An m×n checkerboard is colored randomly: each square is independently assigned red or black with probability 21. we say that two squares, p and q, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at p and ending at q, in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than 8mn. Putnamprobabilityinductiongeometryperimeterexpected valuecollege contests