things are so much more interesting when you learn them on your own than when you learn them at school.
The yearn for understanding really seems to increase with age…
I laughed when he said "one of the greatest mathematicians" and showed his his grinning into the camera
?
@@jAYROCCS1x 12:36
@@John-el5sv i see. thought he meant the guy frm the beginning.
"The world's greatest mathematician: myself"
Humble-bragging or else it's a better way to subvert expectations before revealing truth! Terry Tao looks like someone who would appreciate the joke. 😅
All these numbers are beautiful, but nothing, but this one is weird, but possibily valuable.
Judging from the comments, the Collatz Conjecture could probably have more distinct proofs than the pythagorean theorem! And all of them from engineers, programmers, computer scientists, and amateur physicists with no formal mathematics background! Wonderful!
Not so distinct though. At least 80% of the "proofs" are "Over a given number of steps, the odds of the sequence going down is higher than the odds of it going up". Most of the rest are "It has to hit a power of two eventually".
@@jmodified oh yeah, and let’s not forget the third kind: just look at the last digit, it works for 1,2,3,4, … 8,9. And since every number has those numbers as last digits, it must work for all of them too!
@@AndresFirte Yes, I forgot that one. I think those three cover at least 99%.
The video: "This seems like a really easy problem which is why a lot of mathematicians are curious about it, since it's actually insanely hard" Laymen watching this video: "Pfft, this seems like a really easy problem" Almost like that was why it was interesting in the first place
him: "pick a number, any number." me: "eight.." him: "seven? good choice!"
5... 😭😭
Me: 0. Him: ok, if it's odd, × 3 + 1, if even, ÷ 2 Me: I think you just broke your calculator.
@@KratonWolf yeah. 0 really isn’t even or odd, so your just stuck
@@savathunthewitchqueen8299 and even if you do plug in zero to 3n+1, you go back to one.
Ikr... I picked 4...
I like the amount of people who didn't watch the video for even a moment, and are just here talking about how easy it is to solve 3x+1.
You take my words from my mind :))
I watched it up to 20:57, and had a couple of thoughts along the way. First off, I hit the loop quickly because my chosen number is 4. My thoughts were that this could be considered an exercise in looking at every possible angle of a situation, which both has practical applications, and seems likely to sharpen the analytical way of thinking -- or likely to be frustrating because there is no clear answer other than the loop, without finding an alternate path. A good brain exercise, no question. Second, while looking at the visual ways to consider this, since I'm an occasional artist, I thought mapping it would be a great way to create some drawings or paintings and either add to them, based on what I saw, or call them finished. Either way, it's great for stimulating the mind. And if anyone chose to read all this, it's also fun to think about.
12
The video is 20 minutes. LoL. Of course they not going to watch it.
It's 4
Old video but heres recontextulistion thats pretty neat. Dividing by 2 bit shifts binary numbers to the right. The 4,2,1 pattern is 100 010 001 Hence, a hamming weight (number of non 0 bits in binary number ) of 1 will lead to the 4,2 ,1 pattern, no matter how large the number is. This is the same as the any number of 2^n observation but bear with me. Multiplying by 3 in binary is the same as adding the binary value of itself but bit shifted left by 1(and hence you have this beautiful thing where the bit shift left is the odd process, the bit shift right is the even process). E.g 101 (5) multiplied by 3 is 0101 + 1010 1111 An odd multiple 3 added to by 1 will always either leave the hamming number the same (if the least significant run of ones is size 1 : e.g 010001 + 1 = 010010 Or Will reduce the hamming weight by n-1 where n is the size of rhe least significsnt run of 1s. E.g 011(hamming weight of 2) + 1 / = 100 (hamming weight of 1, hence 2-1 reduction has occured). New runs of 1's in a 3 multiplication will be isolated with size 1 max. Dividing an odd number by 2 will move the least significant run of ones to the least significant bit. This will trigger a termination eventually (with delays only guranteeing a larger reduction in hamming weight) ( not proven) any individual 1s end up in a run of ones before the +1 termination step. Hence, whilst hamming weight may increase temporarily, the overall pattern caused by the +1 termination and the limitation of of new 1 bits tending towards runs of ones, the overall hamming weight will reduce during iteration of the colletz conjecture processes. Hence, the hamming weight tends to 1... guranteeing the 4,2,1 loop. Its not quite a proof. But christ i feel like its close 😅
I like this one. My version was to argue that despite how large the number gets, application of the process results in the number going back to previously checked numbers and everything goes down to 4-2-1, so the conjecture must be true for any natural number
how long did this take
@@kinetik9197 how u mean
★ I love this math problem, it's like my life--difficult and unsolvable yet easy to live with if i purposely close my eyes. ❤️
"Pick a number" "- Seven?" "Seven? Good choise!" "- Looks Back Carefully"
*choice
@@LaniPlayzRoblox *choise
@@keenjoaquin847 *chuse
@@LaniPlayzRoblox choes
@@mrcrunch4635 *cheese
".....that not even the world's best mathematicians have been able to solve. " Me : "Alright, tell what it is, maybe i can solve it. "
I think it's just 10
@@captaincool9636 42. The answer is 42.
@@MP-ut6eb no not at all
@@MP-ut6eb no, you don't know the answer if the best mathematicians don't know it. You're not that guy buddy
@@gbsantana9679 its meme my friend. Its a meme.
Pascal triangle is an example of directed tree It is used to show how much fluctuation carbon shows in a spectrum in chemistry
Or rather it predicts how many fluctuations carbon can shows in its bonding with hydrogen
Nice work Soviets. You got me.
Hitler be like :
@@HottestBrownMan I was watching this video without signing in, but signed in just to like your comment buddy.
The Cold War won't truly be won until the Collatz Conjecture is resolved.
@@akshatvikramsingh8293 thanks mate.
Ngl i hate your facebook page lol
The transition at 12.33 "World's greatest living mathematician ..." was so hilarious. Well played sir.. well played.
12:33
Yeah It Got Me Laughing
@@karlmarx828 i love you 😩😩 My favourite capitalist
He said “ONE OF THE world’s greatest living Mathematician".
It seriously doesn't take effort to write 12:33 But seeing your pfp it makes sense since its shikamaru
I was tought more maths from this video than any of my classes
These people have too much time on their hands... And the power to them !
Shows a picture of himself. “One of the world’s greatest mathematicians…Terry Tao” Then includes Terry. Lol
Yeah that was so good
😂, That's was funny , He's also good scientist tooo
12:33
@@dreamer097 thanks
Haha. I came to look for this. Haha.
"The simplest math problem no one can solve" My math professor: oh, looks perfect for the exam!
🤗 congrats on 300 likes
It can be solved. Edit: So apparently this is an algorithm, so in this case: f(x)={1;2;3;4} or f(x)=[4;1] (those are random numbers, not a solution). This is the correct way to solve problem like this, which is what you learn in algebra classes There is always a solution in math, except you need to write it differently than x=7. Also, if there isn't any possible solution (like delta of an angle being less than zero), then you simply write "no answer" and close the case
@@Phantom-el6oe ?
@@Phantom-el6oe He's not doing an exam, are u dumb?
@@Phantom-el6oe but u cant actually do that
This is actually an interesting concept, the moment the number reaches a number that is a power of 2,it falls all the way down to the 4-2-1 loop.
This is why I believe this conjecture is true for all numbers. The probability is almost certain for a chain to hit a power of two somewhere. But I mean I haven't dealt with numbers quite that high yeeeeet. Maybe someone in the future will prove my thinking wrong
@@kittycat0143 there isn’t meant to be a number that doesn’t hit a power of 2 with this cycle, other than infinity, as we don’t know if it’s odd or even or even a number, but infinity is the only number that can follow this concept WITHOUT entering the 4,2,1 cycle
@@orsonewe3893 the problem is infinity will never exist as a definite number as we can just keep adding 0s at the end and making bigger numbers... therefore this conjecture will never be proven false but theres no possible way to test all natural numbers
@@kittycat0143the problem is really if the number itself hit a loop of itself. 5x+1 for example doesn’t always come back to 1, 13 is already a counter example that it will come back to itself
I think the reason why this works is if we start with 1 and then apply the inverses of both rules and only leaving the integers, eventually, we can get all positive integers one way or another
"Worlds Greatest living Mathematician" I see what you did there.
*there
@@chriswebster24 thanks.. Noted!
Ahaha! That sequence took me a second. Nice one! (12:33)
Could I get some clarification?
@Chinmaye Last name Well now it's obvious. Thanks
Two things I learned from this video: 1. Mathematicians smile gratuitously in front of cameras. 2. Boring subjects become interesting when they're accompanied by animated graphs.
Two to the 68 power?? lots of time on their hands
@@DGill48 Or maybe lots of mathematicians and machines in the world's hands?
Wait til you find out .9 and 1 are equal
3. wHy DoNt PeOpLe GiVe Me A cHaNcE oN yOuTuBe????!!???!!
are u saying math is boring
i have watched this *37* times. its too good.
*37*
"It's a simple spell, but quite unbreakable" ~ Dr. Strange
Perfect.
Yes.
Fitting
And this comment is just like last piece of jigsaw puzzle
I like your style
"Pick a number, any number" Me: 42 "Picks 7 anyways"
Yes I always do thst
Ah, the answer to life and everything.
Ah yes, let me just adjust my super microphone so I can hear you through videos
Yup
I picked nine, not seven >:(
Since I see no = sign, I'd assume the answer would be equal to - 1/3 but that equals zero. A trick question like no want wants to see that 2 to the x = 9 as x equal pi because round off a number does not give a true statement if you are only using a two decimal situation, but some are afraid to expand beyond for digits because it is to much mind set usage?
1,2,4,8,16,32,64,128,256 So long as 3x+1 solves for a multiple of doubling, which will also go on to infinity, then so too must the equation
This problem makes all my life problems seem like child's play. Kinda like having existential dread when you realize how large the universe is.
me to :p
1k square miles ?
@@AnAnonymousMan three, take it or leave it
It makes me feel better when I realise that. Maybe you just have way too much undeserved ego.
@@maxwellsequation4887 When did I ever say it didn't do the same for me? I feel better too.
Me: “tries to do it in negative” “Gets in a loop anyways”
@UC-cuXojkaoATvG21be0s25w 0 x 3 + 1 = 1 And 1 x 3 + 1 = 4 then divide 4 by 2 it’s 2 then divide it again it’s 1 And yeah we’re stuck no matter how you try it
True
This is really dumb 3x+1=3 because u plus the 0 with the 1 = 3x1 I hate math and dont know anything about it but i still clicked on this vid
@@Mango-rl2yg huh?
@@Mango-rl2yg if you meant 3x+1 where x=0, the result would, indeed be zero. Anything times zero is zero, meaning 3•0=0 From there, you add the one, giving you 1 as a result. My apologies if I misunderstood what you were trying to say! ❤️❤️❤️
It's a quirk of base-10. Do it in base-8, base-3, base-40... and you get different, but similar results. It is the (division) and (+1), which is the secret to the quirk... Because (1/3) * (3) != 1... Go back to your "remainders", where they loop down to 1, with powers and +1. 4 + 2 + 1 = 7, which goes Even, Even, Odd, which cancels out the formula of halves and +1 by 3 units. Also, 4=3+1, and 2+1=3, (It's starting to look like the math formula itself.)
It's the same in any number base or representation. You can use Roman Numerals if you want.
Base is irrelevant here. It is just the way we represent numbers in writing, but the actual relations between the numbers does not depend on the base. Think of the base as a language (e.g. German). If we translate a statement in German to, say, French, we are changing the representation we use to write the statement (the language), but the semantic meaning stays the same. Similarly, if we have some arithmetic expression written in base 10, then changing to base 2 only changes the written representation of the numbers and not the meaning of the expression.
It’s a nonlinear dynamical system with a point attractor. It perhaps even describes the heat death of the universe. Conway’s game of life is a simple example.
"use Benford's law for tax evasion" Got it
next they'll invent another law for you to follow
Ffs even when you break the law you gotta follow other laws, huh? ):
@Soul Seeker appropriate name for someone who works at the IRS or on some other auditing team lmao
@@jackrobinson9403 I guess "only break one law at a time" spans between the laws of our society & the laws of mathematics lmao don't try to divide by 0 while smoking weed kids
Most tax verification algorithms are having the same function in the root file patch 🤞
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
Hey what about she it doesn’t need to be a he
@@owenlee913 i really hope you’re being sarcastic
Then he'll solve it and say "how do you like them apples"
@@owenlee913 Well statistically, that kind of very exciting thinking is found in Savants and autism is more common in men. One could say that almost all savants are he.
@@owenlee913 Or a non-binary or a trans kid or someone identifying as a AH-64 Apache helicopter. But probably an AI
12:35 Wow, the world's greatest living mathematician
"One of the world's greatest mathematicians... " *Proceeds to show himself*
I laughed hard that.
Man has some humour
This is the comment I came looking for lol
What's the timestamp for this? I just finished watching but I must not have caught that lol
@@SlimShady-gs8pl 12:36
I love on how people immediately pointed their fingers to the Soviets for an unsolvable problem
I go to Confucius
@@toolaazy And Confucius says
@@YOUNOTSMART I am confusion, this is kansas, why this arkansoo, america eggsplain
@@anmoldeepsingh9281 😭😂🤣😭😂🤣
@@YOUNOTSMART no more numbers jumping on the graph
When it comes to the 3x+1 conjecture, what are we solving for? Sure we could try and prove it true or false but, what are we solving the conjecture for? What are we looking for in the equation?