Reasoning Cubes
A cube is a three dimensional figure, having 8 corners, 6 surfaces and 12 edges.
Number of cubes when a large cube is painted and cut:
If a cube is painted on all of its surfaces with the same colour and then cut into smaller cubes of equal size, then after separation, number of smaller cubes, so obtained will be calculated as under
- Number of smaller cubes = n^3
- Number of smaller cubes with three surfaces painted = 8
- Number of smaller cubes with two surfaces painted = (n-2) *12
- Number of smaller cubes with one surface painted = (n-2) ^2*6
- Number of smaller cubes with no surface painted = (n-2) ^3
Here, n= Number of parts on an edge of the bigger cube after division= Length of edge of bigger cube / length of edge of one smaller cube.
Types of Problems:
The questions asked on cube and cuboids may be of the following types.
Type - I : Several views of a complete cube are given and you have to find which part of the cube lies exactly below a particular part.
Type - II : An opened-up cube is given and you have to predict what it will look like when it is closed into a cube.
Type - III: A cube could not be varnished on or some of its faces with the same colour or different colours and then cut into a certain specified number of identical pieces. Then question of the form- “how many small cubes have 2 faces varnished?” “How many small cubes have only one face varnished?” etc.
FORMAT OF PROBLEMS :
There are two types of problems that appear in exam. At first, you are given several views of a complete cube, and you have to state which part of the cube lies exactly below a particular part. In another type, you are given an opened-up cube, and you have to predict what it will look like when it is closed into a cube.
For Example −
Several faces of a cube are shown below −
Which number would lie opposite to 2?
A − 1
B − 6
C − 5
D − 4
The fundamental approach is as follows -
Type I : A fundamental rule: Opposite cannot be together; Whenever we see a cube, with only three of its faces visible to us, we can never see two opposite faces together. With all this rules, we can easily solve the type of problem discussed above. In the above question where we have to find opposite face of a particular face, we can eliminate those faces which have occurred together with X in any view. Thus, we can eliminate all other choice and remaining will be our answer.
At this point, you should go through the previous paragraph once more and see that you understand the concept. After this, you should try to solve the above example and see if you can apply the concept discussed above. However, you find that you have not understood the concept fully, no problem. Continue reading this section. Things will become clear once you finish the section. With the foregoing fundamental rule at the back of your mind, you can solve the above type of question.
For the question, the rule is sufficient in itself. After that, you can solve it more quickly by three secondary rules.
Solution for above example −
In the given example, we have to find the face opposite 2. Now in the first figure, 2 is appearing along with 1 and 3. It means that neither 1 nor 3 can be opposite to 2. It means that opposite of 1 we can have either 4 or 5 or 6. Similarly, opposite 3 we can have either 4 or 5 or 6. Now, look at the second figure. Here, 3 and 1 occur together with 5. It means that 5 is opposite to neither 3 nor 1. So, it means that either 4 or 6 is opposite 1 and other is opposite 3 so 5 must be opposite 2. Hence 5 is correct answer.
Type II : In this type, we use fundamental rule. This rule helps us to eliminate those combinations where opposite faces are shown in a single view. So it will lead to the elimination of a choice provided we know how to determine which face will be opposite to each other, by looking at the “opened-up cube”. For this purpose, there is a very simple rule using which you can tell by looking at the opened-up cube, which faces will be opposite to each other by just looking at it.
The rule is given below;
Third is opposite rule −
When you want to find out the opposite face of a face (say X), in figure I, II, III; an opened up cube is given. We have to find which faces opposite each other when the cube is closed.
Explanation −
In figure (I), the third figure to A is C. So A is opposite of C. So, D and F will be opposite. B and E will be opposite.
In figure (II), B is third to D, so B will be opposite to D. Similarly, C will be opposite to E and A will be opposite to F.
In figure (III), A is opposite to E, B is opposite F. Hence, C is opposite D.
Steps to solve problem
We can now solve questions of this type. We know how to find the opposite face by looking at an Opened-up cube. We also know that in any view of the cube, opposite faces can’t be together. Hence, combining two rules, we can easily solve problems.
Type - III :
Counting of Cubes (when a varnished solid cube is cut);
In the previous section, we have discussed the problem of finding the opposite face of a cube. There is another type of question related to cubes wherein a larger cube with different colours varnished on different sides, is broken into several smaller cubes and you have to find the number of cubes having only one side varnished or two sides varnished.