Question

Instructions: A cube of 7 cm x 7 cm x 7 cm is kept in the corner of a room and painted in three different colours, one colour on each of the three faces that can be seen. The cube is then cut into 343 smaller, identical unit cubes.

How many of the smaller cubes have at most two faces painted?

• A. 343
• B. 342
• C. 256
• D. 282

Hint:

Only one cube, the very corner, has all three faces painted; every other cube automatically has at most two.

Answer 1

The Correct Option is B

It helps to check this by adding up the pieces directly instead of subtracting, as a way to confirm the answer.

1. Cubes with zero painted faces: 216 (from the earlier count).
2. Cubes with exactly one painted face: 108 (from the earlier count).
3. Cubes with exactly two painted faces: these sit on the edges shared between two painted faces, excluding the single triple-painted corner cube. Each shared edge has 7 unit cubes, one of which is the corner cube (already counted separately), leaving 6 cubes per edge with exactly two painted faces. There are 3 such shared edges: $3 \times 6 = 18$.
4. Cubes with all three faces painted: exactly 1, the corner cube.

Adding the at-most-two categories: $216 + 108 + 18 = 342$.

Checking against the total: $342 + 1 = 343$, which matches the full 343 unit cubes, confirming the count is consistent.

So the correct answer is option B, 342.