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 are given by .

General formula

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:

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.

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:

1. {| class="wikitable defaultright col1center"

! !! 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 different independent real-valued equations.

See also

• Dividing a circle into areas (Moser's circle problem)

• Lazy caterer's sequence

• Pizza theorem

References

External links