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.
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.