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

Step 1: Understanding the Concept.
“At most two faces painted” covers every cube except the ones with all three faces painted, so it is easier to subtract the three-face cubes from the total.

Step 2: Find the cube with three faces painted.
A unit cube has three painted faces only if all three of its coordinates equal 7 at once, since that is the only cube touching all three painted planes together. There is exactly one such cube, sitting at the very corner.

Step 3: Subtract from the total.
Total unit cubes \(= 343\). Cubes with at most two painted faces \(= 343 - 1 = 342\).

Step 4: Final Answer.
342 unit cubes have at most two faces painted, so option B is correct.