Cake number
In mathematics, the cake number, denoted by Cn, is the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. 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.
The values of Cn for increasing are given by
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.
General formula
If n! denotes the factorial, and we denote the binomial coefficients by
and we assume that n planes are available to partition the cube, then the number is:
