Question

Three equr dimensional cubes are joined face to face in a row and are coloured with Blue on
all faces. The first cube is cut into 8 equal pieces, middle one into 27 equal pieces and last
into 64 pieces.
How many cubes have two faces coloured?​

Answer 1

Given :

The first cube is painted from 5 sides and cut into 8 pieces .

The second cube is painted from 4 nd cut into 27 pieces .

The third cube is painted from 5 sides and cut into 64 pieces .

To find :

Number of cubes having two faces coloured .

Solution :

In the $1^{st}$ cube , number of smaller cubes having two faces coloured = 4

In the $2^{nd}$ cube , number of smaller cubes having two faces coloured = 12

In the $3^{rd}$ cube , number of smaller cubes having two faces coloured = 20

Total number of cubes having two faces coloured = 4 + 12 + 20 = 36

Number of cubes having two faces coloured is 36 .

Answer 2

Answer:

Concept:

When a cube is cut into any number of parts it will again have 6 sides. The top and bottom sides will always be there. So these 2 sides are coloured. Now remain 4 sides. The sides which are formed by inner cuttings are assumed to be white and the one which exists on the borders will be coloured. The ones that are on the borders will be 3 coloured except the 4 corners which will be 4 coloured.

Step-by-step explanation:

The number of boxes at the corners will be 4 coloured faces - top, bottom and 2 sides.

The number of boxes on the sides except the corner will be 3 coloured faces - top, bottom and 1 side.

The number of boxes inside the sides will be 2 coloured faces - top, bottom.

Let us say There are total N blocks, Now these boxes can be written as N = X * Y

No of 4 coloured faces will always be 4.

No of 3 coloured faces will be (X - 2) * 2 + (Y - 2)

No of 2 coloured faces will be N - (X - 2) * (Y - 2) - 4

N = 8 blocks, x = 4, y = 2,

4C = 4

3C = (4–2) * 2 + (2–2) = 2 * 2 + 0 = 4 + 0 = 4

2C = 8 - 4 - 4 = 0

N = 27 blocks, x = 9, y = 3

4C = 4

3C = (9 - 2) * 2 + (3 - 2 ) * 2 = 7 * 2 + 1 * 2 = 14 + 2 = 16

2C = 27 - 4 - 16 = 7

N = 64 blocks, x= 8, y = 8

4C = 4

3C = (8 - 2) * 2 + (8 - 2) * 2 = 6 * 2 + 6 * 2 = 12 + 12 = 24

2C = 64 - 24 - 4 = 64 - 28 = 36

Total 3 coloured faces = 4 + 16 + 24 = 20 + 24 = 44

No of 2 coloured faces = 0 + 36 + 7 = 42

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