you guys are already posting in the thread and asking where is the thread moved???
where it is moved?
moving the thread to prep resources section.
Guys common in...its an important last phase...i wld do it in a while..a lil busy..
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)
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
Well last year also same type of a thread was started by a fellow puy but dat was very late jus 10 days prior to the beginning of CAT ...so it wldn't have helped every1...so with roughly 80 days to go to CAT...it's tym to kno some useful tricks dat will help all in cracking CAT...