Cake number

Last updated November 07, 2024

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.

General formula

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

{n \choose 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:^[1]

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).

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.^[1]

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
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.

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References

1 2 Yaglom, A. M.; Yaglom, I. M. (1987). Challenging Mathematical Problems with Elementary Solutions. Vol. 1. New York: Dover Publications.
↑ OEIS: A000125

External links

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

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[Yaglom-Yaglom-1] 1 2 Yaglom, A. M.; Yaglom, I. M. (1987). Challenging Mathematical Problems with Elementary Solutions. Vol. 1. New York: Dover Publications.

[2] OEIS: A000125

[1]

[2]

k n	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

k n	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