First: "The only rule the king has to obey is that eventually he has to call every prisoner in an arbitrary number of times", this means you are screwed, even if you add " until one prisoner answers yes and then the king must either kill them all or set them free depending on whether the answer was correct", then the limit on K is no longer a rule the king must obey, and in fact the King could remove the chalice for the room just for laughs....
Assuming this is not what you meant originally:
This is still "unsolvable" unless you allow for the prisoners and the King to be immortal, if that is the case, then the previously suggested counting solutions are the only options, with the assumption these prisoners do not go mad counting such high numbers for so many years...
Suppose the only limit in flip rate is physical impossibility of the outer rim during the rotation moving faster than the speed of light (there are far more practical limits, which restrict this number further); there is a limited number of times the chalice can be inverted per second. If we multiply this number by the number of seconds in a millennium, and set that to K, then the king effectively has an unlimited K. Either phrase the question allowing for immortals, or accept this riddle is insuperable. If the King has effectively unlimited flips, he simply ensures the chalice is upright every time he calls a prisoner, there is no communication stream, and therefore no way for the chalice to signal anything to the others.
I suggest you rephrase the question to remove any relation to "real world" conditions, as personally after a couple of years of this torture, I suspect I'd just say yes and get it over and done with...
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First: "The only rule the king has to obey is that eventually he has to call every prisoner in an arbitrary number of times", this means you are screwed, even if you add " until one prisoner answers yes and then the king must either kill them all or set them free depending on whether the answer was correct", then the limit on K is no longer a rule the king must obey, and in fact the King could remove the chalice for the room just for laughs.... Assuming this is not what you meant originally: This is still "unsolvable" unless you allow for the prisoners and the King to be immortal, if that is the case, then the previously suggested counting solutions are the only options, with the assumption these prisoners do not go mad counting such high numbers for so many years... Suppose the only limit in flip rate is physical impossibility of the outer rim during the rotation moving faster than the speed of light (there are far more practical limits, which restrict this number further); there is a limited number of times the chalice can be inverted per second. If we multiply this number by the number of seconds in a millennium, and set that to K, then the king effectively has an unlimited K. Either phrase the question allowing for immortals, or accept this riddle is insuperable. If the King has effectively unlimited flips, he simply ensures the chalice is upright every time he calls a prisoner, there is no communication stream, and therefore no way for the chalice to signal anything to the others. I suggest you rephrase the question to remove any relation to "real world" conditions, as personally after a couple of years of this torture, I suspect I'd just say yes and get it over and done with...