(I learned this puzzle from Ravi Vakil.) Suppose you have an infinite grid of squares, and in each square there is an arrow, pointing in one of the 8 cardinal directions, with the condition that any two orthogonally adjacent arrows can differ by at most 45 degrees.
Can there be a closed cycle? (i.e. start at some arrow, move to the square that arrow points to, follow where the arrow there points and so on, and come back to the square you started at).