Cubes 1

An $8 \times 8 \times 8$ cube is made of small $1 \times 1 \times 1$ cubes. How many of these small cubes lie on the surface?

Answer

The total number of small cubes is $8 \times 8 \times 8 = 512$ . Removing the inner $6 \times 6 \times 6$ cube (i.e. $$216$$ hidden cubes) leaves $$512 - 216 = 296$$ cubes exposed on the surface.

Notes and comments

Comment 1: The inner cube of an $n \times n \times n$ is $(n-2) \times (n-2) \times (n-2)$ . For example, inside a $4 \times 4 \times 4$ is a $2 \times 2 \times 2$ core.

Comment 2: All cubes, regardless of size, have exactly 6 faces, 8 vertices, and 12 edges.