Cubes and Dice

Cube and Dice

Cube and Dice

CUBE AND DICE TIPS AND TRICKS

       A cube is a three-dimensional solid object bounded by six sides, with three meeting at each vertex. It features all right angles and a height, width and depth that are all equal ( length = width = height). It has two types:

       1. Standard Cube; and 2. Non-Standard Cube

Important Facts:

       1. A cube has 6 square facesor sides. (Ref. Img 1)

       2. A cube has 8 points (vertices). (Ref. Img 1)

       3. A cube has 12 edges. (Ref. Img 1)

       4. Only 3 sides are visible at a time (called "Joint Sides") and these joint sides can never be on opposite side to each other.

       5. Things that are shaped like a cube are often referred to as ‘Dice’.

       6. Most dice are cube shaped, featuring the numbers 1 to 6 on the different faces.

       7. Addition of number of dots (pips) or numbers from opposite sides of a standard cube or dice is always 7.

       8. Total of two adjacent faces of cube can never be a 7.

       9. 11 different ‘nets’ can be made by folding out the 6 square faces of a cube. (Ref. Img 2)

(Image 1)

       * We can categorise a cube (or a colour cube) after cutting it, in these four categories: (See Image 3)

Types of Problems Based on Cube and Dice:

Tricks and Examples

Question Type 1 : Determining the opposite sides

Question Type 2 : Cutting a Colorful Cube

Question Type 3 : Making big cube by adding small cubes

Question Type 4 : Determining number of cubes placed in stacks

Examples : Question Type 1

Que. 1: This cube is a 'standard cube'. What will be the number on opposite faces of it?

1. Opposite to 1 - ?

2. Opposite to 2 - ?

3. Opposite to 3 - ?

Solution: We know the rule of standard cube - "Addition of number of dots (pips) or numbers from opposite sides of a standard cube or dice is always 7." Hence, the rule = 7-N (N stands for number on facing side). 

1. Opposite to 1 = (7-1) = 6

2. Opposite to 2 = (7-2) = 5

3. Opposite to 3 = (7-3) = 4

Que. 2: Study these cubes and find out the numbers on opposite sides of front facing sides of these.

Solution: To solve this question we'll follow this rule - "Only 3 sides are visible at a time (called "Joint Sides") and these joint sides can never be on opposite side to each other."

* From cube A) and B) - 1, 2, 3, 4 and 5 can never be on opposite side of 3 (common number in cube A & B). Hence the answer will be = 6

* From cube B) and C) - 1, 3, 4, 6, and 5 can never be on opposite side of 5 (common number in cube B & C). Hence the answer will be = 2

* From cube A) and C) - 1, 2, 3, 5 and 6 can never be on opposite side of 1 (common number in cube A & C). Hence the answer will be = 4

Conclusion : 

Opposite to 1 = 4

Opposite to 3 = 6

Opposite to 5 = 2

Examples: Question Type 2

Que : Directions: (Questions 1 to 10) 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.
1. How many cubes have no face painted?

A) 0                                             B) 4

C) 8                                             D) 12

Ans: Cubes have no face painted = Inner Cubes (No Colour): We can find out the total number of cubes without any colour on any side (inner cube) with this formula: (X-2)^3
Implementation of formula: X = 4

(4-2)^3 = 2^3 = 8

2. How many cubes have only one face painted?

A) 8                                             B) 16

C) 24                                           D) 28

Ans: Cubes have only one face painted = Central cubes : In middle of faces & has only one coloured side.

We can find out the total number of cubes with singe colour on any side with this formula: 6(X-2)^2
Implementation of formula: X = 4

6(4-2)^2 = 6(2)^2 = 24

3. How many cubes have only two faces painted?

A) 8                                             B) 16

C) 20                                            D) 24

Ans: Cubes have only two faces painted = Middle Cubes: In middle of edges and have two coloured sides.

We can find out the total number of cubes with singe colour on any side with this formula: 12(X-2)
Implementation of formula: X = 4

12(4-2) = 12(2) = 24

4. How many cubes have only three faces painted?

A) 0                                             B) 4

C) 6                                              D) 8

Ans: Cubes have only three faces painted = Corner cubes : Cubes on corners and have three coloured sides.

A cube can have only 8 cut-corner cubes with colours on three sides. Hence answer will be always the same = 8.

5. How many cubes have three faces painted with different colours?

A) 0                                             B) 4

C) 8                                             D) 12

Ans: Cubes have three faces painted = Corner cubes : Cubes on corners and have three coloured sides.
A cube can have only 8 cut-corner cubes with colours on three sides. Hence answer will be always the same = 8.

6. How many cubes have two faces painted red and black and all other faces unpainted?

A) 4                                             B) 8

C) 16                                            D) 32

Ans: Cubes have two faces painted red and black and all other faces unpainted = 4+4 = 8
7. How many cubes have only one face painted red and all other faces unpainted?

A) 4                                             B) 8

C) 12                                            D) 16 

Ans: Cubes have only one face painted red and all other faces unpainted = Central Cubes of Red Face = 4+4 = 8

8. How many cubes have two faces painted black?

A) 2                                              B) 4

C) 8                                              D) None

Ans:  None

9. How many cubes have one face painted blue and one face painted red? (the other faces may be painted or unpainted?

A) 16                                           B) 12

C) 8                                             D) 0

Ans: Cubes have one face painted blue and one face painted red? (the other faces may be painted or unpainted = 4+4 = 8

10. How many cubes are there in all?

A) 64                                            B) 56

C) 40                                            D) 32

Ans: To find out total number of cubes we use this formula- (X)^3

Implementation of formula: X = 4

(4)^3 = 64

Examples: Question Type 3

Just Opposite to Question Type 2.

Examples: Question Type 4

Finding out the Number of Cubes in a Stack of Cubes

Que. How many cubes are there in this figure:

Ans. (Total numbers of cubes in a line x Number of stack / tower) + ................. 

                              In this example = (6x1)+(5x2)+(4x3)+(3x4)+(5x2)+(6x1) = 6+10+12+12+10+6 = 56
Explanation :

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