# 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 n ≥ 0 are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, …(sequence A000125 in the OEIS)

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. 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.

## General formula

If n! denotes the factorial, and we denote the binomial coefficients by

${n \choose k}={\frac {n!}{k!\,(n-k)!}},$ and we assume that n planes are available to partition the cube, then the number is:

$C_{n}={n \choose 3}+{n \choose 2}+{n \choose 1}+{n \choose 0}={\frac {1}{6}}(n^{3}+5n+6).$ 