Cube and Dice PYQs for UPSC (2011-2024) | Solved Questions & PDF Download

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Q1. The views of a cube following a particular motion are given below: [CSAT 2012]

 

 

What is the letter opposite to A?

(a) H

(b) P

(c) B

(d) M

Solution:

Considering cubes 1 and 2:  B, A, H and M are adjacent to K. Therefore P is opposite to K

Considering cubes 1 and 3:  P, K , H and A are adjacent to B thus M is opposite to B

Considering cubes 2 and 3:  K, M, B and P are adjacent to H thus A is opposite to H

Hence option (a) is correct

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Q2. Six squares are coloured, front and back, red (R), blue (B), yellow (Y), green (G), white (W) orange (O) and are hinged together as shown in the figure below. If they were folded to form a cube, what would be the face opposite to white face? [CSAT 2012]

 

(a) R

(b) G

(c) B

(d) O

Solution:

 

 

Given that,

Six squares are coloured, front and back, red (R), blue (B), yellow (Y), green (G), white (W) orange (O) and are hinged together

So, Green is opposite to Orange, Red is opposite to Yellow and Blue is opposite to White

Thus W is opposite to B

Hence option (c) is correct

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Q3. A cube has six numbers marked 1, 2, 3, 4, 5 and 6 on its faces. Three views of the cube are shown below: [CSAT 2013]

 

 

What possible number can exist on the two faces marked and B, respectively on the cube?

 

(a) 2 and 3

(b) 6 and 1

(c) 1 and 4

(d) 3 and 1

Solution:

Given that,

 

The above diagrams represents that 2, 3, 4 and 6 are adjacent to 1 and 5 is opposite to 1

So, in the place of A and B the number 1 cannot be placed.

Thus by eliminating option (b), (c) and (d) the only option left is (a)

Hence option (a) is correct

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Q4. Each of the six different faces of a cube has been coated with a different colour i.e., V, I, B, G, Y and O. Following information is given: [CSAT 2015]

1. Colours Y, O and B are on adjacent faces.

2. Colours I, G and Y are on adjacent faces.

3. Colours B, G and Y are on adjacent faces

4. Colours O, V and B are on adjacent faces

Which is the colour of the face opposite to the face coloured with O?

(a) B

(b) V

(c) G

(d) I

Solution:

Given that,

1. Colours Y, O and B are on adjacent faces.

2. Colours I, G and Y are on adjacent faces.
3. Colours B, G and Y are on adjacent faces.

4. Colours O, V and B are on adjacent faces.

From the given statement 1 and 4: Y and V are adjacent to O

Statement 2 and 3: I and B are on opposite sides

Thus from statement 1 and 3: G is opposite to O

Hence option (c) is correct

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Directions for the following 3 (three) items:

Rotated positions of a single solid are shown below. The various faces of the solid are marked with different symbols like dots, cross and line. Answer the three items that follow the given figures.

 

 

Q5. What is the symbol on the face opposite to that containing a single dot? [CSAT 2018]

(a) Four dots

(b) Three dots

(c) Two dots

(d) Cross

Solution:

From the given figure:

Front face = 1 dot, Right face = 2 dots, left face = 4 dots, Back face = 3 dots, bottom face = cross and top face = line

Thus back face = 3 dots

Hence option (b) is correct

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Q6. What is the symbol on the face opposite to that containing two dots? [CSAT 2018]

(a) Single dot

(b) Three dots

(c) Four dots

(d) Line

Solution:

From the given figure:

Front face = 1 dot, Right face = 2 dots, left face = 4 dots, Back face = 3 dots, bottom face = cross and top face = line

Left face = 4 dots

Hence option (c) is correct

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Q7. What is the symbol on the face opposite to that containing the cross?[CSAT 2018]

(a) Single dot

(b) Two dots

(c) Line

(d) Four dots

Solution:

From the given figure:

Front face = 1 dot, Right face = 2 dots, left face = 4 dots, Back face = 3 dots, bottom face = cross and top face = line

Top face is opposite to bottom and has line

Hence option (c) is correct

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COLORING OF CUBE/CUBOID

 

Q8. A cube has all its faces painted with different colours. It is cut into smaller cubes of equal sizes such that the side of the small cube is one-fourth the big cube. The number of small cubes with only one of the sides painted is [CSAT 2016]

(a) 32

(b) 24

(c) 16

(d) 8

Solution:

Given that,

A cube has all its faces painted with different colours.

It is cut into smaller cubes of equal sizes such that the side of the small cube is one-fourth the big cube.

Now the figure of one side of the cube

	A	D
	B	C

The smaller cubes not lying on the edges of bigger cube will have only one side painted.

The sides are marked as A,B, C and D

Thus there are 6 faces and 4 sides = 6 x 4 = 24

The number of small cubes with only one of the sides painted is 24

Hence option (b) is correct

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Q9. The outer surface of a 4 cm x 4 cm x 4 cm cube is painted completely in red. It is sliced parallel to the faces to yield sixty four 1 cm x 1 cm x 1 cm small cubes. How many small cubes do not have painted faces? [CSAT 2017]

(a) 8

(b) 16

(c) 24

(d) 36

Solution:

Given that,

The outer surface of a 4 cm x 4 cm x 4 cm cube is painted completely in red.

It is sliced parallel to the faces to yield sixty four 1 cm x 1 cm x 1 cm small cubes.

Thus, total number of layers = 4

All four layers have 16 cubes each

Top layer and bottom layer = All cubes painted

2nd layer = Smaller cubes which are part of bigger cubes are painted, thus 4 cubes are not painted.

3rd layer = Smaller cubes which are part of bigger cubes are painted, thus 4 cubes are not painted.

Total number of cubes not painted = 4 + 4 = 8

Hence option (a) is correct

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Q10. A solid cube of 3 cm side, painted on all its faces, is cut up into small cubes of 1 cm side. How many of the small cubes will have exactly two painted faces? [CSAT 2018]

(a) 12

(b) ৪

(c) 6

(d) 4

Solution:

Given that,

Solid cube of 3 cm side, painted on all its faces, is cut up into small cubes of 1 cm side.

So,

Cubes with 1 side painted = 6

Cubes with 2 side painted = 12

Cubes with 3 side painted = 8

Cubes with no side painted = 1

Therefore total number of cubes with 2 sides painted = 12

Hence option (a) is correct

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Q11. A solid cube is painted yellow, blue and black such that opposite faces are of same colour. The cube is then cut into 36 cubes of two different sizes such that 32 cubes are small and the other four cubes are big. None of the faces of the bigger cubes is painted blue. How many cubes have only one face painted? [CSAT 2019]

(a) 4

(b) 6

(c) 8

(d) 10

Solution:

Given that,

The cube is then cut into 36 cubes of two different sizes such that 32 cubes are small and the other four cubes are big.

Considering the side of original cube = 12

Now,

32 small cubes have sides 3 units each

Similarly, 4 bigger cubes have sides 6 units each

Also, None of the faces of the bigger cubes is painted blue

Therefore total number of cubes painted only one face = 2 x 4 = 8

Hence option (c) is correct

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Q12. Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted?

[CSAT 2019]

(a) 9

(b) 10

(c) 11

(d) 12

Solution:

Given that,

Each face of a cube can be painted in black or white colours.

So the possibilities are

1. No sides painted white, thus all sides painted black

2. 1 side white and 5 sides black

3. 2 opposite sides white and 4 sides black

4. 3 sides with same corner white and 3 sides black

5. 4 sides white and 2 opposite sides black

6. 5 sides white and 1 side black

7. All sides white and no side black

8. 2 adjacent side black and remaining four sides white

9. 2 opposite sides and 1 centre side white and other sides black

10. 2 adjacent sides white and rest 4 sides black.

Therefore total 10 possible ways to paint a cube black or white.

Hence option (b) is correct

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Q13. 125 identical cubes are arranged in the form of a cubical block. How many cubes are surrounded by other cubes from each side? [CSAT 2023]

(a) 27

(b) 25

(c) 21

(d) 18

Solution:

Given that,

125 identical cubes are arranged in the form of a cubical block

So each dimensions = 5 x 5 x 5

Now the middle three cubes or internal cubes are surrounded by other cubes with dimensions = 3 x 3 x 3

Thus total number of internal cubes = 3 x 3 x 3 = 27

 Hence option (a) is correct

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Q14. A cuboid of dimensions 7 cm x 5 cm x 3 cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7 cm x 5 cm, 5 cm x 3 cm, 7 cm x 3 cm respectively. Then the cuboid is cut and separated into various cubes each of side length 1 cm. Which of the following statements is/are correct? [CSAT 2023]

1. There are exactly 15 small cubes with no paint on any face.

2. There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green.

Select the correct answer using the code given below:

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Solution:

Given that,

A cuboid of dimensions = 7 cm x 5 cm x 3 cm

Painted red, green and blue colour on each pair of opposite faces of dimensions = 7 cm x 5 cm, 5 cm x 3 cm and 7 cm x 3 cm

Now,

Statement 1: There are exactly 15 small cubes with no paint on any face.

No paint faces are not on the surface of the cuboids, thus interior cuboids are of 5 x 3 x 1 = 15 cubes

Hence statement 1 is correct

Statement 2: There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green.

The cubes exactly two faces painted would on edges but not on corners.

Edges with blue and green paint are 5 cm x 3 cm

Thus subtracting other corners consisting of 2 from the edge length so the cubes would be 3 + 1 = 4

Hence statement 2 is incorrect

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Q15. What is the least possible number of cuts required to cut a cube into 64 identical pieces? [CSAT 2024]

(a) 8

(b) 9

(c) 12

(d) 16

Solution:

Given that,

Total identical pieces: 64

Now,

64 = 4 x 4 x 4

Total cuts required = 3 cuts in all the dimensions of length, breadth and height

Thus,

minimum number of cuts = (4 - 1) + (4 - 1) + (4 - 1) = 9

Hence option (b) is correct

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