1985 AMC 12 Problem 20

Question

A wooden cube with edge length

n

units (where

n

is an integer

\gt2

) is painted black all over. By slices parallel to its faces, the cube is cut into

n^3

smaller cubes each of unit edge length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is

n?

5

6

7

8

none of these

Answer: D

Concepts:algebraic manipulation basic counting cube geometry

Difficulty rating: 1660

Small hint:

Count the unpainted cubes in the interior cube of side length

n-2

Big hint:

On each face, exactly

(n-2)^2

cubes avoid the edges

Solution:

There are

(n-2)^3

completely unpainted cubes. Each of the six faces contributes

(n-2)^2

cubes with exactly one painted face, so equality gives

6(n-2)^2=(n-2)^3.

Since

n\gt2,

division by

(n-2)^2

gives

n-2=6,

hence

n=8.

Therefore the correct answer is D.