Correct Answer: 18

Explanation:

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.

Correct Answer: 66

Explanation:

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

Correct Answer: 40

Explanation:

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

Correct Answer: 15

Explanation:

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

Correct Answer: 8

Explanation:

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 12 cubes in previous question there are 4 cubes with 2 faces yellow so number of cubes painted two faces only one with yellow and one with blue is 12 - 4 = 8

Correct Answer: 105

Explanation:

Number of cubes removed from top face = 16
Number of cubes removed from bottom face = 4
Number of cubes left = 125 - (16 + 4 ) = 105

Correct Answer: 34

Explanation:

From top face (out of 3 x 3 square face) only one cubes is with one face painted.
From 4 vertical faces each face will give us 6 cubes hence total number of cubes from vertical faces is 6 x 4 = 24.
From bottom face we will get 3 x 3 = 9 cubes
So total number of cubes with one face painted is 1 + 24 + 9 = 34

Correct Answer: 27

Explanation:

Number of cubes with no face painted is
105 - 34 - 24 - 20 = 27
Or else all the 3 x 3 x 3 inner cubes will remain coloured.

Correct Answer: 20

Explanation:

Number of cubes with three coloured face on the top side = 4
Number of cubes with three coloured face on the 2^nd from top side = 4
Number of cubes with three coloured face on the bottom side = 12
Total number of such cubes = 12 + 8 = 20

Correct Answer: 24

Explanation:

Number of cubes with two face painted from the top side (Which is a square of 3 x 3 = 9 cubes ) is 4.
Number of cubes with two face painted from the 2^nd from top side (Which has four edges and edge has 3 such cubes) is 4 x 3 = 12.
Number of such cubes from vertical edges is 4 x 1 = 4
Number of such cubes from bottom face is 4 x 1 = 4
Hence total such cubes is 4 + 12 + 4 + 4 = 24