Prove Undecidability: TM M enters each of its states on Input W?
Hint. We can assume that a Turing machine has exactly one halting state. If a TM visits all of its states, it certainly visits the halting state. Now figure out a way to modify Turing machines so that, if they do halt, they cycle through all their states just before doing so. This gives you a class of TMs that visit every state if and only if they halt.