Question

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?

• A. 1
• B. 2
• C. 3
• D. 4

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.

Answer 1

The Correct Option is A

Step 1: Understand which cube has zero painted faces.
A small cube shows a painted face only if it sits on the outer surface of the big cube. A small cube with no painted face at all must be completely buried inside, with none of its 6 faces touching the outside.

Step 2: Use the standard formula.
When a cube is cut into \(n\) equal parts along each edge, the number of completely inside (fully unpainted) small cubes is given by
\[ (n-2)^3 \]
because you remove one layer of small cubes from each side of the cube along every direction, leaving an inner cube of side \(n-2\).

Step 3: Substitute the value.
Here \(n = 3\), so
\[ (3-2)^3 = 1^3 = 1 \]
This single unpainted cube is the exact centre of the big cube, the one small cube that does not touch any of the 6 outer faces.

Final Answer:
Only 1 smaller cube has no face painted.
\[ \boxed{1} \]