Cake number

Animation showing the cutting planes required to cut a cake into 15 pieces with 4 slices (representing the 5th cake number). Fourteen of the pieces would have an external surface, with one tetrahedron cut out of the middle.

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)

General formula[]

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]

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

Properties[]

The only cake number which is prime is 2, since it requires ${\displaystyle \left(n+1)(n(n-1)+6\right)}$ to have prime factorisation ${\displaystyle 2\cdot 3\cdot p}$ where ${\displaystyle p}$ is some prime. This is impossible for ${\displaystyle n>2}$ as we know ${\displaystyle \left(n(n-1)+6\right)}$ must be even, so it must be equal to ${\displaystyle 2}$, ${\displaystyle 2\cdot 3}$, ${\displaystyle 2\cdot p}$, or ${\displaystyle 2\cdot 3\cdot p}$, which correspond to the cases: ${\displaystyle \left(n(n-1)+6\right)=2}$ (which has only complex roots), ${\displaystyle \left(n(n-1)+6\right)=6}$ (i.e. ${\displaystyle n\in \{0,1\}}$), ${\displaystyle (n+1)=3}$, and ${\displaystyle (n+1)=1}$.^{[citation needed]}

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]

References[]

External links[]

• Eric Weisstein. "Space Division by Planes". MathWorld. Retrieved January 14, 2021.
• Eric Weisstein. "Cake Number". MathWorld. Retrieved January 14, 2021.

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