Question

The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces?

A. 6n^2
B. 6n^2 – 12n + 8
C. 6n^2 – 16n + 24
D. 4n^2
E. 24n – 24

Answer 1

B.
`6n^2-12 n +8`

Explanation:

Given,

The number of smaller cubes =
`n^3`

So, the number of cubes that have no coloured faces. =
`(n-2)^3`,

Note : If a cube painted outside having side n is split into n³ cubes, then the volume volume that is not painted = (n-2)³

Thus, the remaining cubes that have been painted red on at least one of their faces

= Total cubes - cubes with no painted face

`= n^3 -(n-2)^3`

`=n^3 - (n^3 - 8 - 6n^2 +12n)`

`=6n^2-12 n +8`

Hence, OPTION B is correct.