The most difficult - yet so easy to understand - problem in the world!

Take any natural number $n$ . If $n$ is even, divide it by $2$ to get $n / 2$ . If $n$ is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$ . Repeat the process indefinitely. The problem is to prove that no matter what number you start with, you will always eventually reach $1$ .

Don't be fooled by the simplicity of this problem, great mathematicians like Erdős had great trouble proving it. The latter even said: "Mathematics may not be ready for such problems."

#MostDifficultProblem

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

When do you predict that the Collatz conjecture will be solved? @Sharky Kesa @Daniel Liu @Daniel Chiu @Trevor B.

Elliott Macneil - 7 years ago

Possibly in the next decade given the number of young mathematicians rising up and are being seen on Brilliant.

Sharky Kesa - 7 years ago

The Collatz Conjecture? or The Ulam's Conjecture?

Mu'amar Musa Nurwigantara - 7 years, 1 month ago

It's commonly named after Collatz name but it is also named Ulam's conjecture because he used to talk about it in the lectures he gave.

حكيم الفيلسوف الضائع - 7 years, 1 month ago

I did not understand what were you saying about odd numbers. Even one is true but can you explain me the odd process ?

Kushagra Sahni - 7 years, 1 month ago

@Kushagra Sahni – Start with any natural number. If the number is even, divide it by two. If it is odd, triple it and add one. Repeat the process with the new number so formed. Ultimately the number 1 is reached.

Shourya Pandey - 7 years, 1 month ago

@Shourya Pandey – Even one is true. But what about odd ? Let the no. be 5. if we triple it and add one, we will get 16 and then 49 then 148 ......... Which I don't think would approach 1. Or am I not getting the process. Can u explain the process with examples.

Kushagra Sahni - 7 years ago

@Kushagra Sahni – When you got 16, you'll have to use the operation we restricted to even numbers, that is: $n/2$ . And so you will get 8. Repeat it again you'll have 4, then 2, then 1!

حكيم الفيلسوف الضائع - 7 years ago

@حكيم الفيلسوف الضائع – OK. Now I got it very well. Thanks

Kushagra Sahni - 7 years ago

@Kushagra Sahni – You're welcome!

حكيم الفيلسوف الضائع - 7 years ago

@Kushagra Sahni – You have done the operation wrong. Lets call the operation C(x) where x is your starting number.

$C(5) = 16, C(16) = 8, C(8) = 4, C(4) = 2, C(2) = 1$ . And thus the operation ends.

Sharky Kesa - 7 years ago

I also thought this ....one day....Thanks for posting it

Archiet Dev - 7 years ago

couldn't we just consider it true until proven wrong? =)

Gmasha Hussain - 7 years ago

That wouldn't be helpful in any way.

حكيم الفيلسوف الضائع - 7 years ago

Yep, just taking the lazy way out.

Gmasha Hussain - 7 years ago

Wouldn't it be correct if I say it will also reach every time reach 2 .

Utkarsh Dwivedi - 7 years ago

No. If you start on 2 itself, then you will have to carry out the operation n/2 anyway. You have to COMPLETE the sequence and hence end with 1.

Yuxuan Seah - 7 years ago

But by starting with 2 , 2 comes in our sequence. How will you write the sequence - only 1 or 2 , 1. Consider the following example - We take n = 5 .Then how would you write the sequence like this - 5 , 16 , 8 , 4 , 2 , 1 or simply ignoring 5, by - 16, 8 , 4 , 2 , 1. Of course first one 's correct so we could never ever find a sequence except n = 1 in which two does not come.

Utkarsh Dwivedi - 7 years ago

Le problème de Syracuse ;)

Antony Diaz - 7 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$