**Blog-82**

**Dear Students,**

**Today, we will see something new, other than our regular study.**

**We will take 5 colors and 5 cubes. We will paint them in such a way that in some particular position, no color is repeated on any cube to that side. If we look at all the 5 cubes, from the front, we will see that no color is repeated. In the same way, the same situation is there from all the remaining 5 sides. Now we will see the following diagram.**

**Diagram-01**

**The first line of the diagram:**

**Here we can see 5 open cubes. Now we will study each cube. Here 5 different colors are used on each side of the cube but all sides of these 5 cubes have no repetition of any color.**

**The second line of the diagram:**

**Here all above-opened cubes are closed to get the appearance as like cubes. To get the look of the back, left and bottom, these sides are shown at their respective places.**

**The third line of the diagram:**

**This is the diagram of all the 5 colors used with their degree of a vertex, the concept from Graph Theory.**

**Let us see the meaning of the degree of a vertex of each color. Here the pair of opposite sided colors is joined.**

**Now we will see the following chart to clear our idea about the degree of a vertex of each color.**

**1) Orange and Purple colors: These colors are on the opposite sides of cubes 1, 3 and 5.**

**2) Purple and Green colors: These colors are on the opposite sides of cubes 2 and 5.**

**3) Green and Red colors: These colors are on the opposite sides of cubes 1, 2, 3 and 4.**

**4) Red and Blue colors: These colors are on the opposite sides of cubes 3 and 5.**

**5) Blue and Orange colors: These colors are on the opposite sides of cubes 1, 2 and 4.**

**6) Blue and Purple colors: These colors are on the opposite sides of cubes 4.**

**So here each color connected with 6 colors as shown below so each color has the degree of the vertex as 6.**

**1)**

**Orange**

**color: It is connected with cubes 1, 2 and 4 of color**

**Blue**

**and cubes 1, 3 and 5 of color**

**Purple**

**. So the total connection is 3 + 3 = 6.**

**2)**

**Purple**

**color: It is connected with cubes 2 and 5 of color Green, cubes 1, 3 and 5 of color Orange and cube 4 of color Blue. So the total connection is 2 + 3 + 1 = 6.**

**3)**

**Green**

**color: It is connected with cubes 2 and 5 of color**

**Purple**

**and cubes 1, 2, 3, and 4 of color**

**Red**

**. So the total connection is 2 + 4 = 6.**

**4)**

**Red**

**color: It is connected with cubes 1, 2, 3 and 4 of color**

**Green**

**and cubes 3 and 5 of color**

**Blue**

**. So the total connection is 4 + 2 = 6.**

**5)**

**Blue**

**color: It is connected with cubes 3 and 5 of color**

**Red**

**, cube 4 of color**

**Purple**

**and cubes 1, 2 and 4 of color**

**Orange**

**. So the total connection is 2 + 1 + 4 = 6.**

**Special thanks to Jyoti Satpute for excellent drawing work.**

**Study this carefully and enjoy.**