Directions (Q. 61-66): A pile of cubes of equal size is arranged as shown in the figure. Now the block is dipped into abucket full of red paint so that only the surfaces of the block get coloured. Now answer the following questions.

61. How many cubes are there in the pile?1) 110 2) 120 3) 100 4) Data inadequate 5) None of these

62. How many cubes are without any colour?1) 20 2) 14 3) 16 4) 18 5) None of these

63. How many cubes are coloured on one face only?1) 33 2) 26 3) 28 4) 30 5) None of these

64. How many cubes are coloured on two faces only/1) 28 2) 30 3) 36 4) 32 5) None of these

65. How many cubes are coloured on three faces only?1) 14 2) 15 3) 16 4) 17 5) None of these

66. How many cubes are coloured on four faces only?1) 0 2) 1 3) 2 4) Data inadequate 5) None of these

67. Consider the following pictures of a dice:

36

2

26

4

65

4

41

2

What is the number opposite 1?1) 2 2) 3 3) 5 4) 6 5) Can’t sayDirections (Q. 68-72): Read the following information and answer the questions given below:i) Two wooden cubes ‘A’ and ‘B’ are placed adjacent to each other in front of you in such a way that ‘A’ is to your left

and ‘B’ to your right.ii) One pair of opposite faces of cube ‘A’ is painted by the same colour i.e. Red colour. Another pair of opposite faces is

painted by Blue and one of the remaining faces is Yellow and other one is Violet.iii) Only two opposite faces of cube ‘B’ are painted by Blue colour. Remaining pairs of opposite faces are painted in such

a way that opposite face of Brown colour is Green and one of the other two opposite faces is Black and the other isWhite.

68. If Red surface of ‘A’ and Blue of ‘B’ are touching the table and Yellow of ‘A’ and Black of ‘B’ are facing you, then whichcoloured side of ‘B’ is facing Blue side of ‘A’?1) Brown 2) Green 3) White4) Either Brown or Green 5) None of these

69. If Black surface of ‘B’ is kept on the top Red surface of ‘A’, which coloured side of ‘B’ will face the sky?1) White 2) Blue 3) Brown4) Data inadequate 5) None of these

70. If the cubes are rearranged one above the other in such a way that White face of ‘B’ is facing sky and Yellow face of ‘A’is kept above it, then which coloured surface of ‘A’ will be facing you?1) Violet 2) Blue 3) Either Blue or Red4) Either Blue or Violet 5) Data inadequate

CUBES AND CUBOID

71. If ‘B’ is kept to your left with Green coloured surface facing you and ‘A’ kept at your right with Blue surface facing you,then which of the following pairs of colours of ‘A’ and ‘B’ will be facing each other?1) Yellow-Black 2) Yellow-White 3) Black-Violet 4) Violet-White 5) Data inadequate

72. If block ‘B’ is kept behind block ‘A’ in such a way that Brown coloured surface of ‘B’ is facing Yellow coloured face of ‘A’,which colour of block ‘B’ will be to your right?1) Blue 2) Black 3) Brown4) Data inadequate 5) None of these

Directions (Q. 73-76): Read the following informations and answer the questions based on it.I. The length, breadth and height of a rectangular piece of wood are 4 cm, 3 cm and 5 cm respectively.II. Opposite sides of 5 cm × 4 cm piece are coloured in red.III. Opposite sides of 4 cm × 3 cm are coloured in blue.IV. Rest sides of 5 cm × 3 cm are coloured on green in both sides.V. Now the piece is cut in such a way that cubes of 1 cm × 1 cm × 1 cm will be made.

73. How many cubes shall have all the three colours?1) 8 2) 10 3) 12 4) 14 5) None of these

74. How many cubes shall not have any colour?1) No any 2) 2 3) 4 4) 6 5) None of these

75. How many cubes shall have only two colours red and green on their two sides?1) 8 2) 12 3) 16 4) 20 5) None of these

76. How many cubes shall have only one colour?1) 12 2) 16 3) 22 4) 28 5) None of these

Directions (Q. 77-78): Read the following information and answer the questions based on it.I. The length, breadth and height of a rectangular piece of wood are 12 cm, 11 cm and 9 cm respectively.II. The surfaces of size 12 × 11, 12 × 9 and 11 × 9 are coloured by red, blue and green respectively.III. Now they are cut in such a way that cubes of 1 cm × 1 cm × 1 cm are made and no part of the rectangular wood goes

waste.77. How many cubes have only two sides painted?

1) 52 2) 26 3) 104 4) 102 5) None of these78. How many cubes have no side painted?

1) 638 2) 700 3) 900 4) 630 5) None of theseDirections (Q. 79-81): The six faces of a cube are coloured, each with a different colour.I. The white face is between yellow and green.II. The red face is adjacent to brown.III. The green face is opposite the yellow side.IV. The blue face is adjacent to red.V. The yellow face is the top face of the cube.

79. The faces adjacent to white bear the colours1) Yellow, green, brown and red 2) Yellow, brown, blue and green3) Yellow, green, blue and red 4) Can't be determined5) None of these

80. The face opposite the red face is1) green 2) white 3) blue 4) brown 5) None of these

81. The colour of the bottom face of the cube is1) red 2) brown 3) green 4) blue 5) None of theseDirections (Q. 82-95): A solid cube of each side 15 cm has been painted green, blue and yellow on pairs of opposite faces.

It is then cut into cubical blocks of each side 3 cm.82. How many cubes have no face painted?

1) 27 2) 54 3) 36 4) 50 5) 1883. How many cubes have only one face painted?

1) 27 2) 54 3) 36 4) 50 5) 18

84. How many cubes have only two faces painted?1) 27 2) 54 3) 36 4) 50 5) 12

85. How many cubes have only three faces painted?1) 16 2) 18 3) 12 4) 8 5) 6

86. How many cubes have three faces painted with different colours?1) 18 2) 4 3) 16 4) 12 5) 8

87. How many cubes have two faces painted green and blue and all other faces unpainted?1) 12 2) 16 3) 18 4) 8 5) 24

88. How many cubes have two faces painted blue and yellow and all other faces unpainted?1) 16 2) 12 3) 18 4) 8 5) 24

89. How many cubes have two faces painted yellow and green and all other faces unpainted?1) 12 2) 16 3) 18 4) 8 5) 24

90. How many cubes have one face painted blue and other faces unpainted?1) 12 2) 16 3) 18 4) 8 5) 24

91. How many cubes have one face painted yellow and other faces unpainted?1) 12 2) 16 3) 18 4) 8 5) 24

92. How many cubes have one face painted green and other faces unpainted?1) 12 2) 16 3) 18 4) 8 5) 24

93. How many cubes have atleast one face blue?1) 27 2) 54 3) 36 4) 50 5) 60

94. How many cubes have atleast one face green?1) 27 2) 54 3) 36 4) 60 5) 50

95. How many cubes have atleast one face yellow?1) 27 2) 54 3) 50 4) 36 5) 60Directions (Q. 96-111): Read the following information and answer the questions based on it.I. The length, breadth and height of a rectangular wooden block are 10 cm, 8 cm and 6 cm respectively.II. The two opposite surfaces of 10 cm × 8 cm are coloured from outside by red.III. The two opposite surfaces of 8 cm × 6 cm are coloured from outside by blue.IV. The remaining two surfaces of 10 cm × 6 cm are coloured from outside by green.V. Now, the block is cut in such a way that cubes of 1 cm × 1 cm × 1 cm are made.

96. What will be the total number of cubes when block is cut in such a way that cubes of 1 cm × 1 cm × 1 cm are made?1) 380 2) 48 3) 480 4) 188 5) 72

97. How many cubes shall have all the three colours?1) 4 2) 8 3) 12 4) 16 5) 20

98. How many cubes shall not have any colours?1) 72 2) 208 3) 576 4) 384 5) 192

99. How many cubes shall have only one colour?1) 208 2) 192 3) 72 4) 416 5) 104

100. How many cubes shall have only two colours?1) 144 2) 208 3) 104 4) 72 5) 36

101. How many cubes shall have one face painted red and all other faces unpainted?1) 48 2) 64 3) 32 4) 96 5) 24

102. How many cubes shall have one face painted blue and all other faces unpainted?1) 64 2) 48 3) 32 4) 96 5) 24

103. How many cubes shall have one face painted green and all other faces unpainted?1) 64 2) 48 3) 32 4) 96 5) 24

104. How many cubes have two faces painted red and blue and all other faces unpainted?1) 16 2) 24 3) 32 4) 96 5) 48

105. How many cubes have two faces painted blue and green and all other faces unpainted?1) 32 2) 24 3) 48 4) 64 5) 16

106. How many cubes have two faces painted green and red and all other faces unpainted?1) 32 2) 24 3) 48 4) 64 5) 16

107. How many cubes have atleast one face painted red?1) 152 2) 160 3) 136 4) 128 5) 62

108. How many cubes have atleast one face painted red and one face painted blue?1) 16 2) 24 3) 32 4) 40 5) 96

109. How many cubes have atleast one face painted blue and one face painted green?

1) 24 2) 16 3) 32 4) 40 5) 96110. How many cubes have atleast one face painted blue?

1) 88 2) 48 3) 80 4) 96 5) 160111. How many cubes have atleast one face painted green?

1) 104 2) 112 3) 120 4) 88 5) 96Directions (Q. 112-123): A solid cuboid has been painted red, blue and green on the pairs of opposite faces. The cuboid

is then cut into 198 smaller cubes such that 192 cubes are of the same size while 6 others are of bigger size but all the sixcubes are of same size. The cuboid has dimension 10 cm × 6 cm × 4 cm. Cubes of bigger sizes has been cut from the middleof the cuboid.

If two opposite surfaces of cuboid 10 cm × 6 cm are coloured from outside by red, two opposite surfaces 6 cm × 4cm are coloured from outside by blue and rest two surfaces are coloured green.

Then answer the following questions.112. How many cubes have at least one face painted red?

1) 102 2) 48 3) 68 4) 90 5) 52113. How many cubes have only one face painted?

1) 52 2) 8 3) 90 4) 80 5) 60114. How many cubes have only two faces painted?

1) 102 2) 52 3) 90 4) 68 5) 48115. How many cubes have only three faces painted?

1) 4 2) 10 3) 12 4) 8 5) None of these116. How many cubes have no faces painted?

1) 68 2) 102 3) 90 4) 52 5) 48117. How many cubes have at least one face painted blue?

1) 102 2) 68 3) 48 4) 52 5) None of these118. How many cubes have two or more faces painted?

1) 90 2) 102 3) 604) 52 5) None of these

119. How many cubes have at last one face painted green?1) 52 2) 68 3) 1024) 48 5) None of these

120. How many cubes have one or more faces painted?1) 142 2) 150 3) 1584) 60 5) None of these

121. How many cubes have at least two faces painted by red and blue?1) 24 2) 16 3) 84) 48 5) None of these

122. How many cubes have at least two faces painted by blue and green?1) 16 2) 8 3) 244) 32 5) None of these

123. How many cubes have at least two faces painted by green and red?1) 36 2) 28 3) 84) 44 5) None of theseDirections (Q. 124-128): These questions are based on the following information:A cube of side 10 cm is coloured red with a 2 cm wide green strip along all the sides on all the faces. The cube is cut into

125 smaller cubes of equal size.Now answer the following questions based on the above information.

124. How many cubes have three green faces each?1) 0 2) 4 3) 6 4) 8 5) None of these

125. How many cubes have one face red and adjacent face green?1) 0 2) 4 3) 6 4) 8 5) None of these

126. How many cubes have at least one face coloured?1) 25 2) 50 3) 98 4) 100 5) None of these

127. How many cubes are without any colour?1) 0 2) 1 3) 8 4) 27 5) None of these

128. How many cubes have at least two green faces each?1) 8 2) 44 3 ) 63 4) 71 5) None of theseDirection (Q. 129-130): Below is given picture of two cubes. Look at the two cubes carefully assuming the picture to

be correct and answer the questions following it.Here is a picture of two cubes:

# 1 # 2

1) The two cubes are exactly alike.2) The hidden faces indicated by the arrows have the same figure on them.

129. Which figure — , , or — is on the faces indicated by the arrows # 1 and #2?130. Which figure among , , or occurs twice on the cube?131. Study the given figures to answer the questions:

All four layers are exactly same except that each layer is rotated by 90° clockwise from top to bottom in each step.Which of the following statements is true regarding the diagram and information given above?(i) Number of total cubes is less than 26.(ii) Number of total cuboids is greater than 70.(iii) The ratio of cubes to cuboids is 1 : 3.(iv) The number of cubes is greater than the no. of cuboids.

(61-66):

4

4

3

3

3

3

3

3

3

3

3

3

3

22

2222

2 22

2

2

22

22

2

22

22

2

1 1 1

1 1

1

3

23

3

Stack no

2

34

5

1

Note: The numbers on each cube denote the no. of coloured faces.61. 3; Total such cubes = 5((6 + 5 + 4 + 3 + 2) = 10062. 463. 1; 12 from stack 1; 9 from stack 2; 7 from stack 3; 5 from stack 4;64. 4; 11 from stack 1; 5 from stack 2; 5 from stack 3; 5 from stack 4; 6 from stack 5.65. 2; From stack 1; 3 from upper row, 2 from bottom row.

2 : two from upper row3 : two from upper row4 : two from upper row5 : two from upper row and two from bottom row.

66. 3; Obvious from the figure.67. 4(68-72):

68. 4 69. 1 70. 3 71. 5 72. 4(73-76):

73. 1; Three surfaces coloured is constantly 8.74. 4; No. surface coloured = (l - 2) (b - 2) (h - 2) = 3 × 1 × 2 = 675. 2; There are three cubes on each red-green interface (barring corner cubes). So, 4 × 3 = 12 cubes.76. 3; One surface coloured

= 2(l - 2) (b - 2) + 2 (l - 2) (h - 2) + 2 (b - 2) (h - 2) = 2 { 3 × 1 + 3 × 2 + 1 × 2} = 22(77-78):77. 3; Required number of cubes = {(12 - 2) + (11 - 2) + (9 - 2)} × 4 = 26 × 4 = 104

Note that such cubes will be obtained from the edges but excluding the cubes at the corners.78. 4; Such cubes will be obtained from inner part of the rectangular piece of wood.

Required number of cubes = (12 - 2) × (11 - 2) × (9 - 2) = 10 × 9 × 7 = 630(79-81): From the given information we deduce red face is opposite to white face, yellow face is opposite to green face, and

brown face is opposite to blue face.79. 2 80. 281. 3; If yellow face is at the top then green will be at the bottom because yellow face and green face are opposite to each

other.(82-95): Let 3 cm = 1 unit,

then 15 cm = 5 unitsNumber of total cubes = 5 × 5 × 5 = 125Number of cubes only one face painted comprises three types of cubes ie cubes having one face green, one face yellowand the cubes having one face blue.Cubes having only one face green = 2(n - 2)2 = 2(5 - 2)2 = 18.Where n = side of cubeCubes having one face blue =2(n - 2)2 = 2 × 3 × 3 = 18Cubes having one face yellow = 2(n - 2)2 = 2 × 3 × 3 = 18Cubes having only one face painted are 18 + 18 + 18 = 54Cubes of only two faces painted comprises three types of cube1st type: one face blue and one face yellow

4(5 - 2) ⇒ 4(n - 2) = 4 × 3 = 122nd type: one face yellow and one face green

4(n - 2) 4(5 - 2) = 4 × 3 = 123rd type: one face green and one face blue

4(n - 2) 4(5 - 2) = 4 × 3 = 12Cubes having only two faces painted are 12 + 12 + 12 = 36Cubes which have no any face painted are (n - 2) (n - 2) (n - 2)

= (5 - 2) (5 - 2) (5 - 2) = 27 [where n is the side of a cube.]82. 1 83. 2 84. 3 85. 4 86. 5 87. 1 88. 2 89. 1 90. 3 91. 3 92. 393. 4; Cubes having only one face painted blue are 18. Cubes having one face blue and one face green are 12.

Cubes having one face blue and one face yellow are 12.Cubes having painted with three colours are 8. Hence total such cubes are 18 + 12 + 12 + 8 = 50.or 2 × 52 = 50

94. 5 95. 3(96-111): We have, according to the question,

10 cm × 8 cm faces are coloured red.8 cm × 6 cm faces are coloured blue.10 cm × 6 cm faces are coloured green.Total number of cubes = 10 × 8 × 6 = 480Total number of cubes which have one face painted is

(i) red 2(10 - 2) (8 - 2) = 96.(ii) blue 2(8 - 2) (6 - 2) = 48.(iii) green 2(6 - 2) (10 - 2) = 64.Hence 96 + 48 + 64 = 208Total number of cubes which have only two faces painted by(1) red and blue 4(8 - 2) = 24.(2) blue and green 4(6 - 2) = 16.(3) green and red 4(10 - 2) = 32.Hence, 24 + 16 + 32 = 72.Total number of cubes which have no any faces painted, are

(10 - 2) (8 - 2) (6 - 2) = 192Cubes which have only three faces painted by three different colours are 8.

96. 3 97. 2 98. 5 99. 1 100. 4 101. 4 102. 2 103. 1 104. 2 105. 5 106. 1107. 2; [96 + 24 + 32 + 8] or [2 × 10 × 8]108. 3; [24 + 8]109. 1; [16 + 8]110. 4; [48 + 16 + 24 + 8] or [2 × 8 × 6]111. 3; [64 + 16 + 32 + 8] or [2 × 10 × 6](112-123): Here, the number of total cubes = 198

Cubes which have only one face painted byRed = 50; Blue = 16; Green = 24Hence, 50 + 16 + 24 = 90Cubes which have only two faces painted byRed + Blue = 16Blue + Green = 8Green + Red = 28Hence total number of such cubes = 16 + 8 + 28 = 52Cubes which have no any faces coloured, are 24 + 24 = 48.10 cm × 6 cm = red.6 cm × 4 cm = blue.10 cm × 4 cm = green.

112. 1; 50 + 16 + 28 + 8 = 102113. 3; 50 + 16 + 24 = 90114. 2; 16 + 8 + 28 = 52115. 4; 8116. 5; 12 × 4 = 48117. 3; 16 + 16 + 8 + 8 = 48118. 3; 52 + 8 = 60119. 2; 24 + 8 + 28 + 8 = 68120. 2; 90 + 8 + 52 = 150121. 1; 16 + 8 = 24122. 1; 8 + 8 = 16123. 1; 28 + 8 = 36(124-128):

3

3

3 3

3

3 32

22

222

22

2

2 2 22

222 2 2

2

22

2

22

2 2 2

11

1

11

11

11

1 1 1111

1 1 1

1 1 1

1 1 11 1 1

Note: Number assigned to smaller cubes indicates the number of coloured surfaces of each cube. Shaded portion indi-cates red colour whereas unshaded portion indicates green colour.

124. 4125. 1; Such combination is not possible.

126. 3; 125 - (5 - 2)3 = 98[(5 - 2)3 is the no. of those cubes which comprises no coloured surfaces]

127. 4; (5 - 2)3 = 27128. 2; 12(5 - 2) + 8 = 44(129-130):129.130. and

From (1) and the pictured cubes, at least one of and occurs twice.If both occur twice, then the identical cubes look like:

But in this case duplicate faces cannot occur at the arrows, contradicting (2).So either only occurs twice or only occurs twice.If only occurs twice, then the identical cubes look like this:

But in this case duplicate faces cannot occur at the arrows, contradicting (2).So only occurs twice and the identical cubes look like this:

In this the occurs on the right cube at the arrow #2. So, from (2), the occurs on the left cube at the arrow # 1.Thus is on the faces indicated by the arrows. (The occurs on the unmarked face in the last diagram.)

131. (ii); Case I:

A BDC

E FG

L1

No. of cubes in each layer = 6(A, B, C, D, E and F)No. of cuboids in each layer = [ , G, (A + B), (C + D), (E + F) (A + C), (B + D), (C + E), (D + F), (A + C + E), (B + D +F), {(A + C + E) + (B + D + F)} {(B + D + F) + G}, {(A + B) + (C + D), {C + D) + (E + F)}] = 15Hence the four layers have = (6 × 4 =) 24 cubes and (15 × 4 =) 60 cuboids,Case II:Now, consider two layers together:

A BDC

Cuboid

Except the dotted part no combination will give any desired shape.Now, consider the dotted part

A BDC

E F

G

SH

No. of cubes = 1(S)No. of cuboids = [C + D + E + F), (A + B + G + H), (D + B + F + G),

(A + C + E + H)] = 4Now, it can be concluded that for each pair of two layers there are (1 × 3 =) 3 cubes and (4 × 3 =) 12 cuboids [As 3 pairsof two layers can be made.].Case III. When three layers are considered together.

AB

Cuboid

Only 3 cuboids (1 itself + 2 cuboids, each made of three 3 cubes) is present here.Only two cuboids, considering of three layers is possible.Hence 3 × 2 = 6 cuboids.Now, case I + II + III, we get (60 + 12 + 6 = ) 78 cuboidsand (24 + 3 =) 27 cubes.