The Infinite Prisoners with Real Colored Hats Problem
Consider that you have an infinite number of prisoners; each lined up and they all get a distinct real colored hat (think of it as a real number); they can look at everyone else’s hat but their own. They can go free if they guess the color of their hat right; What’s a strategy to maximize the number of prisoners that go free? Beforehand, they discuss with each other and place the infinitely many possibly sequences of real numbers (hat colors — ignore the constraint that I don’t know if we can see uncountably infinite colors of the EM spectrum) into equivalence classes. That is, they look at the Infinitely many possible sequences, and find ones that differ by only finitely many digits and place them in the same group called an equivalence class and call the relation that they differ only by finitely many digits an equivalence relation; they memorize an arbitrary sequence in each equivalence class (and there are infinitely many equivalence classes); this memorization of an arbitrary...