Side-tracked from a slightly related problem, I derived this general set of attributes:

{8, 12n, 6n^2, n^2, n}.

The first 5 situations of this generalisation are:

[8, 0, 0, 0, 0],
[8, 12, 6, 1, 1],
[8, 24, 24, 4, 2],
[8, 36, 54, 9, 3],
[8, 48, 96, 16, 4].

What does this set of attributes describe?

(Attributes #1 & #4 may provide clues. Also, take note of the Title.)

Consider a white cube of size n+2 divided into unit cubes. Paint the outer surface of the cube red.

There are 8 1x1x1 cubes which have 3 red faces.
There are 12n 1x1x1 cubes which have 2 red faces.
There are 6*n^2 1x1x1 cubes which have 1 red face.

Posted by Brian Smith on 2003-05-13 07:11:22