Alternate (but similar) answer again based on determining who is unreliable in the first question
This relies on the ability of the truth teller to never lie, and the liar to never tell the truth. It assumes that if they are unsure they may not answer as the answer they give may break these absolute rules
If I asked "which door is the good door" to the other two guards would their answers be the same?
- If asked to the truth teller he cannot answer as he doesn't know, and therefore may lie if he answered.
- If asked to the liar he cannot answer as he doesn't know, and therefore may tell the - truth if he answered.
- If asked to the unreliable he will answer definitively either way
Now, we can identify a "reliable" guard. If they have not answered we know they are reliable, and if they have answered we know the other 2 are "reliable".
Note that, unlike in Florian's answer in the "Two Guards" version, the problem that we can't know whether the guard can't answer or is still picking the answer, is irrelevant, because once we start talking to the guard while it's picking the guard will tell us to wait.
We can now ask an identified "reliable" guard
On average if repeatedly asked "Is (identify a door here) the good door?" would the other two guards be more likely to say yes.
if the identified door is the good door:
- The truth teller he will say "No"
- The liar he will say "No"
if the identified door is the bad door:
- The truth teller he will say "Yes"
- The liar he will say "Yes"
So if the answer is "No" we go through the door we identified, otherwise, if the answer is "Yes" we go through the other door.