KöMaL 2009 K206

Steve and Charlie are brothers. Their mom bought them petits fours at the confectioner's shop. There are 8 larger cubes of edge 3 cm and 27 smaller cubes 2 cm on edge. Five faces of each cake are covered with icing that has the same uniform thickness on all of them. The bottom of the cakes is not iced. Steve and Charlie would like to divide the petits fours between them so that each gets the same total volume, the cakes are not cut in pieces, and each of them gets both kinds. $a$) Show that these conditions are impossible to meet. $b$) When they see that they cannot divide the petits fours equally by volume, they decide to have equal quantities of icing instead. (They still do not want to cut the cakes, and each of them should get both kinds.) Find all possible ways to divide the petits fours.