# Cake number

In mathematics, the **cake number**, denoted by *C _{n}*, 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 *C _{n}* for increasing

*n*≥ 0 are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, …(sequence A000125 in the 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.

## General formula[edit]

If *n*! denotes the factorial, and we denote the binomial coefficients by

and we assume that *n* planes are available to partition the cube, then the number is:^{[1]}

## References[edit]

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

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