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
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
.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.