A cube is painted Green on four adjoining side faces and Black on the other two faces, which are opposite each other (the top face and the bottom face). The cube is then cut by two evenly spaced cuts parallel to each of its three pairs of faces, which divides every edge into 3 equal parts and turns the big cube into a 3 x 3 x 3 arrangement of 27 smaller cubes of equal size.
How many of the smaller cubes have no face painted at all?
Show Hint
Only the small cube buried at the very centre, with no outer face at all, is fully unpainted; use (n-2)^3 with n = 3.
Step 1: Go back to the layer picture.
Think of the big cube as 3 layers stacked up, a bottom layer, a middle layer and a top layer. Any small cube sitting in the bottom or top layer automatically touches the bottom face or the top face, so it is painted. Only the middle layer can possibly hold an unpainted cube.
Step 2: Look inside the middle layer.
The middle layer itself is a flat 3 by 3 grid of 9 small cubes. The 8 cubes around the border of this grid still touch one of the 4 side faces of the big cube, since they sit along the outer edge of that layer. Only the 1 small cube right at the centre of this 3 by 3 grid does not touch any side face either.
Step 3: Combine both checks.
That centre cube of the middle layer is neither in the top or bottom layer (so it misses the top and bottom paint) nor on the border of its own layer (so it misses the side paint too). It is the one small cube untouched by any outer face.
Final Answer:
Only 1 smaller cube has no painted face at all.
\[ \boxed{1} \]
