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 are given by .
If n! denotes the factorial, and we denote the binomial coefficients by
{n\choosek}=
| n! | |
| k!(n-k)! |
,
Cn={n\choose3}+{n\choose2}+{n\choose1}+{n\choose0}=\tfrac{1}{6}\left(n3+5n+6\right)=\tfrac{1}{6}(n+1)\left(n(n-1)+6\right).
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:
| 0 | 1 | 2 | 3 | Sum | ||
|---|---|---|---|---|---|---|
| 0 | 1 | — | — | — | 1 | |
| 1 | 1 | 1 | — | — | 2 | |
| 2 | 1 | 2 | 1 | — | 4 | |
| 3 | 1 | 3 | 3 | 1 | 8 | |
| 4 | 1 | 4 | 6 | 4 | 15 | |
| 5 | 1 | 5 | 10 | 10 | 26 | |
| 6 | 1 | 6 | 15 | 20 | 42 | |
| 7 | 1 | 7 | 21 | 35 | 64 | |
| 8 | 1 | 8 | 28 | 56 | 93 | |
| 9 | 1 | 9 | 36 | 84 | 130 |
In n spatial (not spacetime) dimensions, Maxwell's equations represent
Cn