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

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

** 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: **

** Central Cube **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.

** Middle Cube **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.

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

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

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

** (i) **Number of cubes with no blurred vane = (n-2)^{3}

** (ii) **Number of cubes with a colored vane = (n-2)^{2} × 6

** (iii) **Number of cubes with two colored pins = (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 panel, we now have some based on the above facts

**Examples of questions will look like **

**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 125 can become small and equal cubes. Answer these questions on the basis of these information.

**1.** By the number of the cubes, whose only color will be colored?

(a) **48** (b) **54**

(c) **42** (d) **64**

2**. **By the number of cubes, which will have only two surface colors?**
(a) **36

** 3. **How many cube are there, whose three surfaces are colored?**
(a) **12

** 4. **How many cubes will there be, whose surface will not be colored?**
(a) **16

** Solution: **here

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

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

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

** 2. ** (a) We know that the middle cube is the same as its two sides are colored, so if the number of middle cubes is determined, then two surface colored total cubes Can know the number of

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

Thus the total number of cubes in the two surface color = 36

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

** 4. **(c) We know that any surface of the inner cube is not colored, so knowing the number of inner cube, the number of intended cubes will be known.

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

Therefore, the cube, whose surface is not colored, that number 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 1 cm smaller cubes is. Answer these questions on the basis of these information.

** 1.** How many cube are there any color on any surface?**
(a) **27

** 2. **How many cube are there, whose three sides are painted?**
(a) **16

** 3. **How many cube are there, whose only surface is colorful?**
(a) **24

** 4. **What is the number of such cubes, whose only two surfaces are painted?**
(a) **26

** 5. **How many cubes will the total from this cube get?**
(a) **64

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

Number of cubes not being colored in any surface

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

** 2. **(c) The total number of the three surface-colored cubes is 8 because the number of cubes being divided into three small cubes is always 8.

** 3. **(a) Total number of cubes having only one surface colored = 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 can be made of small and equal cubes. Answer these questions on the basis of these information.

** 1. **How many cube are there, whose only surface will be colored and it will be red?**
(a) **12

** 2.** What is the number of cubes, whose only color will be colored and it will be green?**
(a) **24

** 3.** How many cube are there, whose two sides will be colored in which one red and the other will be green?**
(a) **24

** 4.** How many cube are there, whose two sides will be colored and both will be green?**
(a) **8

** 5. **How many cube are there, whose surface will not be colored?**
(a) **12

** 6. **How many cube would be, whose two sides are colored and both will be red?**
(a) **4

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

** 1. **(b) We have to find such cubes whose only surface is red. Here we know that a surface is colored in Central Cube only. So let’s find the Central Cube located at 3 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 x 9 = 54

की 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 surface red = 18

** 2. **(c) According to the above rules, the number of cubes dyed with only one surface green is 54 = & lt; 4 = 36 (‘here four colors are filled with green color), hence the number of intended cubes = 36

** 3. **(a) Only two surfaces in which the total number of red and second green colored cubes = 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**
**= 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 contrasting surfaces of a solid cube are colored red, two opposite surfaces are colored blue and the remaining two opposite surfaces are colored with black, its After it is cut 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

** 2. **How many cube are there, whose three sides are painted?**
(a) **4

** 3.** How many cube are there, whose only two surfaces are painted?**
(a) **36

** 4. **How many cube are there, whose only one surface is painted?**
(a) **144

** 5.** How many such cubes are there that no surface is painted?**
(a) **81

** 6. **How many such cubes, whose only color is painted, is also red?**
(a) **32

** 7. **How many cube are there, whose only two surfaces are painted in which one is red and the other is black?**
(a) **12

** 8. **How many such cubes, whose two sides are colored with black?**
(a) **0

** 9. **How much is the length of each side before dividing the solid cube?**
(a) **4 cm

** 10. **How many cube are there, whose surface is blue and the other surface is colored red? (Other surface can be colored or painted without color)**
(a) **24

** Solution:** here

** 1. **(b) We know that the three surfaces are in the corner cube and the number of the Corner Cube is always 8, as well as the opposite surface is colored with different colors, so each surface of the corner cube is different Color will be colored, 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 cubed colored sides of three 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 by the Central Cube

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

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

** 5. **(b) We know that any surface of the inner cube is not colored

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

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

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

= 2x (n – 2) ^{ 2 }= 2x (6-2) ^{ 2}

= 2 x 4 x 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 with only one red and one black color, only two surface colors Total number of cubes to occur = 2 x 2 x (n – 2) = 4 x (6-2) = 16

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

** 9. **(c) Length of each arm

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

**TYPE 5
**

**Cuboid related questions **

** 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

**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)** On either side, whose dimensions are 4 cm × 2 cm, have been colored with blue color.

**(iii)** On either side whose dimensions are 6 cm × 2 cm, Dyed with black color

**(iv)** On either side, whose dimensions are 6 cm × 2 cm, have been reddish.

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

** 1. **How many such cube are there, whose three sides are painted and three surfaces are colored?**
(a) **4

** 2. **How much is the total number of cubes?**
(a) **24

** 3. **How many cube are there, whose only two surfaces are painted?**
(a) **8

** 4. **How many cube are there in which all three colors are engaged?**
(a) **4

** 5. **How many cube are there, whose two surfaces have black color?**
(a) **4

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

** 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.** According to the question, (b) , the diagram above is clear 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, by decreasing the number of total cubes by knowing the central cube and corner cube, the middle cubes will be known, which will be the only dye painted on two surfaces.

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 cube of Corner Cubes,

Therefore, the number of intended cubes = 8

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

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