A hint for the cheese cutting puzzle

Simplify the problem by making the initial cuts all at right angles so that the large cube is sliced into 8 smaller cubes.

Align the cube so that the 8 corners lie at the points $(\pm{1}, \pm{1}, \pm{1}).$ Now consider the plane $x+y+z=0.$

How many of the 8 smaller cubes does this plane pass through?

What can you say about any cubes that this plane does not pass through? How can you alter this plane to try to get more intersections?

For the second part of the problem don't attempt to work out the theoretical maximum number of pieces as this is exceedingly difficult even for 5 cuts! You can, however, consider how many pieces each small piece of cheese can be cut into per slice. You can also experiment with various pre-determined "cutting schemes" to try to find a guaranteed minimum number of possible pieces.