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 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 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).

## General formula

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

${\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!)),}$

and we assume that n planes are available to partition the cube, then the n-th cake number is:[1]

${\displaystyle C_{n}={n \choose 3}+{n \choose 2}+{n \choose 1}+{n \choose 0}={\tfrac {1}{6))\!\left(n^{3}+5n+6\right)={\tfrac {1}{6))(n+1)\left(n(n-1)+6\right).}$

## Properties

The only cake number which is prime is 2, since it requires ${\displaystyle \left(n+1)(n(n-1)+6\right)}$ to have prime factorisation ${\displaystyle 2\cdot 3\cdot p}$ where ${\displaystyle p}$ is some prime. This is impossible for ${\displaystyle n>2}$ as we know ${\displaystyle n(n-1)+6}$ must be even, so it must be equal to ${\displaystyle 2}$, ${\displaystyle 2\cdot 3}$, ${\displaystyle 2\cdot p}$, or ${\displaystyle 2\cdot 3\cdot p}$, which correspond to the cases: ${\displaystyle n(n-1)+6=2}$ (which has only complex roots), ${\displaystyle n(n-1)+6=6}$ (i.e. ${\displaystyle n\in \{0,1\))$), ${\displaystyle n+1=3}$, and ${\displaystyle n+1=1}$.[citation needed]

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]

Cake numbers (blue) and other OEIS sequences in Bernoulli's triangle

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:[2]

k
n
0 1 2 3 Sum
1 1 1
2 1 1 2
3 1 2 1 4
4 1 3 3 1 8
5 1 4 6 4 15
6 1 5 10 10 26
7 1 6 15 20 42
8 1 7 21 35 64
9 1 8 28 56 93
10 1 9 36 84 130

## References

1. ^ a b Yaglom, A. M.; Yaglom, I. M. (1987). Challenging Mathematical Problems with Elementary Solutions. Vol. 1. New York: Dover Publications.
2. ^