Cutting a cube by plane cuts This is an extension of a 3rd grade problem.
How many pieces can one get at most if one cut a unit cube with n plane cuts?
1,2,4,8, ???
And assuming cutting through an area 1 takes time t, what is the least time needed to achieve the maximal pieces for n cuts, asymptotically?
 A: Number of pieces $\frac{n^3+5n+6}{6}$
per Peter C. Heinig  
Define a number of planes in space to be in general arrangement when
1) no two planes are parallel,
2) there are no two parallel intersection lines,
3) there is no point common to four or more planes.  
Suppose there are already n-1 planes in general arrangement, thus defining the maximal number of regions in space obtainable by n-1 planes and now one more plane is added in general arrangement. Then it will cut each of the n-1 planes and acquire intersection lines which are in general arrangement.
These lines on the new plane define the maximal number of regions in 2-space definable by n-1 straight lines, hence this is (Lazy Caterer's sequence)(n-1).  
Each of this regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n)=a(n-1)+(Lazy Caterer's sequence)(n-1).
So,
f(n) = ${n \choose 3} + {n \choose 2} + {n \choose 1} + {n \choose 0}$ = $\frac{1}{6}(n^3+5n+6)$
