Cube and Dice Test

Cube and Dice Test  

Cube & Dice Problems, Aptitude Basics, Practice Questions, 

Answers and Explanations

Important Study Content for 11-plus or Eleven plus exam

Geometry of Cube
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 ‘cubic’.

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)

(Image 2)

:: Problem Solving ::

Things to remember before stepping ahead:

Image 3: Painted Sides of a Cube

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

a.) Central cube (Yellow): 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

b.) Middle Cube (Green): 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)

c.) Corner cube (Blue): 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.

d.) Inner Cube (No Colour): Cubes inside faces & has no coloured side.

We can find out the total number of cubes without any colour on any side (inner cube) with this formula: (X-2)^3

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

Types of Problems Based on Cube and Dice:

Question Type 1 : Determining the opposite sides

Question Type 2 : Cutting a Colorful Cube

Question Type 3 : Making big cube by addingsmall cubes

Question Type 4 : Determining number of cubes placed in stacks

"Elevenplus" / "Eleven Plus Practice Questions and Papers"

Examples: Question Type 1

Que. 1: This cube is a 'standard cube'. What will be the number on opposite faces of it? (नीचे दिया गया घन एक मानक घन है। बताइये कि इसकी विपरित फलकों पर क्या-क्या अंकित होगा?)

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. (दी गई आकृतियों का अध्ययन कर बताइये कि कौनसी संख्या किसके पीछे अंकित होगी?)

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) 1 2

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