Breaking into pieces

Consider the center small cube. (i.e. If the large cube was 3×3 then I am referring to the 1×1 cube that had no exposed faces)

We can divide the 3×3 cube using planes (with each plane corresponding to one cut). If any face of the small 1×1 cube did not lie on a plane, then the the cube would not have been cut out. (i.e. it would not be exposed). To put it simply, each face requires one cut.

Hence at least 6 planes are needed hence 6 cuts. It is achievable: 2 cuts from the top, 2 cuts from the front and 2 cuts from the sides along the black lines.

I assumed that cut meant "with a knife or blade", so I got the right answer. However, if you were to use an appropriately sized tic-tac-toe shaped cookie cutter (not sure if anyone actually makes one) -- assuming the cube material was soft enough, the answer would be 2.

The rubiks cube helps you if you cut along each line in the cube to the bottom u will get 27 cubes as you only need to make 2 vertical cuts 2 horizontal cuts and 2 cuts on the top

The centre cube has to be split from the other cubes that touch all 6 faces. Clearly, only one face can be 'revealed' at a time, so the minimum number of cuts required is 6.

Since you need 27 smaller cubes, the length of each side should be 3 because $3^{3}=27$ .

Imagine this cube in a space in $\mathbb{R^{3}}$ with the regular coordinate axes $x, y,$ and $z$ . Imagine one of the cube's verticies at the origin, and three of the edges along the axes. The planes needed to cut this cube into 27 smaller cubes are: $x=1, x=2, y=1, y=2, z=1, z=2$ .

Therefore, assuming that each cut can go through more than one unit, six cuts are necessary to make $27$ smaller cubes.

Although I would argue that this is an impossible problem ... a Rubik's Cube doesn't actually have a cube in the middle! (Take one apart and see for yourself😄)