Disclaimer: In all the problems below, "mathematician" refers to a perfect logician, not a human.
An infinite number of mathematicians are standing on the positive number line, one at each integer, and they're all facing toward positive infinity. Each wears a hat colored by a real number. The mathematician standing at N can see all hats of mathematicians standing at n > N, but does not know the color of her own or of those on mathematicians behind her. On the count of three, they have to all guess the color of their own hat, all at once, and they won't get to drink coffee ever again unless all but finitely many of them guess correctly. They were allowed to strategize before getting in line and receiving their hats. Can they ensure that they'll continue to enjoy coffee after this ordeal?
Two mathematicians, Green and Brown, are playing a game where they take turns coloring points in the integer lattice on the plane. Whoever is first to color a 2x2 square (four points) all their color wins the game. Green goes first. What is the outcome of the game?