(Moved up since it's solution time.)
So while we're doing the post-mortem on this last puzzle, here's a simpler one. Someone gives you a bandsaw and a cube of wood and asks you to cut it into 27 smaller, identical cubes -- as if you had a Rubik's cube:
and wanted to slice it into its 27 smaller component cubes.
Now, it's obvious that you can do this with six cuts -- two in each dimension -- that solution's trivial. But let's say that, after each cut, you get to reorganize and re-orient your current pieces of wood before making the next cut, to try to maximize the effect of each cut.
How few cuts will you need? And, once again, hold off with the solutions until everyone gets their shot at it.
OPEN FOR SOLUTIONS: OK, feel free to post solutions. And my apologies to the previous commenter whose solution I deleted. I thought I could just undelete it but, apparently, there is no way to undelete comments. My bad.