+1 vote
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A cube of side $12$ cm is painted red on all the faces and then cut into smaller cubes, each of side $3$ cm. What is the total number of smaller cubes having none of their faces painted?

1. $16$
2. $8$
3. $12$
4. $24$
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The big cube is cut into 4 x 4 x 4 = 64 small cubes

There are 8 original corners - 3 painted sides
There were twelve edges on the original cube - each will give two cubes with two painted sides - 24 in all
There were 6 faces - each with four cubes with one painted side - 24 in all

So the number with no painted sides is 64 - 8 - 24 -24 = 8
(The unpainted cubes formed a small cube 2 x 2 x 2).

OR

no of cubes N = 12 / 3 = 4
formula: no of cubes having 0 faces painted = ( N - 2 ) ^ 3 = ( 4 - 2 ) ^ 3 = 2 ^ 3 = 8

similarly formula for 2 face painted = 12*(N-2)
for 1 face painted = 6 * ( N - 2 ) ^ 2

Option B is the Correct Answer.