View source for Cake number

{{short description|Concept in combinatorics}}
[[File:cake_number_3.svg|thumb|Three orthogonal planes slice a cake into at most eight (''C''<sub>3</sub>) pieces]]
[[File:Cake number with 4 cutting planes.gif|thumb|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<sub>n</sub>'', is the maximum of the number of regions into which a 3-dimensional [[cube]] can be partitioned by exactly ''n'' [[plane (geometry)|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<sub>n</sub>'' for {{nowrap|1=''n'' = 0, 1, 2, ...}} are given by {{nowrap|1=[[1]], [[2]], [[4]], [[8]], [[15 (number)|15]], 26, 42, 64, 93, 130, 176, 232, ...}} {{OEIS|id=A000125}}.

== General formula ==

If ''n''! denotes the [[factorial]], and we denote the [[binomial coefficient]]s by
:<math>{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:<ref name="Yaglom-Yaglom">{{cite book|last1=Yaglom |first1=A. M. |authorlink1=Akiva Yaglom |last2=Yaglom |first2=I. M. |authorlink2=Isaak Yaglom |title=Challenging Mathematical Problems with Elementary Solutions |volume=1 |location=New York |publisher=[[Dover Publications]] |year=1987}}</ref>
:<math>
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 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.<ref name="Yaglom-Yaglom"/>

{{Bernoulli_triangle_columns.svg|Cake numbers (blue)}}
The fourth column of [[Bernoulli's triangle]] (''k'' = 3) gives the cake numbers for ''n'' cuts, where ''n'' &ge; 3.

[[File:cake_numbers_visual_proof.svg|thumb|[[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]]:<ref>{{oeis|A000125}}</ref>
{{table alignment}}
:{| class="wikitable defaultright col1center"
! {{diagonal split header|''n''|''k''}} !! 0 !! 1 !! 2 !! 3
! rowspan="11" style="padding:0;"| !! 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 <math>C_n</math> different independent real-valued equations.

== See also ==
* [[Dividing a circle into areas]] (Moser's circle problem)
* [[Lazy caterer's sequence]]
* [[Pizza theorem]]

== References ==
{{Reflist}}

==External links==
*{{MathWorld |id=SpaceDivisionbyPlanes |title=Space Division by Planes |author=Eric Weisstein |access-date=January 14, 2021 |ref= }}
*{{MathWorld |id=CakeNumber |title=Cake Number |author=Eric Weisstein |access-date=January 14, 2021 |ref= }}

{{Classes of natural numbers}}

[[Category:Mathematical optimization]]
[[Category:Integer sequences]]


{{combin-stub}}