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 exactly one colour on them?

• A. 108
• B. 72
• C. 36
• D. 24

Hint:

A cube with exactly one painted face has exactly one coordinate equal to 7 and the other two between 1 and 6.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept.
Using the same coordinate setup as before, a unit cube has exactly one painted face if exactly one of its three coordinates equals 7 (touches exactly one painted plane), and the other two coordinates are from 1 to 6 (so it does not touch either of the other two painted faces).

Step 2: Count for one painted face.
Say x = 7 is the painted coordinate: then y and z can each independently be any of 1 to 6, giving \(6 \times 6 = 36\) cubes.

Step 3: Repeat for the other two faces.
By the same logic, there are 36 cubes with only y = 7 painted, and 36 cubes with only z = 7 painted.

Step 4: Add them up.
Total cubes with exactly one painted face \(= 36 + 36 + 36 = 108\).

Step 5: Final Answer.
108 unit cubes have exactly one painted face, so option A is correct.