A lamington is a piece of cake in the shape of a rectangular prism (rectangular cuboid). Each surface is coated with chocolate icing and coconut. I want to share a lamington with two other friends. The icing is the best bit. The icing on corners is particularly good, because of the high surface/volume ratio.
It's possible to cut three pieces with the same volume and same surface area of icing. But can I divide a lamington into three congruent pieces - where each piece is the same shape (modulo reflection or rotation), and has the same surfaces coated with icing.
BTW: I'm not after an "envy free" algorithm that lets different people cut and choose.