Hailstones (The Collatz Conjecture)

Hey guys, I'm back from camp with a TON of new ideas. I got a bunch of math books as a graduation present (which would be a really mean present if I wasn't myself), and I got some new topics to discuss.

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In the science of the hydrosphere, we all know that water molecules present in air all eventually return to the Earth in the form of precipitation. One of the forms thereof is hail, which forms in heavy storms when liquid rain freezes onto snow pellets and begins to form solid lumps.

Let's say that layers of the atmosphere are marked by numbers, starting with one as the ground. The goal of a hailstone is, obviously to reach one.

The process that determines the path of a hailstone is this:
1) If a hailstone reaches level n, and n is odd, then the hailstone is caught by a gust of wind and shoots up to level 3n+1.
2) If n is even, then there is no wind and the hailstone falls to n/2.
3) This process is continued upon itself until n reaches 1.
For any positive integer from which the hailstone (n) begins, does n reach 1?

This can also be stated as:

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FACTS:

1. The sequence always ends [...] 16, 8, 4, 2, 1.

2. The conjecture has been proved up to about 5476377146882523136.

3. The sequence always returns to one if n is a power of 2.

4. The recurrence is undecidable (it cannot be solved by an algorithm).

5. Once the hailstone reaches 16, the hailstone is set into an infinite loop around (4,2,1).

6. Most diagrams are created in reverse.

Up to 20 iterations. Ends with one in the center.

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Hey guys! It's good to be back. From now on, I'm going to try to have one post every Monday. I have so much to talk about, but I don't want to waste all of my ideas now! We'll see what happens later, I guess.

Stay coolio,

John

NEXT WEEK: More on Game Theory