I believe your first question doesn't quite work, but it's on a useful track.

If A is false, B is true and C is random, A will say 'yes'. B will remain silent, because he cannot say whether C will answer, so A will be speaking falsely.

P.S. Adding to the confusingness of the puzzle, if he uses weak negation, True can say

Not(random would confirm that he is a God)

I think we're better off assuming that the Gods just fall silent when they can't answer.

If the islanders know that either their spot is black or their spot is white, I believe all the black-spotted people will leave the day after all the red-spotted people have left. However, if they do not know for certain that their spot is either red or black, then all the black-spotted people will be stuck on the island forever, not knowing if they personally have a spot that is a different color than everyone else's.

Ok, I'll hide as you prefer.
Alternative first question(s): Ask A whether Random would give the same answer as A would give if asked whether he is God. If A answers he is Random himself. The last question as before.

At first it seems that if there are two or more islanders with a red spot, the announcement of the tourist does not change anything. But in fact, some information is added:
Before the announcement, everyone knew that there is at least one with a red spot.
After the annoucement, everyone knew that everyone knew that there is at least one with a read spot.
This extra information is leading to all n islanders with red spots leaving on the n-th day. (And all the ones with black spots on the day after.)

Again: wonderful puzzle.

What's confusing is what happens when there are three or more red spots--everyone already knows that everyone knows there is a red spot. You have to find a subtler way to characterize what they learn.

If there are three red spots, then before the announcement, everyone knows that everyone knows there is a red spot. After the announcement, everyone knows that everyone knows that everyone knows there is a red spot.

3 spots, BEFORE announcement:

Statement 1: everyone knows that everyone knows there is a red spot

black knows that everone knows there is a red spot
......black knows that black knows there is a red spot ... +
......black knows that red knows there is a red spot ..... +
red knows that everyone knows there is a red spot
......red knows that black knows there is a red spot ..... +
......red knows that red knows there is a red spot ....... +

-> Statement 1 is true.


Statement 2: everyone knows that everyone knows that everyone knows there is a red spot

black knows that everone knows that everyone knows there is a red spot
......black knows that black knows that everyone knows there is a red spot
............black knows that black knows that black knows there is a red spot ... +
............black knows that black knows that red knows there is a red spot ..... +
......black knows that red knows that everyone knows there is a red spot
............black knows that red knows that black knows there is a red spot ..... +
............black knows that red knows that red knows there is a red spot ....... +
red knows that everyone knows that everyone knows there is a red spot
......red knows that black knows that everyone knows there is a red spot
............red knows that black knows that black knows there is a red spot ..... +
............red knows that black knows that red knows there is a red spot ....... +
......red knows that red knows that everyone knows there is a red spot
............red knows that red knows that black knows there is a red spot ....... +
............red knows that red knows that red knows there is a red spot ..... -

-> Statement 2 is not true, some information is still missing and no one will leave the island.



3 spots, AFTER announcement:

With the announcement, statement 2 becomes true, information has been added.

I think that a good way to characterize the new information is "If there were only one person with a red spot, that person would know that they had a red spot". Knowing this conditional is sufficient for the induction, and lacking without the proclamation of the visitor.

But for starting the induction you need more. It is ok with one red spot but why should it also work when you add more red spots?

While I think SpongeBobs logic is correct, it should be possible to explain in a simpler way. Otherwise you should also have to show the same thing for 4 red spots, 5 red spots, ....

Here is my try using induction:
If there is just one red spot, it obviously works.

If there are two red spots, everyone already knows that there is at least one red spot.

But the next level of information comes from the observer (tourist), in the form of making each one of red spots expect the other to leave the next morning. Since the other one doesn't leave, he learns that there is one more red spot except that one.

So, every day he learns that there must be one more, and this information comes from the "expectation" (expectation that other red spots should leave) created by the observer.

So for creating the induction: When you add one more red spot, you delay the expectation of leaving one more day.

Without the observer, this expectation would never be created because there is no reason for any of the red spots to expect anyone to leave. Even if a red spot knows that N other red spots exist, there is still no information relating to the color of his own spot.

What the observer does is turning the rule of leaving next morning into information related to the number of red spots via the expectation.

No. You don't. If there were one person, and this was known, he'd leave. If there were two, then after one day, the two people will know. It goes on. The knowledge of the conditional is sufficient.

Note that this is a conditional. The new knowledge isn't that there is at least one person with a spot, but that if there were only one that person would know.

I'm sure some of us have seen this before, but it's obligatory anyway:

Expecting the other to leave the next morning already involves logical reasoning by the islander based on the information given by the tourist.

Hmm, interesting that you guys think the logical dead-ends are the way to solve it. I posted my "solution" as a joke because I assumed that falling into one of those unanswerable positions meant you failed (hence the "struck by lightning" bit.) Haven't had time to think about it any more, but maybe that is the solution.

I kind of don't think so though, since it seems like ANY question that you pose involving Random causes the same problem, so there wouldn't be a single correct answer if that's the case. Then again, I guess nobody said there had to be.

There are solutions that do not involve stumping the gods. The range of solutions is extremely wide--there are many logically equivalent ways to phrase the same question, and there are many distinct questions.