Consider the plane as a checkerboard, with each unit square coloured black or white in an arbitrary manner. We show that for any such colouring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a multiplicative constant. For the corresponding “finite” problem (NxN checkerboard) we also prove that we can color it in such a way that the above quantity is at most the square root of N log N, for any placement of the line segment.