Question

A cube is a coloured orange on one face, pink on the opposite face, brown on one face and silver on a face adjacent to the brown face. The other two faces are left uncoloured. It is then cut into $125$ smaller cubes of equal size. Now answer the following questions based on the above statements. How many cubes have all the faces uncoloured?

• A. $24$
• B. $36$
• C. $48$
• D. $64$

Answer 1

Answer: (C)

$48$

Let the side of the bigger cube be $5cm$. So, one side of smaller cube is $1cm$.
$\therefore n=\frac{sideofbigcube}{sideofsmallcube}=\frac{5}{1}=5$
Number of smaller cubes having all faces uncoloured = Number of smaller cubes at $2$ Surfaces + Number of smaller cubes at $1$ edge + Number of smaller cubes from inside the cube
$=2\times (n-2{)}^2+(n-2)\times 1+(n-2{)}^3$
$=2\times 9+3+27=18+3+27=48$.