Cake number

Three orthogonal planes slice a cake into at most eight (C₃) pieces

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 Template:Nowrap are given by Template:Nowrap Template:OEIS.

General formula

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

$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!},$

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

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

Properties

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.

Template:Bernoulli triangle columns.svg The fourth column of Bernoulli's triangle (k = 3) gives the cake numbers for n cuts, where n ≥ 3.

Proof without words that summing up to the first 4 terms on each row of Pascal's triangle is equivalent to summing up to the first 2 even terms of the next row

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

Template:Diagonal split header	0	1	2	3		Sum
0	1	—	—	—		1
1	1	1	—	—		2
2	1	2	1	—		4
3	1	3	3	1		8
4	1	4	6	4		15
5	1	5	10	10		26
6	1	6	15	20		42
7	1	7	21	35		64
8	1	8	28	56		93
9	1	9	36	84		130

Other applications

In n spatial (not spacetime) dimensions, Maxwell's equations represent $C_n$ different independent real-valued equations.

See also

• Dividing a circle into areas (Moser's circle problem)
• Lazy caterer's sequence
• Pizza theorem

External links

• Template:MathWorld
• Template:MathWorld

Template:Combin-stub