This document contains 34 pages about cubes, dice, and logical reasoning puzzles. It includes the basics of cubes and dice, examples of cutting painted cubes into smaller cubes and calculating the resulting number and configurations, examples of dice problems calculating totals or numbers on hidden faces, and puzzles involving chess boards and stacked cubes. Formulas are provided for calculating cubes cut from larger painted cubes.

1. www.georgeprep.com
1
Cubes and Dice
Logical Reasoning

2. www.georgeprep.com
2
Basics of Cubes
 3D figure or Object
 All sides are equal
 6 Surfaces
 8 Corners
 12 Edges

3. www.georgeprep.com
3
Painted Cubes
A cube having a side of 6 cm is painted red on all the
faces and then cut into smaller cubes of 1 cm each.
Find the total number of smaller cubes so obtained.
Answer : 216
Formula : (
𝑎
𝑏
)3

4. www.georgeprep.com
4
Painted Cubes
A cube having a side of 6 cm is painted red on all the
faces and then cut into smaller cubes of 1 cm each.
Find the least number of cuts needed.
Answer : 15
Formula : (6-1)*3 = 15

5. www.georgeprep.com
5
Painted Cubes
A cube having a side of 6 cm is painted red on all the
faces and then cut into smaller cubes of 1 cm each.
How many cubes are painted Red on all the 3 faces?
Answer : 8
Formula : 8 Corners

6. www.georgeprep.com
6
Painted Cubes
A cube having a side of 6 cm is painted red on all the
faces and then cut into smaller cubes of 1 cm each.
How many cubes are painted Red on 2 faces?
Answer : 12*4=48
Formula : 12(n-2), along the edges

7. www.georgeprep.com
7
Painted Cubes
A cube having a side of 6 cm is painted red on all the
faces and then cut into smaller cubes of 1 cm each.
How many cubes are painted Red on only 1 face?
Answer : 16*6 = 96 (on all faces)

8. www.georgeprep.com
8
Painted Cubes
A cube having a side of 6 cm is painted red on all the
faces and then cut into smaller cubes of 1 cm each.
How many cubes are not painted at all?
Answer : 4*4*4 = 64

9. www.georgeprep.com
9
Painted Cubes
 Number of cubes with 0 side painted= (𝑛 − 2)3
 Number of cubes with 1 sides painted =6(𝑛 − 2)2
 Number of cubes with 2 sides painted=12(𝑛 − 2)1
 Number of cubes with 3 sides painted= 8

10. www.georgeprep.com
10
Painted Cuboids
 For a cuboid of dimension a*b*c painted on all sides
which is cut into smaller cubes of dimension 1*1*1
 Number of cubes with 0 side painted= (a-2) (b-2) (c-2)
 Number of cubes with 1 sides painted =2[(a-2) (b-2) +
(b-2)(c-2) + (a-2)(c-2) ]
 Number of cubes with 2 sides painted= 4(a+b+c-6)
 Number of cubes with 3 sides painted= 8

11. www.georgeprep.com
11
Problems
A solid cube of each side 8 cm, has been painted red,
blue and black on pairs of opposite faces. It is then
cut into cubical blocks of each side 2 cm.
How many cubes are there in all?
Answer : 64

12. www.georgeprep.com
12
Problems
A solid cube of each side 8 cm, has been painted red,
blue and black on pairs of opposite faces. It is then
cut into cubical blocks of each side 2 cm.
Find the least number of cuts needed.
Answer : 9

13. www.georgeprep.com
13
Problems
A solid cube of each side 8 cms, has been painted
red, blue and black on pairs of opposite faces. It is
then cut into cubical blocks of each side 2 cms.
How many cubes have three faces painted?
Answer : 8

14. www.georgeprep.com
14
Problems
A solid cube of each side 8 cms, has been painted
red, blue and black on pairs of opposite faces. It is
then cut into cubical blocks of each side 2 cms.
How many cubes have two faces painted?
Answer : 12 × 2 = 24

15. www.georgeprep.com
15
Problems
A solid cube of each side 8 cms, has been painted
red, blue and black on pairs of opposite faces. It is
then cut into cubical blocks of each side 2 cms.
How many cubes have one face painted?
Answer : 6 × 4 = 24

16. www.georgeprep.com
16
Problems
A solid cube of each side 8 cms, has been painted
red, blue and black on pairs of opposite faces. It is
then cut into cubical blocks of each side 2 cms.
How many cubes have no face painted?
Answer : 8

17. www.georgeprep.com
17
Problems
A solid cube of each side 8 cms, has been painted
red, blue and black on pairs of opposite faces. It is
then cut into cubical blocks of each side 2 cms.
How many cubes have all the faces painted?
Answer : 0

18. www.georgeprep.com
18
Problems
A solid cube of each side 8 cms, has been painted
red, blue and black on pairs of opposite faces. It is
then cut into cubical blocks of each side 2 cms.
How many cubes have three faces painted with
different colours?
Answer : 8

19. www.georgeprep.com
19
Problems
A solid cube of each side 8 cms, has been painted
red, blue and black on pairs of opposite faces. It is
then cut into cubical blocks of each side 2 cms.
How many cubes have two faces painted red and
black, and all other faces unpainted?
Answer : 8

20. www.georgeprep.com
20
Problems
A solid cube of each side 8 cms, has been painted
red, blue and black on pairs of opposite faces. It is
then cut into cubical blocks of each side 2 cms.
How many cubes have only one face painted red and
all other faces unpainted?
Answer : 8

21. www.georgeprep.com
21
Problems
A solid cube of each side 8 cm, has been painted red,
blue and black on pairs of opposite faces. It is then
cut into cubical blocks of each side 2 cm.
How many cubes have two faces painted black?
Answer : 0

22. www.georgeprep.com
22
Problems
A solid cube of each side 8 cms, has been painted
red, blue and black on pairs of opposite faces. It is
then cut into cubical blocks of each side 2 cms.
How many cubes have one face painted blue and one
face painted red? (the other faces may be painted or
unpainted)
Answer : 16

23. www.georgeprep.com
23
Basics of Dice
Two Types:
Standard Dice :
Type of dice in which the sum of opposite sides is 7
Ordinary Dice:
Type of dice in which the sum of opposite sides is not 7

24. www.georgeprep.com
24
Standard Die
Face of a die
Opposite
face
Adjacent
faces
1 6 2,3,4,5
2 5 1,3,4,6
3 4 1,2,5,6

25. www.georgeprep.com
25
Problems
For a standard dice thrown the sum of all the visible
faces is 16. Fine the number on the face that is
touching the ground.
Answer: 5

26. www.georgeprep.com
26
Problems
Four regular dice are thrown on the ground, the total
of numbers on the top faces of these four dice is 13
as the top faces showed 4, 3, 1 and 5 respectively.
What is the total of the faces touching the ground?
1) 15
2) 14
3) 11
4) Cannot be determined
Answer : 15

27. www.georgeprep.com
27
Problems
Which of the following can be a standard cube?
Answer : 2

28. www.georgeprep.com
28
Possibilities
 When a single die is rolled, the no. of possibilities
is 6.
 When two dice are rolled together, the no. of
possibilities is 6*6=36
 When three dice are rolled together, the no. of
possibilities is 6*6*6= 216.

29. www.georgeprep.com
29
Problems
A dice is numbered from 1 to 6 in different ways. If 1
is adjacent to 2, 4 and 6, then which of the following
statements is necessarily true?
1) 2 is opposite to 6
2) 1 is adjacent to 3
3) 3 is adjacent to 5
4) 3 is opposite to 5
Answer: 3 is adjacent to 5.

30. www.georgeprep.com
30
Problems
What will be the number at the bottom, if 5 is at the
top; the two positions of the dice being as given
below:
Answer : 2

31. www.georgeprep.com
31
Problems
Which colour is opposite to Yellow?
Answer : Violet

32. www.georgeprep.com
32
Problems on stacked cubes
How many cubes are in the stack?
Answer : 56

33. www.georgeprep.com
33
Problems on stacked cubes
How many cubes are in the stack? (SSC Assistant, 2012)
Answer : 9+5+1=15

34. www.georgeprep.com
34
Puzzle
What is the total number of squares on a chess board?
Answer : 204