Your first loop between Start and State 1 only accepts an

__odd__ number of a's. So I was wrong about that one. If you get an even number you're back to where you started from. Try aab, for instance. Follow that with aaab.

That odd number of a's must be followed by a single b, as you rightly point out.

====

OK and then there was loop.

After the b, you can reverse if you find an a instead of the delta that takes you to halt. You'll see this one had me a bit confused back there.

so we've got a tape sort of like this aaaaab

__a__
so T tells us to go

**L**. aaaaa

__b__a

so we land on that b again, which will be followed by the a on this tape (because we're overwriting b with b and a with a at this point). We're in a loop.

And after that a, what would there be on the Tape?

aaaaab

__a__jinglebellsjinglebellsjinglealltheway ... absolutely anything...

Well not absolutely because sigma = {a,b}

If you reach the a after the b, you're stuck forever. So loop a would be for any word

*starting with* ....

Maybe that's the way to do it? Rough work is a piece of the tape, which you execute the moves on, so as to see what's going on? Certainly I've only really gotten this by actually "drawing that tape" for myself. I was stuck before that, as you can easily see.

So I suppose I owe you some thanks for forcing me to find enlightenment.