8 points to the person to solve this puzzle.
(In the spirit of the contest, please refrain from googling answers)
I have a cube with sides of length 3L. I want to use my saw to cut my cube into 27 identical blocks with sides of length of L. The trivial way to do this is with 6 cuts (2 through each face of the cube). If I am allowed to rearrange the pieces after each cut, can I do it in fewer than 6 cuts?
Justify your answer.