Cake number

Summary

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.

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.

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 only cake number which is prime is 2, since it requires   to have prime factorisation   where   is some prime. This is impossible for   as we know   must be even, so it must be equal to  ,  ,  , or  , which correspond to the cases:   (which has only complex roots),   (i.e.  ),  , and  .[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]

 
Cake numbers (blue) and other OEIS sequences in Bernoulli's triangle

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]

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 edit

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

References edit

  1. ^ a b Yaglom, A. M.; Yaglom, I. M. (1987). Challenging Mathematical Problems with Elementary Solutions. Vol. 1. New York: Dover Publications.
  2. ^ OEISA000125

External links edit

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