**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 to all the 5 cubes, from 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**

**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.**

**Second line of the diagram:**

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

**Third line of the diagram:**

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

**Let us see the meaning of degree of 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 degree of 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 bellow so each color has the degree of 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.**