Animation showing the cutting planes required to cut a cake into 15 pieces with 4 slices (representing the 5th cake number). Fourteen of the pieces would have an external surface, with one tetrahedron cut out of the middle.
In mathematics, the cake number, denoted by Cn, is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly nplanes. The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence.
The values of Cn for n = 0, 1, 2, ... are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, ... (sequence A000125 in the OEIS).
and we assume that n planes are available to partition the cube, then the n-th cake number is:[1]
Properties
The cake numbers are the 3-dimensional analogue of the 2-dimensional lazy caterer's sequence. The difference between successive cake numbers also gives the lazy caterer's sequence.[1]