# Cake number

In mathematics, the cake number, denoted by Cn, is the maximum 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.

The values of Cn for increasing n ≥ 0 are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, …(sequence A000125 in OEIS)

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.

Animation showing the cutting planes for cutting a cake into 15 pieces with 4 cutting planes. Fourteen of the pieces have an external surface, with one tetrahedron cut out of the middle.

## 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 number is:[1]

$C_n = {n \choose 3} + {n \choose 2} + {n \choose 1} + {n \choose 0} = \frac{1}{6}(n^3 + 5n + 6).$

## References

1. ^ Eric Weisstein. "Space Division by Planes". MathWorld − A Wolfram Web Resource. Retrieved August 19, 2010.