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 only one face painted, and that face is Black colored?
Show Hint
Only a face-centre cube touches exactly one outer face; check which face-centre cubes sit on the Black top and bottom faces.
Step 1: List out all 27 small cubes by type.
In a cube cut into 3 equal parts on every edge, the 27 pieces split as 8 corner cubes (3 faces painted each), 12 edge cubes (2 faces painted each), 6 face-centre cubes (1 face painted each) and 1 centre cube (0 faces painted). Check the total: $8+12+6+1=27$, which matches the cube we have.
Step 2: Focus only on the group with a single painted face.
The question asks about cubes with just one painted face, so only the 6 face-centre cubes are relevant here, one sitting at the middle of each of the 6 outer faces of the big cube.
Step 3: Filter by colour.
Of these 6 outer faces, 2 are Black (top and bottom) and 4 are Green (the four side faces). So among the 6 face-centre cubes, 2 of them show a Black face and the other 4 show a Green face.
Step 4: Pick out the ones the question wants.
Since the question asks specifically for one face painted and that face is Black, we keep only the 2 face-centre cubes belonging to the top and bottom faces.
Final Answer:
There are 2 such smaller cubes.
\[ \boxed{2} \]
