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Cubes Questions for CAT

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94 easy89 medium97 hard

20 free Cubes questions

1EasyPassage, 4 questions

A solid rectangular block is built from unit cubes with dimensions 5 x 3 x 2. Cubes on the surface have some faces exposed; the painted-cube analogy classifies them by the number of exposed faces.

Question 1

How many unit cubes are used in the block?

  1. A

    25

  2. B

    31

  3. C

    30

  4. D

    60

Question 2

What is the surface area of the block?

  1. A

    60

  2. B

    31

  3. C

    70

  4. D

    62

Question 3

How many unit cubes have exactly 3 faces exposed (the corners)?

  1. A

    12

  2. B

    8

  3. C

    4

  4. D

    6

Question 4

How many unit cubes have exactly 2 faces exposed (the non-corner edge cubes)?

  1. A

    16

  2. B

    12

  3. C

    20

  4. D

    8

Show solution
2EasyPassage, 4 questions

A cuboid of dimensions 5 x 4 x 3 is painted on all six faces and then cut into unit cubes. For a cuboid the edges come in three groups of four (lengths 5, 4, 3) and the faces in three pairs of different sizes, so each category must account for the three distinct dimensions.

Question 1

How many unit cubes are produced in total?

  1. A

    60

  2. B

    12

  3. C

    48

  4. D

    30

Question 2

How many cubes have exactly 2 painted faces (the non-corner edge cubes)?

  1. A

    16

  2. B

    24

  3. C

    20

  4. D

    12

Question 3

How many cubes have exactly 1 painted face?

  1. A

    18

  2. B

    20

  3. C

    22

  4. D

    24

Question 4

How many cubes have no painted face?

  1. A

    8

  2. B

    12

  3. C

    4

  4. D

    6

Show solution
3MediumPassage, 4 questions

Three views of the same cube (faces numbered 1 to 6) are given, each showing the top, front and right faces. View 1: top 1, front 2, right 3. View 2: top 1, front 4, right 2. View 3: top 5, front 1, right 3. Two faces are opposite if they never appear together in any view.

Question 1

Across the three views, which faces is face 1 shown sharing a view with (i.e. adjacent to)?

  1. A

    2, 3, 4 and 5

  2. B

    2, 3 and 6

  3. C

    all other faces

  4. D

    4, 5 and 6

Question 2

Which face is opposite to 1?

  1. A

    5

  2. B

    6

  3. C

    4

  4. D

    2

Question 3

Which face is opposite to 2?

  1. A

    4

  2. B

    6

  3. C

    5

  4. D

    1

Question 4

Which face is opposite to 3?

  1. A

    5

  2. B

    6

  3. C

    2

  4. D

    4

Show solution
4MediumPassage, 4 questions

A cube is to be divided with axis-aligned cuts, distributed across its three dimensions as a, b, c (so a + b + c is the total number of cuts). The number of pieces is (a+1)(b+1)(c+1), and we want to maximise the pieces.

Question 1

With 7 cuts split as nearly equal as possible (3, 2, 2), how many pieces result?

  1. A

    32

  2. B

    40

  3. C

    36

  4. D

    30

Question 2

If the same 7 cuts were split as (3, 3, 1), how many pieces result?

  1. A

    36

  2. B

    30

  3. C

    28

  4. D

    32

Question 3

Which split of 7 cuts maximises the number of pieces?

  1. A

    The most balanced split (3, 2, 2)

  2. B

    All on one dimension (7, 0, 0)

  3. C

    (5, 1, 1)

  4. D

    (4, 3, 0)

Question 4

What is the maximum number of pieces from 8 axis-aligned cuts?

  1. A

    45

  2. B

    48

  3. C

    27

  4. D

    40

Show solution
5HardPassage, 4 questions

A solid 3 x 3 x 3 cube has a straight 1 x 1 tunnel drilled completely through its centre, from one face to the opposite face (removing the 3 central cubes along that axis). The tunnel is open at both ends. Total surface area = external surface + internal tunnel surface.

Question 1

How many unit cubes remain?

  1. A

    24

  2. B

    26

  3. C

    21

  4. D

    27

Question 2

What is the external surface area after drilling (each open end removes one unit square)?

  1. A

    54

  2. B

    50

  3. C

    52

  4. D

    48

Question 3

What is the internal surface area of the tunnel walls?

  1. A

    9

  2. B

    6

  3. C

    8

  4. D

    12

Question 4

What is the total surface area?

  1. A

    60

  2. B

    64

  3. C

    54

  4. D

    66

Show solution
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