A prison, unfortunately, has infinitely many prisoners on death row. But, because it is nice, the appeals court has decided to give the prisoners one last chance.
The prisoners are asked to line up in single file front to back and are each given a hat that is one of a countably infinite number of colours. The prisoners are told this the night before and are not allowed to communicate once they are lined up. The prisoners are told that if they "guess" the colour of their hat correctly they will be set free.
What is the optimal strategy for the prisoners? How many of them can be set free?
Of course, it would seem that there is a "1 in countably infinite" chance of each prisoner actually being set free... so, pretty much as zero as you can get, but is that actually true..?
A closely related, much more famous, but (I feel) slightly less paradoxical problem is the well-known Banach-Tarski “paradox”. We’ll go through these questions and more in a fun-filled choice-fuelled discussion of this curious corner of maths (or not maths, depending on your opinion)!