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 $p$ is some prime. This is impossible for $n>2$ as we know ${\displaystyle n(n-1)+6}$ must be even, so it must be equal to $2$, ${\displaystyle 2\cdot 3}$, ${\displaystyle 2\cdot p}$, 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]

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. ^