# Worked Examples

Directions: A wooden cube is painted black on four adjoining faces and green on two opposite faces, i.e., top and bottom. It is then cut into 27 smaller, identical cubes.

- How many smaller cubes have only one of their faces painted black?
- 1
- 2
- 3
- 4
- 5

- How many smaller cubes have only one of their faces painted green?
- 1
- 2
- 3
- 4
- 5

- How many smaller cubes have only two of their faces painted black?
- 2
- 4
- 6
- 8
- 16

- How many smaller cubes have at least three of their faces painted?
- 8
- 6
- 3
- 2
- 1

- How many smaller cubes have none of their faces painted at all?
- 1
- 2
- 3
- 4
- 5

This problem can be analysed by considering the three horizontal layers, separately.

In the top layer, the central cube has only one of its faces painted green; the four cubes at the corner have three of their faces paintedâ€”one face green and the other two faces black. The remaining four cubes have two of their faces paintedâ€”one green and one black.

In the middle layer, the central cube has none of the faces painted. Four cubes at the corners have two of their faces painted black. The remaining four cubes have only one of their faces painted black.

In the bottom layer, the central cube has one of its faces painted green, and four cubes at the corners have three of their faces paintedâ€”two black and one green. The remaining four cubes have two of their faces paintedâ€”one black and one green.

Answers:

- (4)
- (2)
- (2)
- (1)
- (1)

Directions: A cube has each of its faces painted with different coloursâ€”red, blue or green. No two adjacent faces have the same colour. The cube is then cut into 27 smaller identical cubes.

- How many smaller cubes have the same colour on at least two of their faces?
- 12
- 4
- 8
- 0
- None of these

- How many smaller cubes have three of their faces coloured with different colours?
- 12
- 8
- 4
- 0
- None of these

As no two adjacent faces of the original cube are painted with the same colour, the faces which are painted with the same colour must be opposite to each other. So, it is not possible for a smaller cube to have the same colour on two of its faces.

All cubes at the corners have three faces coloured with different colours. There are eight such corner cubes.

Answers:

- (4)
- (2)

Directions: A solid cube is painted red on two adjacent faces and black on the faces opposite to those painted red. The remaining faces are painted green and the cube is then cut into 64 smaller cubes of equal size.

- How many smaller cubes are painted red on one face and black on the other?
- 24
- 10
- 8
- 0
- None of these

- How many smaller cubes are painted on only two adjacent faces?
- 32
- 24
- 16
- 8
- None of these

- How many smaller cubes have at least one of their faces black?
- 36
- 32
- 28
- 24
- None of these

- How many smaller cubes have at least one of their faces green?
- 32
- 28
- 24
- 16
- None of these

- How many smaller cubes have none of their faces green?
- 24
- 28
- 32
- 16
- None of these

The larger cube is cut into 64 smaller cubes. This implies that the edge of a smaller cube is one-fourth that of the larger cube. We know that the two adjacent faces are painted red and the two faces opposite to these faces are painted black. This implies that the top and the bottom faces are painted green. Thus, there are two edges where the faces painted red and black meet, and each edge has four cubes. Thus, there are totally 8 cubes which have red on one face and black on the other.

All smaller cubes along the edges are painted on two of their faces and the smaller cubes at the corner are painted on three faces. Thus, there are only two smaller cubes on each edge that are painted on two of their adjacent faces. Since the cube has 12 edges, there are totally (12 Ã— 2) = 24 smaller cubes which are painted on only two of their adjacent faces.

There are 4 Ã— 4 = 16 smaller cubes on one face, which are painted black, and 4 Ã— 3 = 12 smaller cubes on the adjacent face, which are also painted black. Thus, there are totally 28 smaller cubes which have at least one of their faces black.

The top and the bottom faces of original cube are painted green. There are 4 Ã— 4 = 16 smaller cubes on each face. So, there are 32 smaller cubes which have at least one of their faces green.

The remaining 64 â€“ 32 = 32 smaller cubes have none of their faces green.

Answers:

- (3)
- (2)
- (3)
- (1)
- (3)

Directions: A solid cube is painted on only three adjacent faces and then cut into 64 smaller cubes of equal size.

- How many smaller cubes have three of their faces coloured?
- 0
- 1
- 2
- 4
- None of these

- How many smaller cubes have only two of their faces coloured?
- 6
- 8
- 9
- 16
- None of these

- How many smaller cubes have only one of their faces coloured?
- 36
- 27
- 16
- 9
- None of these

The cube is painted on three adjacent faces and cut into 64 smaller, identical cubes. In this case, only one corner cube is painted on three of its faces.

Each pair, out of three coloured adjacent faces, has only one common edge. Thus, there are three edges, along which two coloured faces meet. There are three smaller cubes along each such edge which are painted only on two of their faces. Hence, there are totally 3 Ã— 3 = 9 smaller cubes which are coloured on two of their faces.

There are 9 smaller cubes on each coloured face which are coloured only on one of their faces. Thus, there are totally 9 Ã— 3 = 27 smaller cubes which are coloured only on one of their faces.

Answers:

- (2)
- (3)
- (2)

Directions: A solid cube is painted silver, purple and pink on its three sets of opposite faces and then cut into eight smaller cubes of equal size.

- How many smaller cubes have at least one of their faces painted silver?
- 8
- 6
- 4
- 2
- None of these

- How many smaller cubes have at least two of their faces painted with different colours?
- 2
- 4
- 6
- 8
- None of these

The opposite faces of the original cube are of the same colour. Thus, there are two opposite faces on the original cube which are coloured silver. There are four smaller cubes on each face. Hence, all the 8 cubes have silver colour on at least one of their faces.

There are 8 smaller cubes at 8 corners. All these smaller cubes are coloured on three of their faces with different colours.

Answers:

- (1)
- (4)

- How many smaller cubes have three of their faces coloured?
- 2
- 8
- 9
- 12
- None of these

- How many smaller cubes have two of their faces coloured?
- 10
- 12
- 16
- 18
- None of these

- How many smaller cubes have at least one of their faces coloured?
- 24
- 30
- 35
- 33
- None of these

- How many smaller cubes have none of their faces coloured?
- 0
- 1
- 2
- 4
- None of these

- How many smaller cubes have only one of their faces coloured?
- 1
- 3
- 7
- 9
- None of these

- (2)
- (3)
- (4)
- (3)
- (4)

Directions: Each of the faces of a cube is painted with different colours. The face painted red is opposite to the one painted green. The face painted blue is between the red and green coloured faces. The face painted yellow is adjacent to the one painted pink. The face painted white is adjacent to the yellow coloured one, and the green face is facing down.

- What is the colour of the face at the top?
- Blue
- White
- Red
- Yellow
- Pink

- What is the colour of the face which is opposite to the one coloured pink?
- White
- Green
- Blue
- Yellow
- Red

- What is the colour of the face which is opposite to the one coloured blue?
- Red
- Yellow
- White
- Pink
- None of these

- Which are the four colours on the faces adjacent to yellow coloured face?
- Red, White, Blue, Pink
- Green, White, Blue, Pink
- Red, Pink, Blue, Green
- White, Pink, Red, Green
- Blue, Pink, Red, Green

Hint:

Face ABEHâ€”Green CDGFâ€”Red

CBEFâ€”Blue ABCDâ€”White or Pink

AHGDâ€”Yellow EFGHâ€”Pink or White

Answers:

- (3)
- (1)
- (2)
- (4)

Directions: A cube of 3 cm edge is painted red on all of its faces. It is then cut at equal distances, at right angles, four times vertically (top to bottom) and twice horizontally (along the sides) as shown in the figure, where the lines represent the cuts made. Study the diagram and answer the following questions:

- How many smaller cubes have three of their faces painted red?
- 64
- 4
- 12
- 8
- None of these

- How many cubes have two of their faces painted?
- 4
- 8
- 12
- 6
- None of these

- How many cubes have only one of their sides painted red?
- 9
- 6
- 1
- 4
- None of these

- How many cubes have none of their sides painted?
- 1
- 4
- 0
- 4
- None of these

Here n = 3.

Therefore, number of cubes with 3 sides painted red = Number of corner cubes = 8.

Number of cubes with 2 faces painted = (n â€“ 2) Ã— 12 = (3 â€“ 2) Ã— 12 = 12.

Number of cubes with 1 face painted = (n â€“ 2)^{2} Ã— 6 = (3 â€“ 2)^{2} Ã— 6 = 6.

Number of cubes with no face painted = (n â€“ 2)^{3} = (3 â€“ 2)^{3} = 1.

Answers:

- (4)
- (3)
- (2)
- (1)

Directions: A cuboid of dimensions (6 Ã— 4 Ã— 1 cm) is painted black on two of its faces with dimensions (4 Ã— 1 cm); green on the faces with dimensions (6 Ã— 4 cm); and red on the faces with dimensions (6 Ã— 1 cm). The cuboid is then cut into smaller cubes of dimensions (1 Ã— 1 Ã— 1 cm) each.

- 1. How many smaller cubes are formed?
- 12
- 24
- 4
- 6
- None of these

- How many smaller cubes have at least three of their faces coloured with different colours?
- 12
- 10
- 8
- 4
- None of these

- How many smaller cubes have four faces coloured, and two faces without any colour?
- 4
- 16
- 10
- 8
- None of these

- If the smaller cubes having both black and green coloured faces are removed, then how many cubes will remain?
- 20
- 16
- 8
- 4
- None of these

- How many cubes have two of their faces coloured green and the remaining faces without any colour?
- 4
- 16
- 10
- 12
- None of these

- Twenty-four small cubes are formed.
- The four cubes at the corners have each of the three colours, at least on one of their faces.
- All the four cubes at the corners have four faces coloured and two faces without any colour.
- If the cubes having both black and green faces are removed, then 16 cubes remain.
- Eight cubes that are not along the edges have two faces painted green, and the remaining four faces without any paint on them.

Answers:

- (2)
- (4)
- (1)
- (2)
- (5)

Directions: A cube of dimensions 4 cm is painted black on one pair of opposite faces and green on another pair of opposite faces. The remaining one pair of opposite faces is left unpainted. Now, the cube is cut into 64 smaller cubes of dimensions 1 cm each. Analyse the pattern of smaller cubes.

- Number of smaller cubes with three faces painted = 0 (because each corner cube shares a face which is unpainted).
- Number of smaller cubes with two faces painted = Number of smaller cubes present at the corners + Number of smaller cubes with two faces painted along the 4 edges = 8 + (n â€“ 2) Ã— 4 = 8 + 8 = 16.
- Number of smaller cubes with one face painted = Number of cubes with one face painted along the 8 edges + Number of cubes with one face painted on the four faces = (n â€“ 2) Ã— 8 + (n â€“ 2)
^{2}Ã— 4 = 2 Ã— 8 + 4 Ã— 4 = 16 + 16 = 32. - Number of smaller cubes with no face painted = Number of cubes with no face painted on the two unpainted faces + Number of cubes with no face painted inside the cube = (n â€“ 2)
^{2}Ã— 2 + (n â€“ 2)^{3 }= 4 Ã— 2 + (2)^{3 }= 8 + 8 = 16.

Directions: A cube of dimensions 4 cm is painted red on one pair of adjacent faces and green on another pair of adjacent faces. The remaining two adjacent faces are left unpainted. Now, the cube is cut into 64 smaller cubes of dimensions 1 cm each. Analyse the pattern of smaller cubes.

- Number of smaller cubes with three faces painted = Number of smaller cubes at two corners = 2.
- Number of smaller cubes with two faces painted = Number of smaller cubes with two faces painted at four corners + Number of smaller cubes with two faces painted along the 5 edges = 4 + (n â€“ 2) Ã— 5 = 4 + 2 Ã— 5 = 4 + 10 = 14.
- Number of smaller cubes with one surface painted = Number of smaller cubes with one face painted on the four faces + Number of smaller cubes with one face painted along the 6 edges + Number of smaller cubes with one face painted at two corners = (n â€“ 2)
^{2}Ã— 4 + (n â€“ 2) Ã— 6 + 2 = 4 Ã— 4 + 2 Ã— 6 + 2 = 16 + 12 + 2 = 30. - Number of smaller cubes with no faces painted = Number of smaller cubes with no face painted inside the big cube + Number of cubes with no face painted on the two faces + Number of cubes with no face painted along one edge = (n â€“ 2)
^{3}+ (n â€“ 2)^{2}Ã— 2 + (n â€“ 2) = (2)^{3}+ (2)^{2}Ã— 2 + 2 = 8 + 8 + 2 = 18.