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All numbers lead to one

jasondavies.com

95 pointszeratul15 comments

The Collatz Conjecture nearly killed me. Literally.

For three months I was consumed with the conjecture. I slept, ate, and breathed it. I was sure I was on to a successful line of attack, using some sort of inverse tree approach mixed with a density argument. One day, while working on the conjecture as usual, I heard cars honking but couldn't see what the fuss was. I turned around and realized I had just ran a red light at a busy intersection going 40 MPH.

I haven't thought about the problem since.

I can't believe that. Unless you write a blog post with the "trip" detailed. Pretty please ;-)

FYI, Jason Davies has been doing great work with Mike Bostock on the d3 visualization library, which is used in this example: http://mbostock.github.com/d3/

If you do this for long enough eventually all your friends will stop calling to see if you want to hang out ( http://xkcd.com/710/ )

I've read somewhere that no other mathematical problem in history has wasted so much time of such brilliant minds. I wonder if it's true ...

For any mathematics problem, we can never know whether studying it is wasted time.

For example, look at number theory. For centuries, it was without utility. Then, suddenly, it got practical applications, making all the time spent on it wasted time :-)

Similarly, the Collatz problem may seem useless enough, but what if, in a few millenia, someone applies it to physics or to sociology?

That's a valid point.

However, everyone whose work does not directly or indirectly lead to an applicable (or otherwise useful) result has been wasting their time.

Although I guess that's true for anything.

Destinations are not always more important than the journey that leads to them.

Your analogy is off. With number theory, even if no one had an external use for it, progress was (is) made, new theorems were proved, so time was not wasted in that regard. The problem with the Collatz conjecture is whether the effort spent on it is actually generating any insights into the problem.

But there is 'progress'. Looking at the Wikipedia page:

"The proof of the conjecture can indirectly be done by proving the following:

- no infinite divergent trajectory occurs

- no cycle occurs

thus all numbers have a trajectory down to 1.

In 1977, R. Steiner, and in 2000 and 2002, J. Simons and B. de Weger (based on Steiner's work), proved the nonexistence of certain types of cycles."

I am placing 'progress' in quotes because one cannot measure progress in maths. Before one has a proof, we cannot know whether existing approaches are true dead ends or whether they just need that one extra insight.

This is a simple visualization of http://en.wikipedia.org/wiki/Collatz_conjecture - and from the Wikipedia article :

Paul Erdős said, allegedly, about the Collatz conjecture: "Mathematics is not yet ripe for such problems." and also offered $500 for its solution.

Which is a very strong indication that this will be a tough nut to crack... Even if many people are given it as year 7 homework.

Cheers for this, I didn't know the mathematical basis for it, but our eldest child brought this "puzzle" home for year 7 homework a while ago. We spent the weekend experimenting with loads of bits of paper on the floor coming to the conclusion that once you land on a base 2 number, you have a path directly back to 1.

He learned binary in a weekend and we had a fun few days hacking math :)

Don't you mean on an even number? Or just a power of 2? Or are you talking about a base 10 number that's composed of all 1's and 0's? (All natural numbers are base 2 numbers...)

Also related: http://xkcd.com/710/

Sorry, I mean a power of 2 :)

It's very interesting to watch in binary, actually.

You first check if the LSB is 1. If not, you right shift it until it is. Once the LSB is 1, you add it to itself shifted left by one bit, then increment.

Watching the bits go by and shrink over time reminds me of cellular automata in a way.

Someone claimed to have solved this:

http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2...

Then withdrew it, and is currently busy trying to fix the problem.

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