Cake number

Short description: Greatest number of regions into which a cube can be partitioned by n planes

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 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 n = 0, 1, 2, ... are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, ... (sequence A000125 in the OEIS).

General formula

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

[math]\displaystyle{ {n \choose k} = \frac{n!}{k!(n-k)!}, }[/math]

and we assume that n planes are available to partition the cube, then the n-th cake number is:

[math]\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}(n+1)\left(n(n-1) + 6\right). }[/math]

Properties

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

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:

k n	0	1	2	3		Sum
1	1	—	—	—		1
2	1	1	—	—		2
3	1	2	1	—		4
4	1	3	3	1		8
5	1	4	6	4		15
6	1	5	10	10		26
7	1	6	15	20		42
8	1	7	21	35		64
9	1	8	28	56		93
10	1	9	36	84		130

Other applications

In n spatial (not spacetime) dimensions, Maxwell's equations represent [math]\displaystyle{ C_n }[/math] different independent real-valued equations.

External links

• Eric Weisstein. "Space Division by Planes". http://mathworld.wolfram.com/SpaceDivisionbyPlanes.html. 
• Eric Weisstein. "Cake Number". http://mathworld.wolfram.com/CakeNumber.html.