Collatz Conjecture ;; Or I'm Crazy ;; Oh, and Suggested Reading Part 1

I originaly put this on Facebook, before I was using Blogger. How disappointing to think I use to do such a thing, oh well. In conversation with my girlfriend today, I remembered this (anyone else think it's odd that things like this come up in our every day conversation?) and decided it would be much better on my Blog instead of.. pfft.. Facebook. I think if I was to rewrite this it would be much more lengthy with a few parts expanded but I decided to leave it as it is.

Also, before you get reading I would like to request that people actually read my 4100 word post. It can be found here. May I suggest reading it in sections? For this posts suggested reading, I would like to suggest you read the first four paragraphs. I would like to say the first five, almost six actually, but I think four is a reasonable amount. It looks to be a small read for me, the first four or five that is, however the seventh paragraph is where the story starts. That why I think up to six should be read. It's all up to you, that is all I have to say about it. If you want to read more, go for it. If you want to read less, I would love for you to do so also. If you wish to read it all, please drop a comment but only if you had any thoughts, questions, misunderstandings, or anything to add through the entire time. Even just one moment of, "Hmm, but I think..." must be noted in order for you to consider your reading worth while. If you don't your reading has been a waste possibly, and I know you wouldn't enjoy that. Have a nice day; or ever better than leaving you could take a glance at this first. Then have a nice day ^__^

I've spent a couple hours.. wait. Now that I look at the time a few hours on xkcd and was sent a link to this comic http://xkcd.com/710/ and of course wanted to understand what it was based on so Googled it and found a Wikipedia page, was quick to get into a discussion on how it worked and by the end the person who I was explaining it to had take a leave of absence from their computer but was kind enough to leave their MSN signed in so I may continue typing but now I have the desire to post it here.
http://en.wikipedia.org/wiki/Collatz_conjecture
In the Wikipedia article there is a mention of someone offering $50 for a solution to this.. misfortunate he's dead?
A long time ago when my mind was wandering during math class the pattern of even and odd numbers when you add, subtract, multiply and divide them occurred to me. I think I may of almost found a useful .

Anyways, the rules are if the number is even you divide by two. If the number is odd you multiply by three then add one. No matter what number you start with, you will (theoretical) always end up with a 1). This is limited whole numbers larger than 0 (1, 2, 3, 4, 5... etc)

This through this before reading.. I'm curious to see other people's ideas? Once I’m done I'm going to tag a handful of people who may have a slight chance of being interested if I remember.

First I will look over the second rule, regarding odd numbers. When you take an odd number and multiply it by three you will always run into an odd number. By adding one you will have an even number. This leads you to always having an even number after an odd number, every time you have an odd number the number increases, however now to look at the first rule.

Every time you have an even number you divide by two, this will reduce the number. It will not reduce the number more than the second rule has in the previous step (ex. 5 -> 16 - > 8) but it will have a 1 in 2 chance of being even again. This is because every second number that is even (2, 8, 12, 16.. etc) divided by two is an odd number and that leaves every first number to be even. This means the number will reduce again. After you do this there is another 1 in 2 chance that the number you now have will be even and then reduce.

To recap and probably simplify.. this means when ever you have an odd number the next number will be even. Then you have a 1 in 2 chance of the number being even. If the number is even it will go down again.

Back to the example with more numbers and some explanation.. 5 -> 16 -> 8 . This is an odd, then an even, then another even. If you continue it.. 8 -> 4 -> 2-> 1 After the 8 is another even number, then another, then a 1. The larger the number you start with, the higher the chance is that it will be odd more often until you run into one of the numbers that are 2^x (2, 4, 8, 16, 32, 64, 128, 246.. etc) The reason the chance of you running into one of these numbers is less when starting higher numbers is the higher you go the more spaced out these numbers are. 8 and 16 have a difference of 8 only, however the difference between 128 and 236 is 128, and if you continue this pattern a few times you can run into 32,768 and 65,536 having a difference of 32,768