Math Forum - Project of the Month, November 1996
#### A Math Forum Project

# November POM - Winner

## Jeffrey Chang

BN High School, somewhere...

**From: Hai-Feng Chang (bnhs13a@mail.erols.com)**
It is taken that n is integral, because otherwise the cube would be
exceedingly difficult, if not impossible, to divide in the aforesaid way.

a) No matter how long the side, only the corner unit cubes will have
three sides of paint on them, because this is the most number of corners
revealed for any unit cube in a cube with side larger than 1. There are
always eight corner unit cubes (in cubes with sides larger than 1), and
so the formula is f(n) = 8, where f(n) --> 3 painted sides, unless n = 1,
where f(n) = 0. (There is only one cube for a unit cube itself, with 6
painted sides).

b) The edges of the cube, not including the corners, will have two of
their sides painted, those cubes having only two sides revealed. There
are twelve edges for each cube. However, each edge of n includes the two
corner cubes, which must be subtracted because of their painting of three
sides. Thus, g(n) = 12(n-2), where n is larger than 1 (for reasons
explained already in a)).

c) The center unit cubes, of each face, will only have one surface
painted, because this is the only one revealed. These center cubes do
not include those of the edge, or the corners. Thus, there are n-2 by
n-2 of these, for each face, and six faces of the large cube. h(n) = 6 *
(n-2)^2, where n is larger than 1 (for reasons explained already in a)).

d) The remaining cubes are not painted, as the most exposed unit cube in
a cubical structure would only be the corner, revealed on three sides.
These three, two, and one sides have already been covered, and so the
remainder are unpainted unit cubes. There are n^3 total unit cubes
present, and so the number of these unpainted unit cubes is j(n) = n^3 -
6 * (n-2)^2 - 12(n-2) - 8, where n is necessarily larger than 1 (for the
above explained reasons, in a)).

### My Comments

Jeffrey explains each part of the problem very well from a visual
viewpoint, and it's written very clearly. Breaking the solution up into
parts makes it a lot easier to read, and you can refer back to the other
parts as necessary.
Return to Main Page