Question

How many smaller cubes have all the surfaces uncolored?
$A)0$
$B)9$
$C)18$
$D)27$

Answer 1

Hint: First we have to define what the terms we need to solve the problem are.
Since there are six sides in the cube and so six times of the length is multiplied and the smaller color is that we need to find the uncolored surface, so there are two restrictions that the cube is uncolored and in all the surfaces.
Formula used: $A = 6{a^2}$(surface of the cube)
Total number of cubes \[ = {n^3}\]
Cubes with $3$sides colored$ = 8$
Cubes with $2$sides colored$ = 12(n - 2)$
Cubes with $1$sides colored$ = 6{(n - 2)^2}$
Cubes with no sides colored$ = {(n - 2)^3}$

Complete step by step answer:
For the inside surface cube (restriction) we have $ \Rightarrow {(n - 2)^3}$, Since the surface of the cube formula is $A = 6{a^2}$whereases a is the length that is in the sides of each edge if the cube is placed.
Since in this problem a cube has both surfaces that are inside surface and outside surface;
Also, we need only the smaller cubes with no colored surface and all the restriction is that side will need to be inside the presented cube; and hence the formula for this problem is $ \Rightarrow {(n - 2)^3}$
Since the restriction is surfaced uncolored so that we will need to find $n$if the inside uncolored
Since the cube is assumed to stack to be composed of $5$horizontal layers, and that is the number of cubes with no surface colored.
Thus $n = 5$and hence substitute in the above equation we get $ \Rightarrow {(n - 2)^3}$$ = {(5 - 2)^3}$
Further solving we get $ \Rightarrow {(n - 2)^3} = {3^3} = 27$
(If the cube is colored on one side then we get Cubes with $1$sides colored$ = 6{(n - 2)^2}$and thus we get $6{(n - 2)^2} = 6{(5 - 2)^2} \Rightarrow 6 \times 9 = 54$)

So, the correct answer is “Option D”.

Note: Since if all sides are colored then we generally said that Cubes with $3$sides colored$ = 8$ and the Total number of cubes is given as \[{n^3}\]
If the cube is colored on two sides then we get $12(n - 2) = 12(5 - 2) \Rightarrow 36$