** Cube **The length, breadth and height of a cube are identical, i.e. whose length, breadth, height is equal, it is called a cube.

There are always (a) Six pane (b) Twelve edges (c) Eight corners in a cube.

** Cuboid **The length, breadth and height of a cuboid is always different, and the other characteristics are similar to the cube. That means there are also six panels, twelve edges and eight corners.

**Determination of the cube based on the colors: **

It is located right in the center of each panel and only one surface is colored. In the picture below the central cube is displayed with the number 3.

It is located in the middle of every edge and its only two surfaces are colorful. The picture shows the middle cube with the number 1.

It is located on each corner and its three sides are colored. The figure shows the corner cube with the number 2.

is located in the middle of the central cube of each panel and its surface is not colored.

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**Important formula** for calculating the number of colorful or colorless panes

(i) Number of cubes with no colourless surface = (n-2)^{3}

(ii) Number of cubes with one colored surfaces = (n-2)^{2} × 6

(iii) Number of cubes with two colored surfaces = (n-2) × 12

(iv) The number of cubes having three colored pins = 8

where n = the number of the same cubical volumes in each column of each surface.

**Now checkout some examples** based on the above facts

**Type 1**

**⇒ instructions (example 1-4)** All the surfaces of a cube are painted with the same color and it has been cut in such a way that results in 125 small and equal cubes. Answer below questions on the basis of these information.

**1.** The number of the cubes with only one color?

(a) 48 (b) 54

(c) 42 (d) 64

**2. **The number of cubes, which will have only two colored surface ?

(a) 36 (b) 48

(c) 54 (d) 64

** 3. **How many cube are there, whose three surfaces are colored?

(a) 12 (b) 16 (c) 24 (d) 8

** 4. **How many cubes will there be, whose surface will not be colored?

(a) 16 (b) 24 (c) 27 (d) 36

** Solution: **here

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1. (b) We know that only one surface of Central Cube is colored, so if we find the number of Central Cube, then the number of cubes with one colored surface will be known.

∵ Central Cubes = 6 (n – 2)^{2 }= 6 (5-2)^{2 }= 6 × 9 = 54

So the total number of cubes with only one surface colored = 54

2. (a) We know that the middle cube are those whose two sides are colored, so if the number of middle cubes is determined, then total number of cubes with two colored surface will be known

∵ Middle Cube = 12 (n – 2) = 12 (5-2) = 12×3 = 36

Thus the total number of cubes with two surface colored = 36

3. (d) We know that the three surfaces are colored only at Corner Cube and the number of Corner Cube is always 8. Thus the total number of Three colored cubes = 8

4. (c) We know that any surface of the inner cube is not colored, so if the number of inner cubes is determined, then total number of cubes with no colored surface will be known.

∵ Inner cube = (n-2)^{3} = (5-2)^{3} = (3)^{3} = 27

Therefore, the cube, whose surface is not colored would be 27.

**Type 2 **

**⇒ instructions (example 1-5):** All surfaces of a 4 cm cube have been painted red and it was cut to convert into smaller cubes of 1 cm. Answer these questions on the basis of the above information.

** 1.** How many cube are there having no color on any surface?

(a) 27 (b) 8 (c) 36 (d) 6

** 2. **How many cube are there, whose three sides are painted?

(a) 16 (b) 12 (c) 8 (d) 24

** 3. **How many cube are there, whose only one surface is colorful?

(a) 24 (b) 28 (c) 32 (d) 12

** 4. **What is the number of such cubes, whose only two surfaces are painted?

(a) 26 (b) 24 (c) 22 (d) 16

** 5. **How many total cubes you will get from this cube ?

(a) 64 (b) 27 (c) 48 (d) 36

** Solution :** 1. (b) as here n = 4;

Number of cubes not being colored on any surface

= (n-2)^{3 }= (4-2)^{3} = 2^{3} = 8

2. (c) The total number of the three colored surfaces cubes is 8 because the number of cubes having 3 colored surfaces is always 8.

3. (a) Total number of cubes having only one colored surface= 6 (n – 2)^{2} = 6 (4-2)^{2} = 6×4 = 24

4. (b) Total number of two surface colored cubes = 12 (n – 2) = 12 (4-2) = 24

5. (a) Total number of cubes = (4)^{3} = 4 × 4 × 4 = 64

**Type 3**

**⇒ instructions (example 1-6):** Two opposite surfaces of a cube are reddish and all the remaining surfaces are painted green. Now it has been cut in such a way that 125 small and equal cubes can be made. Answer these questions on the basis of these information.

** 1. **How many cube are there, whose only one surface is colored and it will be red?

(a) 12 (b) 18 (c) 24 (d) 27

** 2.** What is the number of cubes, whose only surface will be colored and it will be green?

(a) 24 (b) 27 (c) 36 (d) 48

** 3.** How many cube are there, whose two sides will be colored in which one will be red and the other will be green?

(a) 24 (b) 27 (c) 36 (d) 48

** 4.** How many cube are there, whose two sides will be colored and both will be green?

(a) 8 (b) 12 (c) 16 (d) 20

** 5. **How many cube are there, whose surface will not be colored?

(a) 12 (b) 18 (c) 24 (d) 27

** 6. **How many cube would be there, whose two sides are colored and both will be red?

(a) 4 (b) 8 (c) 16 (d) not one

** Solution: **Here n = 3 / number of all small cubes

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1. (b) We have to find such cubes whose only one surface is red. Here we know that one surface is colored in Central Cube only. So let’s find the 3 Central Cube located on the two panels, then we will get the desired cube.

So the total number of Central Cube = (n – 2)^{2 }= 6 (5-2)^{2} = 6 × 9 = 54

Number of Central Cube located on six panels = 54

Number of Central Cube located on two panels = (54/6) × 2 = 18

Hence the number of cubes dyed with only one red surface = 18

2. (c) According to the above rules, the number of cubes dyed with only one surface green is 54/6 × 4 = 36 (here four surfaces are colored with green color), hence the number of intended cubes = 36

3. (a) Total Number of cubes whose only two surfaces are colored out of which one is red and second is = 2 × 4 × (5 – 2) = 8 × 3 = 24

4. (b) The total number of cubes having two surfaces painted with green color = 4 (n – 2) = 4 (5-2) = 4×3 = 12

5. (d) The total number of non-colored cubes = (n – 2)^{3} = (5-2)^{3} = (3)^{3} = 27

6. (d) Total number of cubes with two surfaces painted red colored

= 0 × (5 – 2) = 0 × 3 = 0 which means none

**⇒ Note:** When any two opposite panels and the remaining other four panels are colored with different colors, the total number of two surface cubes painted in two opposite panels will always be zero.

**Type 4 **

**⇒ instructions (example 1-10):** Two opposite surfaces of a solid cube are colored red, two opposite surfaces are colored blue and the remaining two opposite surfaces are colored with black, After cut it into 276 small, solid cubes of 7 cubic cm. Answer these questions on the basis of these information.

** 1.** How many cube are there whose three surfaces are colored with different colors?

(a) 10 (b) 8 (c) 6 (d) 12

** 2. **How many cube are there, whose three surfaces are painted?

(a) 4 (b) 6 (c) 8 (d) 10

** 3.** How many cube are there, whose only two surfaces are painted?

(a) 36 (b) 48 (c) 24 (d) 16

** 4. **How many cube are there, whose only one surface is painted?

(a) 144 (b) 88 (c) 96 (d) 102

** 5.** How many such cubes are there that no surface is painted?

(a) 81 (b) 64 (c) 7 (d) 81

**6. **How many such cubes are there, whose only one surfaces is painted of red color?

(a) 32 (b) 84 (c) 36 (d) 81

** 7. **How many cube are there, whose only two surfaces are painted in which one is red and the other is black?

(a) 12 (b) 16 (c) 24 (d) 28

** 8. **How many such cubes are there, whose two sides are colored with black color?

(a) 0 (b) 4 (c) 6 (d) 8

** 9. **How much is the length of each side before dividing the solid cube?

(a) 4 cm (b) 5 cm (c) 6 cm (d) 8 cm

** 10. **How many cube are there, whose one surface is blue and the second surface is red? (Other surface can be colored or painted without color)

(a) 24 (b) 28 (c) 26 (d) 12

** Solution:** here

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1. (b) We know that there are three surfaces in the corner cube and the number of the Corner Cube is always 8, as well as the opposite surfaces are colored with different colors, so each surface of the corner cube is colored with different Color , so the total number of cubes dyed with three different colors = 8

2. (c) The three surfaces of Corner Cube are colored so the number of cubes having three colored surfaces = 8

3. (b) We know that the two surfaces are colored in the middle cube

∴ middle cube = 12 (n – 2) = 12 (6-2) = 12×4 = 48

Therefore, the total number of cubes dyeing only two surfaces = 48

4. (c) We know that only one surface is colored in the Central Cube

∴ central cube = 6 (n – 2)^{2} = 6 (6-2)^{2} = 6×16 = 96

So the total number of cubes having one surface colored = 96

5. (b) We know that no surface of the inner cube is colored

∴ inner Cube = (n -2)^{3} = (6-2)^{3} = 64

Thus the total number of cubes not colored on any surface is 64 = 64

6. (a) We know that only one surface of Central Cube is colored but here only two opposite surfaces of the cube having red color , so the total number of cubes with only one surface painted red color

= 2 × (n – 2)^{2 }= 2 × (6-2)^{2}

= 2 × 4 × 4 = 32

7. (b) We know that only two surfaces are colored in the middle cube, but in the question only two opposite surfaces are red and two opposite surfaces are black, so the cube having two surfaces colored with one red and one black color.

Total number of cubes to occur = 2 × 2 × (n – 2) = 4 × (6-2) = 16

8. (a) The total number of cubes having only two surfaces that are painted in black color = 0 × (n – 2) = 0 because black is only on two opposite surfaces which do not touch the sides.

9. (c) Length of each arm

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10. (a) While the other surface is colored or colorless, the cube which has one surface red and one surface blue will be 24.

**Type 5 **

** Example:** A rectangular block whose dimension is 6 cm × 4 cm × 2 cm, if converted to small cubes of 1 cm in diameter, how many cubes will be available?

(a) 81 (b) 64 (c) 36 (d) 48

Solution: (d) Number of intended cubes = 6 × 4 × 2 = 48

**Type 6 **

**⇒ instructions (example 1-5):** Please carefully study the information given below and answer the questions based on these.

**(i)** There is a rectangular wooden block whose length is 6 cm, width is 4 cm and height is 2 cm.

**(ii)** Both sides, whose dimensions are 4 cm × 2 cm, have been colored with Blue color.

**(iii)** Both sides whose dimensions are 6 cm × 2 cm, colored with Black color.

**(iv)** Both sides, whose dimensions are 6 cm × 2 cm, colored with Red color.

**(v)** This block has been cut in such a way that it has been converted into equal cubes of size 1 cm.

** 1. **How many such cube are there, whose three sides are colored and three surfaces are not colored?

(a) 4 (b) 24 (c) 8 (d) 16

** 2. **How much is the total number of cubes?

(a) 24 (b) 48 (c) 96 (d) 12

** 3. **How many cube are there, whose only two surfaces are painted?

(a) 8 (b) 16 (c) 24 (d) 32

** 4. **How many cube are there which contains all three colors ?

(a) 4 (b) 8 (c) 16 (d) 32

** 5. **How many cube are there, whose two surfaces have black color?

(a) 4 (b) 8 (c) 6 (d) not one

** Solution: **The diagram given below is clear that here each surface has 24 cubes,

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** 1. **(c) We know that only the three surfaces are colored in the Corner Cube and the three surfaces are colorless, as well as the number of Corner Cube is always 8.

Therefore, the number of intended cubes = 8

**2.** (b) According to the question, It is clear in the above diagram that each layer of rectangular block has 24 cubes here, so the number of intended cubes = 24 × 2 = 48

Or the number of total cubes by another method = 6 × 4 × 2 = 48

** 3. **(c) We know that only two surfaces of the middle cube are colored. Thus, calculate the number of central cube and corner cube and subtract it from the number of total cubes we will get total number of the middle cubes, which will be the only number of cubes having two surfaces painted.

Therefore the number of intended cubes = 48 – (8 × 2 + 8) = 48 – 24 = 24

** 4. **(b) Cube which is colored with three colors will be the Corner Cubes, Therefore, the number of intended cubes = 8

** 5. **(d) By the above diagram it is clear that there will not be any such cube whose two surfaces are colored with black color.

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