In this article, at the end of page 6, it is given the following puzzle,
An evil wizard has threatened a village where an infinite number of gnomes reside. The wizard will cast a spell that will cause a hat to appear on the head of every gnome. Each hat will either be red or blue, but each gnome will be unable to see that hat on his or her head. The wizard will leave the gnomes alone only if only a finite number of gnomes guess the color of the hat on their heads incorrectly. The gnomes can strategize before the wizard puts the hats on their heads, but they cannot talk or communicate with each other once the hats are on their heads. The gnomes have very good eyesight and can see the hat of every other gnome. The wizard can listen to the gnomes strategize and choose the most evil possible placement of hats. What should the gnomes do?
[solution, if you can find it, requires the Axiom of Choice.]
I have been thinking this puzzle since yesterday, but I couldn't come up with any solution, so what is best strategy(ies) that can gnomes choose so that the wizard will leave them alone ?
Edit:
Note that there are infinitely many gnomes, not necessarily countably infinite.Morever, the puzzle does not say that a gnome that guesses his/her hat wrong will be kill right away, so there is no way s/he can learn whether his/her choice was true or not before everyone else made their guess.
Call two ways to colour the hats equivalent if they differ in only finitely many places. Next, form a choice function on the set of such equivalence classes. If each gnome assumes the colouring chosen from their equivalence class is correct, this guess will differ from the true one in at most one place, so is equivalent to it. Since they all work from the same hypothesis, only finitely many are wrong.