## 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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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

Datum objavljivanja: Prije mjesec

0:21 It's not that "Mathematics is not yet ripe enough for such questions"...it's the fact that we can't measure or quantify something by lacking the cognitive capacity of doing so. On a case basis it might also be a useless and/or stupid question...asked from the point of view and/or understanding of an individual that is unable to asses its own limitations. Humans have this tendency to think they know more they actually do...and we see that throughout human history through our predictable actions. Understanding we cannot understand some things...it's a monumental achievement. I simply know that I don't know...and I'm okay with that. Edit: 1 may most probably be the universal balance just like everything around a nucleus core will find itself at position 1(one as in the prime order(measured numerically by humans) of universal balance). As an example...let's absurdly assume you could just poor water on earth enough to cover everything. The surface of the water will be perfectly balanced around the gravitational force coming from the center of the planet...and so everything will balance to the universal order of 1. Through force/energy you can change that order...but no matter what it will always come back to 1.

watching this video at 11PM and literally feeling my brain melting... love it.

answers 42

Pff easy, he said there were rules, no restrictions, just add 103 everytime you get to one

How come the number range doesn't start at 0. That would solve the problem, wouldn't it, as 0 would be an even number?

Well, 0 is a trivial loop. But that doesn't say anything about other numbers, though.

-7 - the cycle of the number -7

Hi! Why not use negative numbers?

I found the solution. I´ll publish in an academic paper soon. Working on it.

If you notice, there is a pattern where every turn, or equation change in the sequence is signaled by a prime number. It doesn't just simply reduce from an even number to a simple odd number. Each odd number encountered in any path is also a prime number. The "chain" will continue to rise in value until a prime number lower than the starting value is encountered, and then it will reduce down to 1, the lowest prime number. That was my observation doing about 20-30 samples, if anyone found anything different, I'm open to hear it, I'm not saying this is concrete, just what I have observed. The looping effect stems from the fact that 1 and 2 are both prime numbers. Since 2 is both prime and even, it creates a barrier from continuing the equation past 4, since the only multiples of 4 are, 1, 2, and 4. Taking the smallest prime number (1) and applying it to the "growing" equation of 3x+1, then the result is 4. For instance, you will never encounter the number 27, unless you start with this number. Why? there is no integer value that can satisfy the equation 3x+1 = 27. There are also a multitude of even numbers that will never be encountered, unless that number is used as a starting value. What about 54? Since 54/2 is 27? There is also no integer value that satisfies 3x+1 = 54. I haven't done enough tests to prove anything, but I observed that any even number that reduces to an odd number that is not prime is also not present. Even though this appears to be a linear equation, this is very misleading. In a linear equation, we assume any value along that line can be a solution that satisfies X. Because we can only use integers, and not all real numbers, this limits the number of possible solutions to any given situation. So you are starting the problem with the false assumption that 3x+1 is a linear equation (which it normally is), when you are limiting the values of x to integers. When you begin a problem with false assumptions, you can't possibly hope to have a real answer. Point? You can say with pretty strong confidence that any multiple of any number that satisfies the equation 3x+1= 0 is not defined by the problem. Therefore, the "solution" to this problem does not lie in the range of possible values for X. So the problem itself becomes undefined. Take 0 for example. 3(0)+1 returns 1. The smallest prime number. If we then use the rules to apply this 3(1)+1, we again get 4. If we assume x = 0 for each equation, then the first returns 1, and the second returns 0. We would assume then that the answer would lie somewhere between 1 and 0, and since we can only have integer values of x, then it becomes undefined.

it makes sense to me, its basically like a tree. where the roots eventually invert to make a tap root so to speak. while the positive numbers branch out like a tree. also like a tree the tap root goes deep and slims down eventually. the divergence happens at zero.

One two ka four Four two ka one My name is Vikram

3×+1=10

Simple

This will go to infine , and you can it check :) ( try chenge first and see what hapen if give result to 2 :) def col(n): sp = [n] if n < 1: return [] while n > 1: if n % 2 == -100: n = 3 // n - 1 else: n = n * 2 sp.append(n) print(n) while True: col(int(input())) check via this def col(n): sp = [n] if n < 1: return [] while n > 1: if n % 2 == 0: n = n // 2 else: n = 3 * n + 1 sp.append(n) for i in sp: print(n) while True: col(int(input()))

We're nothing but smart apes. I hope we go extinct soon.

Some problems were never meant to be solved. Stop trying to be God, humans. Just die.

I believe the issue here is that we are trying to using decimals. There is a solution described dividing by 16, by 8, by 2....that's hex, octal, and binary. What if these numbers were translated to binary and then worked through? Just a thought.

@Rob AdkinsThe representation method you use changes nothing about the properties of numbers. Also, the binary representation was explicitly used in the video 17:37, and while it's definitely interesting there's nothing that can easily be said about it that couldn't otherwise.

Interestingly, 4,2,1, are the first 3 weights of binary numbers. A decimal 4 written in binary is 100. Again, interesting that the whole number is 100, which is 3 variables plus 1 logic high..3x+1.

Not 3/2, it’s 1/6th

-1/6th

Click and clack the tappit brothers said a long time ago, the answer is always 1.

The Universe is like this math problem

I dont see what is signifigant about patterns set by arbitrary formulae

Why is it a +1 though? Where or what is that 1 and why not any other number? Or are all the other problems solved and this is the only 1? Lol.

You could also add -1 or +3 if you want, as long as it's an odd number. If it's an even number, including 0, then the answer is really simple. 3x+1 is arguably the simplest variation of the problem for which we have no definite answer.

Well, I don't know if this means anything but the premise here is strongly related to that of a pyramid... Think of how a pyramid is laid out - 1 stone is at the top, then you have 2 stones below that, which is 3 in total, then below that you have 4 stones, which is 7 stones in total then the layer below that has 8 stones, which is 15 total stones, then below that is 16 stones, which is 31 total stones and so on - but there is always that 1 stone on the very top which is why any number of stones you chose always ends up at 1.. Of course a pyramid is a geometric object so this math riddle obviously has ties to geometry, weather or not if that means something, I don't know? Below is a simpleton "math diagram" of what I'm trying to imply er say here, lol. 1 2 ----3^ 4 ----7^ 8 ----15 16 -----31 1 1+2=3 3+4=7 8+7=15 16+15=31 32+31=63 It's the total sum of stones needed to construct a perfect pyramid.. It doesn't matter what number of stones you choose because you will always end up at the top of the pyramid... It's basically the reverse engineering of a pyramid.. Look, I'm not a math nerd, lol... It's difficult trying to explain abstract ideas but I was just watching a video where pyramids where discussed, a light went off in my head and it made me think of this video and I associated the sum of stones it takes to build a perfect pyramid with this math riddle and it matches perfectly. Now I don't know if geometry has anything to do with this or if this was a coincidence - or even an answer - but I thought it was fascinating to say the least..

This guy likes saying arbitrarily.

An awesome equation which will be solved by future guys soon or an AI 😉

imagine how you would solve this if the numbers were negitive - hmmm how's that for an math problem :D

i wonder would you get the same result or diffrent outcome :D ???

What happens if you do this with Grahams number? Wouldn't that go to infinity?

4

How is that sequence, if it takes random quantity of operations of dividing the number by 2 to bring it to odd number?

A even number divided by 2 can be an odd number, 6/2 = 3

It’s just a function that tapers down into a pool. That pool being a cycle that math itself created and perceived by humans of this age are not capable of breaking down and seeing further into, once math becomes evolved enough to break the cycle down, we as humans will discover that the exact same thing will take place, it is only in comparison to another intellectual age that we have evolved, in its own system function perceiving from within from a mind of its own era there will seem to be no answer, but on the brink of evolution, inspiration and awe fuels change and new perception on the very material existence of consciousness itself.

Try Pi

integer

what about zero? 0/2 = 0

If you look at the graph upside-down it will go up instead of down. Problem solved.

the name of this channel is based off the Latin word which means truth

What is the most fascinating thing about this video is that I understand nothing of it, but I still find it super interesting and entertaining. You (and your team) are (an) amazing presenter(s)!

Jeremy Bearimy

So this is calculus

3x+1=life

Wait, what’s the problem?

Really amazing. It's like organic math😅

Your fuel for science is so contagious, Derek! Absolutely love your work.. one of a kind creator. Hands down!!

answer is 21

Numbers are interesting on their own. But why are humans obsess with them, the numbers?

Somewhat understanding these videos makes me feel like a mathematical genius when I'm getting loaded with my friends.

If ever I believe my work is done Then I'll start back at one !

Nah nah, my question is, why does 3x+1 even exist in the first place?

I ve always hated maths and now I hate it even more.

"communism is the riddle of history solved, and it knows itself to be this solution. "

I don't need it, but it's mildly interesting

beautiful

10:36 you're welcome people

lowest number must be zero

hello. I no every write and speak English. You give 1,57!!!

10:38 is he okay??

this stuff hurts my brain

But what is the problem?

MATHEMATICIANS: 3X +1 NOT POSSIBLE TO SOLVE ME: X=-1/3 PROBLEM SOLVED * DROPS PEN*

The simplest way to avoid spending your life trying to solve this is by only applying it to the numbers 1 through 10 or technically 0 through 10 but zero doesn't do anything. any number that comes before 1-10 doesn't matter eg: 438 only the last digit matters while the 430 is completely meaningless because being an odd or even number is defined by the last digit.

@TheRealLori What doesn't matter for what?

@Релёкс84 but that doesn't really matter

Sure, but the value of the other digits come into play as soon as you start dividing by 2, which starts happening no later than on the second step.

You can't prove it true because it's not math

Now I'm wondering. What if plants are more smarter than us humans? I feel like it's somehow connected with math. Okay blame the Soviets for this question.

Most interesting when I feel like a challenge I will definitely try this until then I'll work on my Hot Rod, customizing is a challenge on its own because you can't buy the parts you have to create them which I've done to every inch of my hot rod. Measure two or three times then cut shape and fit all other pieces interlock with each other makes it stronger than what you can buy on the showroom floor and looks much better because it's Customized! Better than original did the same with my Harley Davison 1950 vintage totally customize it road around on it, 22 years without opening up the cases just a valve job and replace the rings which I did in the kitchen with my ex-wife helping me worked out great! some people have an Eye for customizing most people don't majority don't. that's what I do along with electrical wiring with no mistakes! From San Diego California ( Customindesign.)

this seems similar to the halting problem (proven to be "undecidable" by Alan Turing) Edit: nevermind, you touched on that later lol. I need to remember to watch the whole video *before* I comment

my math teacher told me to watch this lmao-- thanks for the homework fjsfakllfsdflksjf

Even in trying in Base 8 it still gets to 4, 2, 1 cycle But, if you ignore actual numbers and think about even distribution of stuff without numbers then if follows logic path.... The sequence of "stuff" are measured in discrete units of Z starting with nothing (0) and increasing by Z, with a repeating sequence Where Z-Z is nothing and X is cumulative of all Z, and Y is half of X it looks like: Whole numbers only X is the start value in sequence, X can be even or odd Y is always even (proof) if X is even and X=2Y then Y is even Z is always odd and a set value of 1, so Z-Z = nothing and Z+Z=2 if X = Y+Y then X is even (Xeven) If X = Y+Y +/- Z then X is odd (Xodd) If Xeven then Y becomes new X in series. if Xodd then X+X+X+Z becomes new X in series. The sum of X+X+X+Z will always be Xeven, then the disposition of Xodd's will always become even. Even numbers can be expressed as the disposition of two equal lower value Xodd's Xeven = 2Y can be expressed as (3X+Z)+(3X+Z) or 6X+2Z. If Xeven numbers can be expressed as the disposition of lower Xodd, and the disposition of lower Xodds are even, and Xevens are always halved (divided by 2) then all Whole numbers 0->infinity will eventually reduce to the 8, 4, 2, 1, 2, 1, 2, 1, 2, 1 loop unless we introduce imaginary numbers into sequence. then again I am Kinda tired and my eyes are blurry and burning....

boycott Beijing 2022 Olympics ! Countries engaged in committing genocide don't deserve the honor of hosting an Olympic Games.

Mathematicians waste their time with this kind of nonsense, and people in Humanities are the ones everyone makes fun of.

ok but whats the point, wheres the real world application of multiplying odd numbers by 3 and adding 1 and dividing even numbers by 2

Learning, expanding knowledge. It might lead nowhere, but the next problem solved might be the gateway to humanities future of abundance. Point being to ignore the problems of they seem pointless means sticking with the status quo and as we know that is the ultimate in destructive conservative futility.

This is hilarious, it's like a riddle for math geeks in their language, lol.. It's like watching Rain Man trying to figure out "Who's on first?" lol

what if i pick the initial number to be Zero? Zero is a even number, so the rule says we divide it by two. So, 0 / 2 = 0. Hence we break the loop of odd number and the end result will always be zero.

I wonder how many students have quit mathematics in dispair over not being able to solve problems like this.

My smaet ass "its x4"

Gravity

In other words it's true

I created another loophole for Collatz Conjecture. Which when you start off from 21 you end up still on 21, I also apologize for creating another loophole.

isnt infinity a loop?

Great video but what happened here 13:55 ?

Would decimals make a difference?

I suppose someone has tried this with decimal numbers?

Since when are decimal numbers odd or even?

19:46 - *We have no right to have solutions to all of these other problems*

do it with 350

This equation sounds like a law of infinite. And unlike the laws that humankind made, we broke trillions of them in a matter of seconds under 3x+1. Would everyone be sinners then if this equation for some reason is a law of infinite justices that even our own justice system violated?

Isn’t it 10

So, why 7 and not other nine numbers? Why do I always see such thing when people are choosing a number from 1 to 10 and it's most of the time 7? I DON'T NEED SLEEP I NEED AN ANSWER!

I LOVE THIS PROGRAM!

Amazing. My father showed me this when I was 10. That was 59 years ago. Now I instantly see patterns - in life and in peoples behavior. I see it in the traffic, and in crowd movements. Thanks, Verisatium Team.

@DJ-murlock Animation ............GIGGITY...........

You're 69 now? Nice

Is it possible to proof any prime number comes back to the original 4 2 1 sequence? Because of that is true we'd only need to proof any starting number eventually comes to a prime

That doesn't sound any easier.

Sometimes I wish I wasn't such a dumbass, So I too can understand anything

Surely you can take comfort from the fact that when it comes to this notorious unsolved problem, nobody understands it. Here, even the world's greatest mathematicians are "dumbasses".

Another great video

The alleged sort arespectively heap because gemini psychophysically employ as a bad resolution. guttural H habitual, heavenly heavy hellish colt

My question is: why is that a “problem”? It is just a patrern

Humans can live happily ever after without 3x+1

it makes me so proud that most mathematicians mentioned from the same country were hungarians in this video. such a small country yet there are many smart people who pave the path going forward, even though i lowkey didnt understand anything in the vid :'D

Seems like someone made it up as a joke to have people spin their wheels. If you can continually divide an even number result by 2, and the setup has a propensity to produce an even number, of course it's going to get smaller. More simply, look at the last digit. Any odd number x3+1 is going to create an even number. Any even number x3+1 will make an even number in the next round anyways.

It's not that simple. You could very well have a sequence of increasing odd numbers separated by even numbers. Some starting numbers reach thousands of times their initial value that way, and there's no reason to think it's impossible to grow to infinity.

lol fractrum what a load of rubbish

the logo of the video

its 10

Does it only work with whole numbers tho

Just gonna throw out square root of 421 Billion

AH... My head, hurts

The number q is not in this loop 😎