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 only one face painted, and that face is Black colored?

• A. 1
• B. 2
• C. 4
• D. 6

Hint:

Only a face-centre cube touches exactly one outer face; check which face-centre cubes sit on the Black top and bottom faces.

Answer 1

The Correct Option is B

Step 1: Recall the four kinds of small cubes in a 3 by 3 by 3 cut.
When a cube is cut into 3 equal parts on each edge, every small cube falls into one of four positions: a corner cube (touches 3 outer faces), an edge cube (touches 2 outer faces), a face-centre cube (touches only 1 outer face) or the one centre cube (touches no outer face).

Step 2: Note which outer faces are Black.
The question tells us Black is painted only on the top face and the bottom face, while the four side faces are Green.

Step 3: Find the cubes that touch exactly one Black face.
A cube that touches only one outer face must be a face-centre cube, since corner and edge cubes always touch 2 or 3 outer faces. On the top face, the face-centre cube sits in the middle of that face and does not touch any of the 4 side faces, so it shows exactly one painted face, and that face is Black. The same is true for the face-centre cube of the bottom face.
The face-centre cubes of the four Green side faces do exist too, but each of those shows a Green face, not Black, so they do not count here.

Step 4: Count them.
Only the top face-centre cube and the bottom face-centre cube qualify, so there are 2 such cubes.

Final Answer:
There are 2 smaller cubes with exactly one face painted Black.
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