Question

A cube of side 12 cm is painted with 3
colours, red, blue and green such that the
opposite faces have the same colour. It is
then cut into 216 equal smaller cube?
How many cubes are unpainted on all
sides?

Answer 1

Answer

There are two ways in which a cube can be painted on 3 faces

1 - All the three faces are painted one after the other. (Cube has six faces, let give them number from 1 to 6. Suppose 1 and 6 are on opposite faces .5 and 3 are on opposite faces and 2&4 are on opposite faces. So in this case paint can be done on 1,4,6 or 1,5,6 or 2,3,4 or 3,4,5 and so on) .

After cutting in 216 small cubes, each face will have have 36 cubes. Therefore number of cubes painted exactly one face will be 84.

2 . Thia is another way in which paint can be done on three faces So in this case paint can be done on 1,2,3 or 1,3,4 or 4,5,6 or 2,3,6 and so on.

Therefore in this case no cubes having one face painred will be 75

So least 75

And largest 84

If you have any confusion, you should try by drawing the cube with given numbers on a sheet of paper.