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**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 :