Regarding the busy beaver function, what's a simple rule to prove that the machine runs forever in one direction? I believe there's something about the machine not backtracking far enough before it repeats a state?
As an example, prove that this machine doesn't halt, where $A$ is the initial state, and the machine starts with an all-$0$ tape:
A B C
------------
0 | 1RB 1LA 1LB
1 | 1RA 0RC H
Obviously I'm looking for as general a rule as possible.
The technique I was thinking of is explained here:
http://www.cogsci.rpi.edu/~heuveb/research/BB/status/nonhalters/simpleloop.html