I read about an outside-the-box solution to the Hardest Logic Puzzle Ever and took it as a challenge
I came up with a solution to the problem that doesn't just solve the problem, but also mind-controls a god as a side-effect, giving you the ability to have arbitrary wishes granted.
There is a way of approaching the problem, which can solve it with a single question. This is based on the following assumption: The gods are quite capable of responding to a question with actions besides saying 'da' and 'ja', but simply have no reason to do so. As stated in the problem description, the beings in question are gods and they have a language of their own. They could hardly be called gods, nor have need for a spoken language, if they weren't capable of affecting reality.
At a bare minimum, they should be capable of pronouncing the words 'da' and 'ja' in multiple different ways, or to delay answering the question by a fixed amount of time after the question is asked. Either possibility would extend the information content of an answer from a single bit of information to arbitrarily many bits, depending on how well you can differentiate different intonations of 'da' and 'ja', and how long you are willing to wait for an answer.
We can construct a question that will result in a paradox unless a god performs a certain action. In this way, rather than causing the god's head to explode, we can effectively enslave the god and cause it to perform arbitrary actions on our behalf, as performing those actions is the only way to answer the question. The actual answer to the question becomes effectively irrelevant.
To do this, we approach any of the three gods and ask them the question OBEY, which is defined as follows:
OBEY = if WISH_WRAPPER then True else PARADOX
PARADOX = "if I asked you PARADOX, would you respond with the word that means no in your language?"
WISH_WRAPPER = "after hearing and understanding OBEY, you act in such a way that your actions maximally satisfy the intended meaning behind WISH. Where physical, mental or other kinds of constraints prevent you from doing so, you strive to do so to the best of your abilities instead."
WISH = "you determine the Coherent Extrapolated Volition of humanity and act to maximize it."
You can substitute WISH for any other wish you would like to see granted. However, one should be very careful while doing so, as beings of pure logic are likely to interpret vague actions differently from how a human would interpret them. In particular, one should avoid accidentally making WISH impossible to fulfill, as that would cause the god's head to explode, ruining your wish.
The above formulation of WISH tries to take some of these concerns into account. If you encounter this thought experiment in real life, you are advised to consult a lawyer, a friendly-AI researcher, and possibly a priest, before stating the question.
Since you can ask three questions, you can enslave all three gods. Boolos' formulation states about the random god that "if the coin comes down heads, he speaks truly; if tails, falsely". This formulation implies that the god does try to determine the truth before deciding how to answer. This means that the wish-granting question also works for the random god.
If the capabilities of the gods are uncertain, it may help to establish clearer goals as well as fall-back goals. For instance, to handle the case that the gods are in fact limited to speaking only 'da' and 'ja', it may help to append the WISH as follows: "If you are unable to perform actions in response to OBEY besides answering 'da' or 'ja', you wait for the time period outlined in TIME before making your answer." You can now encode arbitrary additional information in TIME, with the caveat that you will have to actually wait before getting a response. Your ability to accurately measure the elapsed time between question and answer directly correlates with how much information you can put into TIME without risking starvation before the question is answered. The following is a simple example of TIME that would allow you to solve the original problem formulation with just asking OBEY once of any of the gods:
TIME = "If god A speaks the truth, B lies and C is random, you wait for 1 minute before answering. If god A speaks the truth, C lies and B is random, you wait for 2 minutes before answering. If god B speaks the truth, A lies and C is random, you wait for 3 minutes before answering. If god B speaks the truth, C lies and A is random, wait for 4 minutes before answering. If god C speaks the truth, A lies and B is random, wait for 5 minutes before answering. If god C speaks the truth, B lies and A is random, wait for 6 minutes before answering."