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 increasing are given by

General formula

If ''n''! denotes the factorial, and we denote the binomial coefficients by
:$= \frac ,$
and we assume that ''n'' planes are available to partition the cube, then the ''n''-th cake number is:
:$C_n = + + + = \tfrac\left(n^3 + 5n + 6\right) = \tfrac\left(n+1) (n (n-1) + 6\right).$

Properties

The only cake number which is prime is 2, since it requires $\left(n+1) (n (n-1) + 6\right)$ to have prime factorisation $2 \cdot 3 \cdot p$ where $p$ is some prime. This is impossible for $n > 2$ as we know $\left(n (n-1) + 6\right)$ must be even, so it must be equal to $2$, $2 \cdot 3$, $2 \cdot p$, or $2 \cdot 3 \cdot p$, which correspond to the cases: $\left(n (n-1) + 6\right) = 2$ (which has only complex roots), $\left(n (n-1) + 6\right) = 6$ (i.e. $n \in \$), $(n+1) = 3$, and $(n+1) = 1$.

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'' ≥ 3.

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

References

External links

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Mathematical optimization

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