A cube is taken and painted with yellow,blue and green colours on opposite faces.It is now cut into 64 smaller cubes of equal size. how maany cubes do not have any colour on them:

1) 64 2)27 3)8 4)1

Solution given:

There are 8 cubes which donot have any colour on them.

This question was asked in the model test paper (olympiad) test paper no.2 qno.13.

Please elaborate the solution .

A painted cube is cut into 64(= 43) smaller cubes. Thus there will be

a) Smaller cubes with three faces painted (cubes at the corner) = 8

b) Smaller cubes with two faces painted (cubes at the edge) = (4 – 2) × 12 = 24

c) Smaller cubes with one faces painted (cubes at the faces) = (4 – 2)2 × 6 = 24

d) Smaller cubes not painted

∴ Numbers of such not painted cubes = 64 – (8 + 24 + 24) = 8

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