A large 2x2x2 wooden cube can be sawn into eight 1x1x1 cubes with three cuts.

A 3x3x3 cube can be sawn into twenty-seven 1x1x1 cubes with six cuts.

How many cuts are needed for a 4 x 4 x 4 cube?

For an n x n x n cube?

Some of the faces of 1 x 1 x 1 cubes made like this

are freshly-hewn (FH), the rest are pre-cut (PC).

There are seven categories that each 1 x 1 x 1 cube could fall into:

Faces PC 0 1 2 3 4 5 6

Faces FH 6 5 4 3 2 1 0

How many of each type are there starting with a 2 x 2 x 2 cube?

With a 1 x 1 x 1 cube?

3 x 3 x 3?

n x n x n?

We know the 3 x 3 x 3 cube can be cut into

27 1 x 1 x 1 cubes with six cuts.

Can the number of cuts be reduced

by rearranging the pieces after each cut?