A solid cube of side 6 units is first painted on all its faces and then cut into smaller cubes of side 2 units. How many of the smaller cubes will have paint on exactly two faces?

Introduction / Context:
This question tests your understanding of three dimensional geometry and how surface painting behaves when a cube is cut into smaller cubes. Problems of painted cubes are standard in aptitude exams and help you visualise edges, faces and corners in a structured way.

Given Data / Assumptions:

• A large cube has side length 6 units.
• The cube is painted on all six faces.
• It is then cut into smaller cubes, each of side 2 units.
• We need the number of small cubes that have paint on exactly two faces.

Concept / Approach:
When a painted cube is cut into smaller equal cubes, cubes on corners have three painted faces, cubes along edges (but not at corners) have two painted faces, cubes on faces but not on edges have one painted face, and cubes completely inside have no paint. The count of cubes with exactly two painted faces is therefore the count of edge cubes excluding corners.

Step-by-Step Solution:
Step 1: Side of the big cube = 6 units. Side of each small cube = 2 units.Step 2: Number of small cubes along each edge = 6 / 2 = 3.Step 3: Total number of small cubes = 3 * 3 * 3 = 27.Step 4: For a cube cut into n segments along each side, the number of edge cubes with exactly two painted faces is 12 * (n - 2).Step 5: Here n = 3, so the count = 12 * (3 - 2) = 12 * 1 = 12.

Verification / Alternative check:
Draw a 3 × 3 × 3 grid representing the smaller cubes. On each of the 12 edges of the cube, there are 3 small cubes. The corner cube on each edge has three faces painted and is not counted for exactly two faces, leaving one middle cube per edge with exactly two painted faces.Total edge cubes with two faces painted = 12 edges * 1 middle cube per edge = 12, confirming our formula based result.

Why Other Options Are Wrong:
30 and 24 are too large and would imply more than one middle cube on each edge, which is impossible with only 3 cubes per edge.8 would match the number of corner cubes (each with three painted faces), not cubes with exactly two painted faces.

Common Pitfalls:
Learners sometimes confuse cubes with two faces painted and those with three faces painted.Another mistake is to count face center cubes or internal cubes as having two faces painted.A clear mental or drawn model of edges, faces and corners avoids such errors.

Final Answer:
The number of smaller cubes with paint on exactly two faces is 12.