Cubist Cuts

A $3 \times 3 \times 3$ cube may be reduced to unit cubes ($1 \times1 \times1$ cubes) in six saw cuts if you go straight at it.

If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer? Answer the same question with a $4 \times 4 \times 4$ cube:

What about a cube of any size (an $n \times n \times n$ cube)?

This problem is taken from "Sums for Smart Kids" by Laurie Buxton, published by BEAM Education. To obtain a copy call the BEAM orderline on 020 7684 3330 quoting product code SMAR. (Price: £13.50 plus handling and delivery.)