Finite State Machines are awesome! Resume from reboots and automation is possible even on older Powershell versions without Workflow support. Setting state and transitions resuming where it left off is huge for getting PC's set up. Unfortunately can reimage due to no vl reimaging rights but scripting is totally possible...and with it creating it's own scheduled task to run on startup it does what it needs, then reboots resumes and deletes the task when done.

It should be possible to prove the collatz conjecture by mapping the unit digit transitions between numbers, namely into a finite state machine. From there we could use predicates and quanitifiers to prove, by process of exclusion, that for any given combination of 10s digit and 1s digit, no number can transition to anything but whats specified in the state machine assuming that number equals x in x3+1 or x/2

Ipso facto, a series of equations proving by process of elimination, that state machines transitions are the only allowable ones, would prove the collatz conjecture by proving the fsm is a valid representation for any given integer N.

I'm actually working on it now but I don't know enough about modular arithmetic and predicate logic to write a proof. I just have the state diagrams on some dot matrix paper at the moment.

If anyone wants to beat me to it, feel free.

So for example any number ending in 13, will, after x3+1, end in 40.

Any number ending in 40 will end in 20. Any number ending in 20 will end in 10, which will end in 5 as the unit digit.

It's easier to prove in the single digit case, and the finite state machine for that is already written, at least on paper.

I'll post pictures when I get a chance.