If a 3D-cake is cut by $n$ planes yielding the maximum number of pieces, then what is the number of pieces with the cake crust? It is known that a 3D-cake can be cut by $n$ plane cuts at most into $N$ pieces, defined by Cake Number $N=\frac {1}{6}(n^3+5n+6)$. However, some of the pieces would have a crust of the cake as one of their boundaries. How many pieces are with crust? (Entire surface of the cake is assumed to be crust).
To get the maximum number, the cutting planes are in general position.

Consider the 2-D surface (crust) of the cake. Let the intersection of the cutting planes and the crust be denoted as cutting circles. Remove 1 point that is not on any of the cutting lines. Project what remains onto the plane. We now have $n$ circles in general position cutting the plane. We know that the number of regions it divides it into is

$$n^2-n+2.$$
Number of pieces with crust $= ( n^2+n+2)/2$