A simple game with surprising mathematical properties

1. Pick a natural number.
2. If the number is odd, then multiply by 3 and add 1.
3. If the number is even, then divide by 2.
4. Repeat from 2 with the result

You get a series of number.

Apparently, all series ever tested end up with the number 1 (you can continue from there, of course).
If you begin a series with 1 you get a loop (get back to the number you started with). Apparently, this is the only loop ever discovered.
As a result all series have a repeating pattern from a certain point on: 4, 2, 1, 4, 2, 1.....

Mathematicians cannot seem to prove that no matter which number is used as the initial number you will end up with 1. They cannot find a counter example either.

Here are some series:

1. 1, 4, 2, 1,...
2. 2, 1, 4, 2, 1,...
3. 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...
4. 4, 2, 1, 4, 2, 1,...
5. 5, 16, 8, 4, 2, 1, 4, 2, 1,...
6. 6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...
7. 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...
8. 8, 4, 2, 1, 4, 2, 1,...
9. 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...
10. 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...

Some interesting questions

• If we try an infinitely large number of numbers, what will we get more, jumps up (that is a step for an odd number) or slides down (that is a step for an even number)?
• How does the series of the length of series up to the first number 1 look like? Does this series have some special or surprising property?