The puzzle is explained as follows: an evil wizard has captured infinitely many gnomes, and is announcing they would soon – after infinite time for preparation – challenge their captives by assigning each of them a black or white hat, in such a way that each is constrained not to see their own, but all other's hats; and such that neither of them can communicate with another.

Every gnome must secretly write down the color of their own hat. The gnomes are to be released if and only if finitely many of them answer incorrectly. Is there a strategy by which the gnomes can always emerge victorious?