Question

A cube is coloured blue from three adjacent faces and yellow from other three faces and then cut into 729 smaller cubes of equal size. How many cubes have only one face coloured yellow and other faces uncoloured?

• A. 49
• B. 196
• C. 147
• D. 294

Hint:

When identifying cubes with only one coloured face, count the cubes that are not located on the edges or corners of the coloured faces.

Answer 1

The Correct Option is C

We are given that a cube is painted blue on three adjacent faces and yellow on the other three faces. The cube is then cut into 729 smaller cubes of equal size.
Step 1: Understanding the structure of the cube The cube is cut into 729 smaller cubes, so each smaller cube has a side length 1 cm.
Since the cube has been divided into smaller cubes, it must be a \( 9 \times 9 \times 9 \) arrangement (since \( 9^3 = 729 \)).
Step 2: Identifying cubes with one yellow face The smaller cubes that have only one face coloured yellow are located along the faces of the cube, but not on the edges or corners where other faces would also be coloured.
There are 3 faces coloured yellow, and each yellow face has a square grid of \( 9 \times 9 = 81 \) cubes.
The cubes on the edges and at the corners of these faces will have more than one yellow face coloured, so we exclude them.
The number of cubes with only one yellow face on each yellow face is \( (9-2) \times (9-2) = 7 \times 7 = 49 \) cubes per face.
Since there are 3 yellow faces, the total number of cubes with only one yellow face is: \[ 49 \times 3 = 147 \] Thus, the correct answer is \( \boxed{147} \).