Chapter 8.1, Problem 107E

Cube A $3\times 3\times 3$ cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure.

(a) Determine how many unit cubes of the $3\times 3\times 3$ cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces.

(b) Repeat part (a) for a $4\times 4\times 4$ cube, a $5\times 5\times 5$ cube, and a $6\times 6\times 6$ cube.

(c) Write formulas you could use to repeat part (a) for an $n\times n\times n$ cube.