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 at least three faces painted?

• A. 4
• B. 6
• C. 8
• D. 10

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.

Answer 1

The Correct Option is C

Step 1: Work out the highest number of faces any small cube can show.
On a cube, exactly 3 faces meet at a single corner, and no more than 3 faces of the big cube ever meet at one point. So the most painted faces a small cube can carry is 3, and that only happens for a small cube that sits exactly at a corner of the big cube.

Step 2: Since 3 is the maximum, "at least three" means "exactly three."
Because no small cube can show more than 3 painted faces, the group of cubes with at least 3 painted faces is the same as the group with exactly 3 painted faces, which is the corner cubes.

Step 3: Count the corners of a cube.
Every cube, big or small, has 8 corners. So there are 8 small cubes sitting at the 8 corners of the big cube, and each one touches 3 outer faces of the big cube (some Black, some Green depending on the corner, but that detail does not matter here).

Final Answer:
8 of the smaller cubes have at least three faces painted.
\[ \boxed{8} \]