Power of variants of Turing machines
I was under the impression that if a Turing machine can perform the same actions and satifies unrestricted access to unlimited memory they're all pretty much equivalent. 


Hint 1: So your first step should be to determine whether a and b have, or can obtain, unrestricted access to unlimited memory. 

Hint 2: If yes, then try to emulate the missing pieces of your preferred Turing machine model.

Hint 3: If no, then try to show that the halting problem (with empty tape as input) is decidable.