The riddle is impossible to solve with absolute ceetainty and I can prove it.
If Alice an Bob are not allowed to strategize, relying only on pure logic would not grant them to follow any of the strategies proposed by any of the solutions. This is true because I have proposedas long as a coherent and logical framing of this riddle which does not guarantee any solution exists, like the one I proposed here. Unless I committed some logical fallacy in my framing, one must admit that a perfect logician may actually use the same framing as the one I proposed, given that my framing can be obtained by perfect logics alone. Even if the strategies proposed in any of the solutions are also logical, Alice and Bob would not be guaranteed to automatically follow any of them without strategizing first or somehow choosing to follow the same logics by total serendipity.
If there was only a unique strategy that can guarantee a win, or if all the possible strategies that would guarantee a win would have the same outcome (e.g. Alice finding that Bob has 8 bars on the morning of the 4th day), Alice and Bob being perfect logicians would be able to identify and implement this unique strategy (or decide to implement any of the equivalent strategies) without previously agreeing with each other, as this would be the most logical thing to do to be freed. The fact that multiple successful strategies exists to solve the riddle on different days already implies that Alice and Bob can't know for sure which of these strategies the other will implement. One could argue that the strategy which would guarantee the earliest freeing time would be the best choice and should therefore automatically implemented by Alice and Bob. But given that with any of these strategies Alice and Bob gain information over time and don't know in which day they would gather the necessary information to solve the riddle, I argue that it's not possible for them to identify with absolute certainty a strategy that is a priori superior to the others. In addition to that, given that in theory multiple strategies can be devised to solve the problem and given that perfect logicians are not omniscent beings but mere humans, logics suggests that both Alice and Bob needs to account for the possibility that the other person might devise a better strategy than any of the strategy that they are currently thinking about. In other words, blindly deciding to choose a strategy without aligning with the other prisoner first would be the same as making a random choice. Assessing the probability of success of this random choice is also not possible (because Alice and Bob cannot know for sure how many successful strategies the other is thinking about), which means that choosing to follow a strategy may be just as good or possibly even worse than just randomly pick one of the two numbers proposed by the logician. And in any case, even if deciding to blindly follow a strategy would somehow guarantee a better chance of success than randomly picking one of the two numbers, success would still not be guaranteed with absolute certainty.
Therefore the riddle is unsolvable with absolute certainty.
