How many cubes in the top layer have three red faces each?

Only two faces are coloured is when cubes are at the edges (baring the corner cubes)
If no cubes have been removed then on each edges we will get 3 cubes that has exactly 2 faces coloured, hence total number of such cubes = 12 x 3 = 36, because we have 12 edges.
Out of these 3 cubes are removed hence required number of cubes = 36 - 3 = 33

In any plane, leave 4 sides cube and select (3 x 3 x 3) inter section. But
the cubes 2 x 2 x 1 give 2 less cube because that part we are already removed.
No. of cubes = (3 x 3 x 3) - 2 = 25.

Total no. of cubes = 5³ = 125,
Some cubes from different corners are removed and the number removed cubes are 2, 3, 4 and 4.
Remaining number of small cubes:
= 125 - 2 - 3 - 4 - 4 = 125 - 13 = 112

For least number of cuts 120 = 4 x 5 x 6 i.e number of cuts must be 3, 4 and 5 in three planes in this case number of cubes on a face is either 6 x 5 = 30 or 6 x 4 = 24 or 4 x 5 = 20 cubes . And number of cuboids on an edge is 4 or 5 or 6
In this case we have to use red and green twice and same colour should be on opposite faces then required cube is given by 4 edges (but not corner), maximum number of cubes one edge is 6 - 2 = 4 so required number of cubes is 4 x 4 = 16

For least number of cuts 120 = 4 x 5 x 6 i.e number of cuts must be 3, 4 and 5 in three planes in this case number of cubes on a face is either 6 x 5 = 30 or 6 x 4 = 24 or 4 x 5 = 20 cubes . And number of cuboids on an edge is 4 or 5 or 6
Maximum number of cuboid with red colour is possible when cube is painted with red colour in 3 sides with minimum number of common edges (which is equal to 2)
Hence required maximum value is 6 (5 + 5 + 4 - 2) = 72
For minimum number of such cuboid Red colour is used only once and minimum number of cubes in that case is 20
Hence required ratio is 72 : 20 = 18 : 5

Number of cubes with 3 face coloured = 4 (Bottom cubes) + 8 top cubes + 4 (column cubes) = 16

Initial total number of cubes = 343,
Number of cubes removed = 27
Smaller 27 cubes painted blue
Exposed faces of original big cube (3 faces with 9 cube on each face i.e total 27 cubes) painted with black
Since 7 corner (Vertices) of bigger cube is untouched hence they are painted with three faces.
Now consider the corner from where we have removed 3 x 3 x 3 cubes,
After removed 3 new corners of the bigger cube will be generated that will be painted with 3 faces and 8 corners from smaller cube of 3 x 3 x 3 painted with 3 faces.
So the such total number of such cubes is 7 + 3 + 8 = 18.

Initial total number of cubes = 343,
Number of cubes removed = 27
Smaller 27 cubes painted blue
Exposed faces of original big cube (3 faces with 9 cube on each face i.e total 27 cubes) painted with black
In original big cube number of faces with one colour is 3(6 -2)² = 48 (here we have considered only 3 untouched of big cube)
But here we have removed a cubes of the form of 3 x 3 x 3 and again put it back so out of three new exposed faces of big cube we will have 4 cubes in each face that is painted with one colour hence number of cubes from these three surfaces is 3 x 4 = 12
Now consider out of 3 x 3 x 3 cubes we will have 6 cubes (one in each face ) which are painted only one face.
Hence total number of cubes = 48 + 12 + 6 = 66

Initial total number of cubes = 343,
Number of cubes removed = 27
Smaller 27 cubes painted blue
Exposed faces of original big cube (3 faces with 9 cube on each face i.e total 27 cubes) painted with black
Without any changes number of cubes with no face colour is given by (6 - 2)³ = 64
Now because of removal of 3 x 3 x 3 cubes from one of the corner from each face that were not painted earlier got exposed and will get painted, so from 3 x 3 x 3 cubes 4 x 3 = 12 cubes got painted, and a similar number from 3 exposed faces of big cube got painted.
Total number of cubes with no face painted is 64 - 12 - 12 = 40

Initial total number of cubes = 343,
Number of cubes removed = 27
Smaller 27 cubes painted blue
Exposed faces of original big cube (3 faces with 9 cube on each face i.e total 27 cubes) painted with black
Out of 27 small cubes from 3 x 3 x 3, outer 26 cubes are 1st painted with blue and then it is kept back with original cube and painted with yellow so out of 26 cubes only 5 edges will give us cubes with both the colours and number of such cubes are 12