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 smaller cubes are formed in all?

• A. 27
• B. 25
• C. 18
• D. None of these

Hint:

Two cuts parallel to each face split every edge into 3 equal parts, so the number of small cubes is (edge parts) cubed.

Answer 1

The Correct Option is A

Step 1: Work out how the big cube is cut.
The cube is cut by two evenly spaced cuts parallel to each pair of opposite faces. Two cuts along one edge split that edge into 3 equal parts, so along each of the three directions, length, breadth and height, the edge is divided into 3 equal parts.

Step 2: Apply the counting rule for a cut cube.
When a cube is cut so that each edge is divided into \(n\) equal parts, the total number of small cubes formed is \(n^3\). This is because the big cube is a 3 dimensional shape, so the split along the length, the split along the breadth and the split along the height all act together, and their counts multiply rather than add.
Here \(n = 3\), so the total number of small cubes is
\[ 3 \times 3 \times 3 = 3^3 = 27 \]

Step 3: Rule out the other options.
Option (b), 25, and option (c), 18, do not match a cube where every edge is split into 3 equal parts, they would only be correct if the cuts were uneven or if some pieces were fused back together, which the question does not say. Since \(3^3=27\) is an exact match, option (d), none of these, is also ruled out.

Final Answer:
The big cube breaks into 27 smaller cubes in all.
\[ \boxed{27} \]