Cake number

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, ... .

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 n-th cake number is:

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 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.

The fourth column of Bernoulli's triangle (k = 3) gives the cake numbers for n cuts, where n &ge; 3.

The sequence can be alternatively derived from the sum of up to the first 4 terms of each row of Pascal's triangle:


 * {| class="wikitable defaultright col1center"

! !! 0 !! 1 !! 2 !! 3 ! rowspan="11" style="padding:0;"| !! Sum ! 0 ! 1 ! 2 ! 3 ! 4 ! 5 ! 6 ! 7 ! 8 ! 9
 * 1 || — || — ||  — ||   1
 * 1 || 1 || — ||  — ||   2
 * 1 || 2 || 1 ||  — ||   4
 * 1 || 3 || 3 ||  1 ||   8
 * 1 || 4 || 6 ||  4 ||  15
 * 1 || 5 || 10 || 10 || 26
 * 1 || 6 || 15 || 20 || 42
 * 1 || 7 || 21 || 35 || 64
 * 1 || 8 || 28 || 56 || 93
 * 1 || 9 || 36 || 84 || 130
 * }

Other applications
In n spatial (not spacetime) dimensions, Maxwell's equations represent $$C_n$$ different independent real-valued equations.