That's an interesting insight, and it may be useful as part of the solution, but it doesn't solve the puzzle. The question is how to determine the identity of the three gods with three questions.
Personally, I think it's interesting that they are gods. Does this mean that they can answer questions about what WILL happen? E.g. you could ask a god what the next god will answer. If the answer is correct, that's the true or random god. If the answer is wrong, then it's the false or random god. But then we get recursive rather quickly if we ask the question twice in a row. :)
I suspect they are mentioned to be gods to suggest that they may be asked extremely complicated questions, which they will interpret from a purely logical standpoint. ie. Not human.
Starting from this premise, imagine this sequence of questions (it's not the answer, btw):
1. What is the answer to the next question? A: 1, 0
2. What was the answer to the previous question? B: 1, 0
3. What was the answer to the previous question? C: 1, 0
000-impossible: no false god
001-C is the false god, but can't distinguish the other two
010-impossible: no true god.
011-C is the true god, but you can't distinguish between the false and random god
100-A is the false god, but can't distinguish the other two
101-impossible: no true god.
110-A is the true god, but you can't distinguish between the false and random god
111-impossible: no false god
This is interesting: we have eliminated 4 possibilities, 000, 111, 101 and 010. Not only that, now we have a whole new universe to converse about - we can ask the gods about these patterns!
The remaining possibilities show some symmetry: 001 and 100, 011 and 110.
So, thinking aloud, start with these questions:
1. Will the result of my questioning be either 001 or 100?
2. Will the result of my questioning be either 011 or 110?
The answers won't give us much information because they can be anything. But we can "break symmetry" and ask them like this instead:
1. Will the result of my questioning be either 001 or 011?
2. Will the result of my questioning be either 100 or 110?
...and I feel like I'm very close but I'm missing something.
Personally, I think it's interesting that they are gods. Does this mean that they can answer questions about what WILL happen? E.g. you could ask a god what the next god will answer. If the answer is correct, that's the true or random god. If the answer is wrong, then it's the false or random god. But then we get recursive rather quickly if we ask the question twice in a row. :)