**Painting a Stack of Cubes**

The learners build “Three-Dimensional” cubes for studying and analyzing the patterns of the cubes that are painted on their number of sides.

Students usually paint the “Three-Dimensional” cubes of small units so that they can be used to make a large painted cube. All the sides of a cube are equal; that is the “Length”, “Breadth” and “Height” of a cube is the same. A “Cube” has “6 Sides”, “8 Vertices” and “12 Edges”. A “Dice” is also a type of cube with certain numbers on it. There are two types of “Dice” namely “Standard Dice” and “Ordinary Dice”.

**About cubic structures**

A cube is said to be a “Three-Dimensional” structure where the “Length”, “Breadth” and “Height” are equal. The “Length”, “Breadth” and “Height” make the sides of the cube. There are 6 sides of a cube; those are the “Front”, “Back”, “Right”, “Left”, “Top” and the “Bottom”. It has a total number of 8 vertices four of which are on the top and the remaining four on the bottom. “Vertices” are referred to as the corner points of a cube. The “Cubes and Dices” consists of a total number of 12 edges. “Edges” refer to the lines that join the different “Vertices”. It can be said that “Dice” is a type of cube that has numbers on it. The learners paint as well as build “Three-Dimensional” cubes of small units to make a larger painted cube. The students study as well as analyze the patterns of the cubes based on the number of sides painted of every unit of the cube.

Students usually paint the “Three-Dimensional” cubes of small units so that they can be used to make a large painted cube.

**Examples of Painting Stack of Cubes**

The study of “Cubes and Dices” can be illustrated with a set of examples. Consider a large cube that is blue on every face; it is divided into 216 identical pieces. The questions are:

- What will be the number of smaller cubes which will have exactly 3 sides painted?
- What will be the number of smaller cubes which will have exactly 2 sides painted?
- What will be the number of cubes that will have one of its sides painted?

For solving such a cube-based problem the researcher has to be familiar with certain terms such as “Edges”, “Corners” and “Faces”. It is noted that a cube has “12 Edges”, “8 Corners” and “6 Faces”. Considering “N” to be the number of cuts made on a large cube then “N+1” is going to be the number of pieces made out of a large cube.

Considering this information a figure can be developed. Based on the figure it can be derived that having 3 faces painted on the cube there will be 8 of them based on the numeric.

Based on the next question it is seen that there are 12 edges and 2 sides painted which makes it have 4 smaller cubes at the edge. Therefore it states that 12 multiplied by 4 will make 48 cubes.

The formula used here is “N multiplied by N multiplied by N” multiplied by “N-2”. Based on this formula, in the cube where only one face is painted, there will be 16 small cubes. Therefore 16 small cubes multiplied by 6 will give 96 smaller cubes.

In order to paint a cube with 6 colours where it is required to provide the possible number of ways, a remarkable approach is considered to the constructive process of counting. This includes fixing the first colour with one face through the assumption of the cube to be fixed in a position that is related to the face of similar color along with going for painting the other faces with the rest 5 colors.

This goes as 1 . 5 ..6 ! = 6/6 . 5 . 3 !

The fixing is similar to the assumption that 6 options that are there in the start for the starting color ends up with a similar bunch of 6 types of color arrangement along with the cardinality of 30.

If it is assumed that each of the elements of that bunch is basically a permutation or arrangement of the 6 different colors over the cube, every 6 options specify the same sets.

As in this particular situation the problem case is small along with the visualization of the fixing not causing the problem is obvious.

However in general, these problems of counting in constructive method,

whether there is {a,b,c,d,e} for the arrangement, the progression be shown in the same order of abcde and cdeab in fact the count of the same set?

How to prove that nothing is missed in this method ? i.e. the constructed set through the method can be surjective? It is appreciable if there is an explanation about the relation of this domain.

**Conclusion**

Throughout this research paper, the researcher has illustrated the different aspects of painting cubes as per their sides and requirements based on their identities. The researcher has elaborated the different advantages of performing this work as well as the practical applications of this work as examples. The researcher has also shown the categories of cubes as per their sizes and mathematical areas through elaborating their formative descriptions. Besides that, the formative explanations are being done as per their applications and requirements.