Cake number - Alchetron, The Free Social Encyclopedia

In mathematics, the cake number, denoted by C_n, is the maximum 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.

The values of C_n for increasing n ≥ 0 are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, …(sequence A000125 in the OEIS)

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.

General formula

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

( n k ) = n ! k ! ( n − k ) ! ,

and we assume that n planes are available to partition the cube, then the number is:

C n = ( n 3 ) + ( n 2 ) + ( n 1 ) + ( n 0 ) = 1 6 ( n 3 + 5 n + 6 ) .

References

Cake number Wikipedia

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