In mathematics, the cake number, denoted by C<sub>n</sub>, is the maximum of the 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. It is the 3D analogue of the lazy caterer's sequence.
The values of C<sub>n</sub> for are given by .
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 n-th cake number is:
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.
The fourth column of Bernoulli's triangle (k = 3) gives the cake numbers for n cuts, where n ≥ 3.
The sequence can be alternatively derived from the sum of up to the first 4 terms of each row of Pascal's triangle:
In n spatial (not spacetime) dimensions, Maxwell's equations represent different independent real-valued equations.