Cube With 6 Cuts

A cube is made from a block of white pine, 6 cm on each edge. It is painted a nice barn red. After the paint dries, the cube is cut into 27 little cubes, each one 2 cm on each edge. This can be done with six cuts of the saw, right? Now, a question: Starting over, after making a saw cut, you are allowed to rearrange the pieces any way you want before making the next cut. Following this rule, can you make the 27 little cubes in fewer than six cuts of the saw?

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No matter what you do, there will be one little all-white cube at the center. This little cube has six white faces; how can you cut these six faces in fewer than six cuts?

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