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'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 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?
"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.
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.
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.
He learned binary in a weekend and we had a fun few days hacking math :)
Also related: http://xkcd.com/710/
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.
Then withdrew it, and is currently busy trying to fix the problem.