## Small Cubes in a Cube

Arrange n^3 small cubes into one large cube.  Then paint one or more of the large cube's faces.  How many small cubes are there (or "What is n"), and which large faces were painted if we find:
1. Exactly 52% of the small cubes have paint on at least one face.
2. Exactly 36% of the small cubes have paint on at least one face.
3. Exactly 8% of the small cubes have paint on at least one face.
Source: Alan O'Donnell.
Solutions were received from Jeremy Galvagni, Ravi Subramanian, Joseph DeVincentis, Al Zimmermann, David Bachtel, Sven Mettepennigen, Maurizio Morandi, Radu Ionescu, Larry Corrado, and Philippe Fondanaiche.
Jeremy Galvagni's solution:

Part 1. n=5, paint 3 sides in a U shape 65/125=.52

Part 2. n=5, paint 2 adjacent sides 45/125=.36
or n=10 paint 4 sides in a ring 360/1000=.36

Part 3. n=25 paint 2 opposite sides 1250/15625=.08

Method:
First note all the percents reduce to fractions with 25 in denominator. n must be a multiple of 5.
There are 8 ways of painting 1,2,3,4 or 5 sides of the large cube. These paint the following numbers of small cubes:
(a)1 side n^2
(b)2 opposite sides 2n^2
(c)2 adjacent sides 2n^2 - n
(d)3 sides in a U 3n^2 - 2n
(e)3 mutually adjacent sides 3n^2 - 3n +1
(f)4 sides in a ring 4n^2 - 4n
(g)4 sides unpainted adjacent 4n^2 - 5n + 2
(h)5 sides 5n^2 - 8n + 4

ways (e), (g), and (h) can be discarded because they will never be multiples of 5 if n is a multiple of 5 (and so the fraction will never reduce enough.)

for each of the remaining ways, set up the equation (x)/(n^3) = p where x is the desired letter (a, b, c, d, f) and p is the desired proportion.

the solution(s) to this equation give the n for the pattern, if n is a whole number: problem solved.

Example: (c) gives (2n^2 -n)/(n^3) = .52
.52n^2 - 2n +1 = 0
n=3.255, n=.5907 so cannot be a way of painting the cube in part 1
but
.36n^2 - 2n + 1 =0
n=5, n=.55 so it is a way of painting the cube in part 2 if n=5

There are only 15 cases to check, the 4 that work are the solutions to the puzzle.

Mail to Ken