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 three faces meeting at the corner facing into the room). The cube is then cut into 343 smaller, identical unit cubes.

How many of the smaller cubes do not have any face painted?

• A. 125
• B. 180
• C. 144
• D. 216

Hint:

A cube has zero painted faces only if none of its three coordinates lie on the outer painted layer; that gives a 6x6x6 block.

Answer 1

The Correct Option is D

Step 1: Set up coordinates.
Label each unit cube by its position (x, y, z), with x, y, z each running from 1 to 7 along the three edges of the big cube. The three painted faces are the three outer faces, say x = 7, y = 7 and z = 7 (the three faces visible from inside the room).

Step 2: Understanding the Concept.
A unit cube gets paint on a face only if it sits on one of these three outer layers. A unit cube has zero painted faces only if none of its coordinates touch 7, meaning x, y and z must each be chosen from 1 to 6.

Step 3: Count the unpainted cubes.
There are 6 valid choices for x, 6 for y and 6 for z, so the count is \(6\times6\times6=216\).

Step 4: Final Answer.
216 of the smaller cubes have no paint on them at all, so option D is correct.