How to prove that a problem is undecidable by using the Halting problem?
When you reduce problem A to problem B, you show that B is at least as hard as A. (It's sort of like a proof by contradiction: if B were any easier, then A would become just as easy.) And undecidability is about as hard as you can get, in computer science.

So what you want to do here is:


Assume you have an oracle for the "fifth state problem"
Show that you can solve the Halting Problem with this oracle


It doesn't matter how long it takes or how many times you call the oracle, for this problem (since all you care about is decidability)

The Halting Problem is undecidable
Therefore, the "fifth state problem" is also undecidable


Here are a few hints to get you started:


You can reorder the states in a Turing machine without really changing anything important
If a TM has less than five states, you can add new "dummy" states that are never used, without really changing anything important
A TM never needs more than one halting state
The question "does this TM halt on empty input?" is a variant of the Halting Problem that is also undecidable