82-5 Colors and 5 Cubes - Fun with mathematics- graph theory

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.

Anil Satpute