Cubes 'Ice'

Lets brush up some painted cube funda 

We assume the cube is divided into n^3 small cubes.

no. of small cubes with ONLY 3 sides painted : 8( all the corner cubes )

no. of small cubes with ONLY 2 sides painted : 

A cube is painted on 2 sides means, it is on the edge of the bigger cube ,and we have 12 edges, each having n cubes. but since the corner cubes are painted on 3 sides, we need to neglect them. so in effect, for each side we will have (n-2) small cubes with only 2 sides painted. thus, then number is, 12 * (n-2)

no of small cubes with ONLY 1 side painted :

for each face of the cube ( 6 faces ) we have (n-2)^2 small cubes with only one side painted. and we have 6 faces in total.  so th number is, 6*(n-2)^2

no of small cubes with NO sides painted :

if we remove the top layer of small cubes from the big cube we will end up a chunk of small cubes with no sides painted.  this number will be equal to, (n-2)^3.

Also, remember for Cuboids with all different sizes, the following are the results:

a x b x c (All lengths different) 

Three faces - 8 (all the corner small cubes of the cuboid)

Two faces - There are two (a-2) units of small cubes on one face of the cuboid and there is a pair of such faces. Hence, number of such small cubes corresponding dimension a of the cuboid = 4(a-2).

Similarly, for others.

So, total with two faces painted = 4(a - 2) + 4(b - 2) + 4(c - 2) 

One face - Since each face of the cuboid is a combination two different dimensions, hence for the face which is a combination of a and b dimensions, the number of small cubes is 2* (a-2)(b-2) [multipliesd by 2 because there are 2 such faces for the combination]

Similarly, for others.

So, total with one face painted = 2(a - 2)(b - 2) + 2(a - 2)(c - 2) + 2(b - 2)(c - 2) 

Zero faces - The entire volume of small cubes except for two cubes in each of the rows and columns will not be painted at all. hence this is the simplest ... 

(a - 2)(b - 2)(c - 2) 

You can put different integer values for number of small cubes producing different edge lengths of cuboid to get varied results.

To verify for a cube, put a=b=c=L, you get

Three faces - 8  Two faces - 12(L - 2)  One face - 6(L - 2)^2  Zero faces - (L - 2)^3