Cake number

In mathematics, the cake number, denoted by C_n, 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_n 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).

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

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

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
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 ${\displaystyle C_{n}}$ different independent real-valued equations.