# 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

When you find yourself back at 1 and change the rules and divide 1 by 2 and divide every subsequent number by 2 because 0 is a even number in the spectrum it brings you to the number 7.450581e-9 according to my calculations. Which could possibly mean nothing..... Prove me wrong

The number 12 ends odd

ı hate you !!!

MANY PEOPLE'S

SHARES BUY OUT AT COST.

I made it to 8:35 before I started to get confused

Of all of your videos, this one blows my mind the most. Whenever I think I am smart, I remember this video and the people in it, and I feel stupid again. Thanks for that. :) Great video!

I watched for ten minutes, still never heard anyone say what problem they were trying to solve. I get the arbitrary rules of multiply if odd, divide if even, but what is the actual problem?

4....................................................................................................................................

“Pick a number” Me: *Picks 0* 😎

No wonder why stock and crypto market will crash at the end :)

The answers 35

3x+1=3x+1 ez

Zufall ist es nicht haha... Chaos steht über Raum und Zeit 3,6,9 als Flussdiagramm? schauts eher aus wie ein Kreis, ist es aber nicht^^

Looking for a counter example, PI comes to my mind. According to the definition only integers are even or odd. I guess if a number is odd when there is a remainder after being divided by 2, then every real number not part of integers would be odd. With 3 x + 1 algorithm, some of these number would grow until being eventually big enough to be integers themselves... For those who are infinite fractions, it means they would grow until infinity, and thus, like PI, would be on different trees. And the conjecture would be false. Interesting topic!

That's not surprizing nor interesting, as all irrational numbers, which stastistically speaking are 100% of real numbers, would trivially diverge to infinity under those rules. The interesting part is when limiting your view to integers, and asking yourself if they all follow the trend of falling down to 1, or if exceptions exist.

"this stupid little problem, how come we cant solve it", bitch i dont even get the problem, my brain is smoking i dont think its a stupid little problem or simple

Finally a way to stall a lesson.

Math does not have a proof for all true statements. I does not have a proof for all false statements. I would expect - numbers to be different after all -x- =+ but a +x+ = + the 2 sides of the number line are not equal in operations. What about complex numbers?

If you end up on a square number you always end up on 1

I doubt no one noticed this pattern but when it's falling into the 4-2-1 loop it's going down the long thread of binary. If any starting number eventually hits any number that can be created by 2^n it will follow along this giant chain. I find that very fascinating and would even change the visual aspect as a single pillar in the middle with those changes funneling into it the binary numbers. I then wonder if this has to do with the fact that 3x+1 is the simplest multiplication and addition pattern that can give you odd or even numbers of different origins, and if we change the division of 2 into 3 and the 3x+1 into a slightly differentbut similar formula, will the new pillar in the middle turn into a 3^n loop that always leads to one instead of a binary 2^n loop. Finding a pattern there would be interesting too

Infinite

In that we are dealing with odd and even numbers. It's final state will be as 1,2 respectively saying the numbering sequence is the result of only odd and even (1,2) therefore 4,2,1. For negative you will have to take into account of a double negative becoming a positive with odd and even numbers.

Maybe this equation shows that no matter how hard you try to overcome nature and decipher everything around you, but something's always out of human hands.

The Turing machine part broke me...

alright you've earned my sub

But I didn’t want to pick 7

Well I already suck at math

13:40 it's not me, it's Patricia

The video: 3x + 1 The simplest math problem no one can solve Me: *4x*

Basic problem is abiding by 1+1=2, the root of mathematics. The universe doesn't use this and neither does your mind. Maths is a guide to uncovering the face of reality - bit like dot-matrix porn, and it requires contempt for it's limitations.

when I saw the problem as "3x + 1" my mind went to "solve x" but seeing it in the video as "3N + 1" instead makes more sense to me I actually set 3x + 1 to = 0 and solved that I might not be the best at math I still love it so shush

Why do You divide by 2? I don't get it. Just keep applying the fórmula

Dividing by to when even is part of the formula. I'm sure you're smart enough to understand why it's not interesting if you don't do that.

I came to find this video again because I thought I stumbled on this unique sequence, but Instead I found something that I think works similar... in that you divide any number by 2 and if it has a remainder square it, then divide by 2, and then repeat

Its funny how a educational video can make you feel stupider

What’s the problem that they are trying to figure out...maybe you left that part out?!?

It’s the difference between reality and the illusion of reality

369 is the answer !

10958¹⁰⁹⁵⁸

I can see the foreseeable future and someone on twitter will tweet "how about you 3x+1 some bitches bro".

imagine the universe growth is following this theory.

its just how the universe builds planets and height maps.

I wonder if they tried this with 5x +1

If the last number (unit) is 1-9 the outcome will always be the same no matter the other numbers it will obviously be larger but it will only matter about the unit, the size of the number would be irrelevant

I don't understand the problem?

That's my internet speed graph dude

My dumbass was sitting like "the answer is 4x"

can you use a computer code that will do it automatically?

Why is this a problem ?

1 = 3x+1 JA DIT IS WAAR

if only numbers were words...

Obviously adding something (1) that cant be divided by 2 will end at exactly this numer (1) lol

can we use decimal

Isn't this proof of infinity? The answer to the equation is infinity.

What if we take a negative number?

To me, the Collatz Conjecture is a beautiful analogy that we are all random numbers in a Supreme Sequence. We will go through seasons of x3+1 where life flourishes and also /2 where life cuts us to the core... we will eventually peak at a point and then fall downwards until reaching One. The One. The Creator. This is when we enter "The Loop" also knows as Eternal Life. 2,000 years ago God became Man and was brutally murdered on the Cross for the rebellion of mankind in order for you and I to enter The Loop of Collatz' Conjecture. Repent and Trust in Jesus. Amen.

when you get to heaven can you ask big G what the number is

Just a comment from someone not in a math major... why are we so fixated with multiplying it by 3? Why not multiply it by 9 or 2 or anything else? Many mathematicians may have wondered before what if they changed 3, which is an odd number into 2, an even number. p.s. Don't hate me for this. Again, I'm someone not in a math major.

For some other multipliers the result is easily provable.

great video! btw, which model/brand of the phone is he using at 6:41, anyone?

Whatever goes up must come down.

If we let skynet solve the problem then we have solved another.

So what about "3X+1" where X=PI?

Binfords law can detect irregularities in ballots, except in the 2020 election that is………..

0

Awesome

Where's the "="?

There's nothing random about the stock market. It is highly manipulated by central banks, governments, corporations and financial whales.

I'm sorry I can't get over this... how the hell do most numbers start with 1? This is breaking my brain and I feel like it shouldn't work, because there's always just another number and it's not like the number of potential options is changing. There's only 9 options so shouldn't it be 1/9th for each option with a slight exception in zero?

They don't, except in certain situations. For city sizes in the US, for example, there are many small towns, fewer mid-sized cities, and very few large cities. When you have a distribution like that where the higher the number is the less frequently it occurs, then you can get a Benford's Law distribution or something close to it. This is what you would expect because there are more 10,000-19,999 population cities than 20,000-29,999, and more 100,000-199,999 than 200,000-299,999. Many things have distributions like this. City/town populations are a very close fit. Anything closely related to that, like city land area or number of voters per city will also be a good fit. Others are company employee numbers, personal income and wealth, and animal species sizes. There are many small companies and few huge ones. There are many poor people and few ultra rich. There are many small animal species and few large ones. Of course none of those is a perfect fit because there is some clustering - the size distributions are not entirely smooth. Other things have other distributions. When rolling two dice, you get a lot of 7s and not many 2s and 12s. Male heights in inches almost all start with 6 or 7.

If from 1 every other number is odd, and from 2 every other is even, 0 is nether even or odd and both even and odd. 0÷2=0 is infinite, 0×3+1=1 falls into 4,2,1 cycle

10

X = 1