My goal here is present this riddle and then add a twist which invalidates a portion of the classic solution. I'd like to see what solutions people come up with. The riddle below is copied from here.
Sailing through a thick fog, you come upon a mysterious island shrouded in mist. A towering volcano in the center of the island pierces the clouds, billowing smoke into the sky. You land your boat and set out to ascend the peak. After an arduous climb, you approach the volcano summit, where lava glows red within a vast crater.
Here, you are approached by three gods.
On the summit of this remote volcano, you realize a few things through divine intervention.
First, you know that one of the three gods always tells the truth, another always lies, and the third will respond to questions randomly. Therefore, let us call the gods True, False, and Random.
The gods speak a different language. They understand all languages perfectly well, but only answer questions with either ja or da, the words for yes and no. You do not know which god is which, and you do not know which word means yes and which word means no.
Finally, you have an existential problem on your hands. You may ask three yes-or-no questions, each one directed to only one god, and only that god will answer with either ja or da. If you can determine the identities of the three gods, they will send you on your way with their blessing, and you can be assured of a prosperous and fulfilled life. If you fail to determine the identities of the gods, however, they will be less generous in their treatment. The volcano pit smokes and glows red beside you.
With your three questions, how do you figure out which god is True, which is False, and which is Random?
STOP HERE TO AVOID SPOILERS BY IMPLICATION
See here for the "classical" solution.
I'd like to add a simple twist. What happens if you're not allowed to ask any of the gods a question about themselves?
At some point I'll give my answer in the comments but I'd like to see what people come up with.