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 at least three faces painted?
Show Hint
A cube has only 8 corners, and 3 faces is the most that can ever meet at a point, so at least three painted faces means the 8 corner cubes.
Step 1: Take the total and remove the other three groups.
Out of the 27 small cubes, we already know 1 centre cube has 0 painted faces, 6 face-centre cubes have 1 painted face each, and 12 edge cubes have 2 painted faces each.
Step 2: Add up what is accounted for.
\[ 1 + 6 + 12 = 19 \]
These 19 small cubes have 0, 1 or 2 painted faces, so none of them can belong to the "at least three faces painted" group.
Step 3: Subtract from the total of 27.
\[ 27 - 19 = 8 \]
Whatever is left over, 8 small cubes, must be the ones with 3 painted faces, and since 3 is also the maximum possible for any small cube here, these are exactly the cubes with at least 3 faces painted, which is the same set as the corner cubes.
Final Answer:
8 smaller cubes have at least three faces painted.
\[ \boxed{8} \]
