Question

After all the faces of a cube are painted, the cube is cut into 64 small cubes of equal dimensions. How many small cubes have
(i) no face painted?
(ii) 1 face painted?
(iii) 2 faces painted?
(iv) 3 faces painted?​

Answer 1

Answer:

For a cube of side n*n*n painted on all sides which is uniformly cut into smaller cubes of dimension 1*1*1,

Number of cubes with 0 side painted = (n-2)3

Number of cubes with 1 side painted = 6(n - 2)2

Number of cubes with 2 sides painted = 12(n-2)

Number of cubes with 3 sides painted = 8 (always)

Given, the cube is separated into 64 small cubes.

So, n3 = 64

⇒ n = 4

By formula:

Number of cubes with no side painted = (n - 2)3

= (4 - 2)3

= 23

= 8

Hence, ‘8’ is the correct answer.