![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Cake_number_3.svg/220px-Cake_number_3.svg.png)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Cake_number_with_4_cutting_planes.gif/220px-Cake_number_with_4_cutting_planes.gif)
In mathematics, the cake number, denoted by Cn, 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 Cn 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
[edit]If n! denotes the factorial, and we denote the binomial coefficients by
and we assume that n planes are available to partition the cube, then the n-th cake number is:[1]
Properties
[edit]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]
![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Bernoulli_triangle_columns.svg/330px-Bernoulli_triangle_columns.svg.png)
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:[2]
- kn
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
[edit]In n spatial (not spacetime) dimensions, Maxwell's equations represent different independent real-valued equations.
References
[edit]- ^ a b Yaglom, A. M.; Yaglom, I. M. (1987). Challenging Mathematical Problems with Elementary Solutions. Vol. 1. New York: Dover Publications.
- ^ OEIS: A000125
External links
[edit]- Eric Weisstein. "Space Division by Planes". MathWorld. Retrieved January 14, 2021.
- Eric Weisstein. "Cake Number". MathWorld. Retrieved January 14, 2021.