In Greek mythology, Achilles was a hero of the Trojan War. Some 2500 years ago, philosopher Zeno of Elea included him into one of his paradoxes. Achilles was quite the runner... or maybe the paradox shows otherwise.
Suppose Achilles was in a 100 meter foot race with a tortoise. The tortoise is given a 25 meter head start. It would seem obvious that when Achilles eventually starts running, he will easily overtake the tortoise and then comfortably reach the finish line (with time to spare). Looking at it logically though, one could wonder how Achilles can ever beat the tortoise.
The ‘Bottle Imp Paradox’ is similar to the Unexpected Hanging, but it's a fun one.
“You are offered the opportunity to buy, for whatever price you wish, a bottle containing a genie who will fulfill your every desire. The only catch is that the bottle must thereafter be resold for a price smaller than what you paid for it, or you will be condemned to live out the rest of your days in excruciating torment.
Obviously, no one would buy the bottle for 1¢ since he would have to give the bottle away, but no one would accept the bottle knowing he would be unable to get rid of it. No one would buy the bottle for 2¢ either: he would be unable to sell it since we just established that no one would take it for 1¢. Similarly, no one would buy it for 3¢, and so on. However, for some reasonably large amount, it will always be possible to find a next buyer, so the bottle will be bought.” source
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the ‘surprise hanging’ can't be on a Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on a Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.
He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.” source
An interesting paradox! There are many variations to this story. For example, another popular version is about a surprise examination in class. Resolutions from varying perspectives have been suggested, but the different schools of thought have yet to agree on the correct one.
