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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.

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    Found it, http://math.stackexchange.com/questions/50085/can-a-rectangle-be-cut-into-5-equal-non-rectangular-pieces which also has a link to a discussion on MO.2011-07-28

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There's a solution if the "pieces" don't have to be contiguous: Cut out cubes from each of the corners, with side length half the smallest side length of the lamington. Each of the remaining rectangular prisms can simply be cut into three congruent pieces with congruent icing. Distribute $3n$ of the cubes, and divide the $8-3n$ remaining cubes into three pieces along their $3$-fold symmetry axis.

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    Yes, great answer. Thanks Joriki! I wonder i$f$ there's an answer that will yield contiguous pieces...2011-07-30