reasoning - cube and cuboid

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Reasoning - Cube and CuboidReasoning - Cube and Cuboid

Nowadays, questions based on ‘Cubes and Cuboids’ are being asked in almost every competitiveNowadays, questions based on ‘Cubes and Cuboids’ are being asked in almost every competitiveexam. These problems are very frequent in various competitive exams.exam. These problems are very frequent in various competitive exams.

The methods described below are simple yet elegant. They should be very easy to understand andThe methods described below are simple yet elegant. They should be very easy to understand andwith a little practice you should master them. A cube is a three dimensional solid having 6 faces, 12with a little practice you should master them. A cube is a three dimensional solid having 6 faces, 12edges, and corners. All edges of a cube are equal and all the faces are square in shape. That is aedges, and corners. All edges of a cube are equal and all the faces are square in shape. That is asolid figure which has 6 faces; each face being a square is called solid figure which has 6 faces; each face being a square is called a cubea cube. If each of the six faces. If each of the six facesbe a rectangle, it is called be a rectangle, it is called cuboidcuboid. A cuboid is also called a . A cuboid is also called a rectangular parallelepipedrectangular parallelepiped..

IntroductionIntroduction

The questions asked on cube and cuboids may be of the following types.The questions asked on cube and cuboids may be of the following types.

Type IType I

Several views of a complete cube are given and you have to find which part of the cube lies exactlySeveral views of a complete cube are given and you have to find which part of the cube lies exactlybelow a particular part.below a particular part.

Type IIType II

An opened-up cube is given and you have to predict what it will look like when it is closed into aAn opened-up cube is given and you have to predict what it will look like when it is closed into acube.cube.

Type IIIType III

A cube could not be varnished on or some of its faces with the same colour or different colours andA cube could not be varnished on or some of its faces with the same colour or different colours andthen cut into a certain specified number of identical pieces. Then question of the form- “how manythen cut into a certain specified number of identical pieces. Then question of the form- “how manysmall cubes have 2 faces varnished?” “How many small cubes have only one face varnished?” etc.small cubes have 2 faces varnished?” “How many small cubes have only one face varnished?” etc.

Types of ProblemsTypes of Problems

There are two types of problems that appear in exam. At first, you are given several views of aThere are two types of problems that appear in exam. At first, you are given several views of acomplete cubecomplete cube, and you have to state which part of the cube lies exactly below a particular part. In, and you have to state which part of the cube lies exactly below a particular part. Inanother type, you are given an another type, you are given an opened-up cubeopened-up cube, and you have to predict what it will look like when, and you have to predict what it will look like whenit is closed into a cube.it is closed into a cube.

For ExampleFor Example − −

Format of ProblemsFormat of Problems



Several faces of a cube are shown below −Several faces of a cube are shown below −

Which number would lie opposite to 2?Which number would lie opposite to 2?

AA − 1 − 1

BB − 6 − 6

CC − 5 − 5

DD − 4 − 4

The fundamental approach is as follows −The fundamental approach is as follows −

Type IType I

A fundamental rule: Opposite cannot be together;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 oppositeWhenever we see a cube, with only three of its faces visible to us, we can never see two oppositefaces together. With all this rules, we can easily solve the type of problem discussed above. In thefaces together. With all this rules, we can easily solve the type of problem discussed above. In theabove question where we have to find opposite face of a particular face, we can eliminate thoseabove question where we have to find opposite face of a particular face, we can eliminate thosefaces which have occurred together with X in any view. Thus, we can eliminate all other choice andfaces which have occurred together with X in any view. Thus, we can eliminate all other choice andremaining will be our answer.remaining will be our answer.

At this point, you should go through the previous paragraph once more and see that youAt this point, you should go through the previous paragraph once more and see that youunderstand the concept. After this, you should try to solve the above example and see if you canunderstand the concept. After this, you should try to solve the above example and see if you canapply the concept discussed above. However, you find that you have not understood the conceptapply the concept discussed above. However, you find that you have not understood the conceptfully, no problem. Continue reading this section. Things will become clear once you finish thefully, no problem. Continue reading this section. Things will become clear once you finish thesection. With the foregoing fundamental rule at the back of your mind, you can solve the abovesection. With the foregoing fundamental rule at the back of your mind, you can solve the abovetype of question.type of question.

For the question, the rule is sufficient in itself. After that, you can solve it more quickly by For the question, the rule is sufficient in itself. After that, you can solve it more quickly by threethreesecondary rulessecondary rules..

Solution for above exampleSolution for above example − −

In the given example, we have to find the face opposite 2. Now in the first figure, 2 is appearingIn the given example, we have to find the face opposite 2. Now in the first figure, 2 is appearingalong with 1 and 3. It means that neither 1 nor 3 can be opposite to 2. It means that opposite of 1along with 1 and 3. It means that neither 1 nor 3 can be opposite to 2. It means that opposite of 1we can have either 4 or 5 or 6. Similarly, opposite 3 we can have either 4 or 5 or 6. Now, look at thewe can have either 4 or 5 or 6. Similarly, opposite 3 we can have either 4 or 5 or 6. Now, look at thesecond figure. Here, 3 and 1 occur together with 5. It means that 5 is opposite to neither 3 nor 1.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. HenceSo, it means that either 4 or 6 is opposite 1 and other is opposite 3 so 5 must be opposite 2. Hence5 is correct answer.5 is correct answer.



Some Quicker RulesSome Quicker Rules

Now you must have understood the basic trick of solving such questions. The trick is that youNow you must have understood the basic trick of solving such questions. The trick is that youshould eliminate those choices which are not possible. For this, you take help of fundamental ruleshould eliminate those choices which are not possible. For this, you take help of fundamental rulewhich says that if two faces are opposite to each other, their simultaneous occurrence in one viewwhich says that if two faces are opposite to each other, their simultaneous occurrence in one viewof the cube is not possible. However, in today’s time-precious competitions, merely the concept willof the cube is not possible. However, in today’s time-precious competitions, merely the concept willnot do. You must be able to solve a question quickly. There are some secondary rules for solvingnot do. You must be able to solve a question quickly. There are some secondary rules for solvingquestions.questions.

Rule IRule I − Let us call that figure X, the opposite of which you have to find. Suppose in any one view − Let us call that figure X, the opposite of which you have to find. Suppose in any one viewof the cube, X appears with Y and Z. Along with a third figure (say A), then X will be opposite A.of the cube, X appears with Y and Z. Along with a third figure (say A), then X will be opposite A.

So as for example, you have to find the face opposite 2. (This is our X). Now, 2 appears in oneSo as for example, you have to find the face opposite 2. (This is our X). Now, 2 appears in onefigure along with 1 and 3. (Y and Z). Also 1 and 3 appear together in one more figure, along with 5.figure along with 1 and 3. (Y and Z). Also 1 and 3 appear together in one more figure, along with 5.(That is A). Hence 2 must be opposite 5.(That is A). Hence 2 must be opposite 5.

Rule IIRule II − We have to find opposite face of ‘X’. Suppose that in any one view of the cube, X appears − We have to find opposite face of ‘X’. Suppose that in any one view of the cube, X appearswith Y and Z. Now, suppose Y and Z don’t appear together in any more views, but they appearwith Y and Z. Now, suppose Y and Z don’t appear together in any more views, but they appearseparately in two or more different views. Then the common figure between the two more views inseparately in two or more different views. Then the common figure between the two more views inwhich Y and Z appear separately, will be the figure opposite X.which Y and Z appear separately, will be the figure opposite X.

Rule IIIRule III − Let’s call the figure X, the opposite of which you have to find. Now, suppose X appears in − Let’s call the figure X, the opposite of which you have to find. Now, suppose X appears intwo views and in these two views four different figures are seen with X. Then the only figure nottwo views and in these two views four different figures are seen with X. Then the only figure notseen with X in these two views must be opposite to X.seen with X in these two views must be opposite to X.

SummarySummary − You have to keep the fundamental rule at the back of your mind and then apply three − You have to keep the fundamental rule at the back of your mind and then apply threesecondary rules for quick answers. The entire approach can be summarised by following diagram.secondary rules for quick answers. The entire approach can be summarised by following diagram.

Type IIType II



In this type, we use fundamental rule. This rule helps us to eliminate those combinations whereIn this type, we use fundamental rule. This rule helps us to eliminate those combinations whereopposite faces are shown in a single view. So it will lead to the elimination of a choice provided weopposite faces are shown in a single view. So it will lead to the elimination of a choice provided weknow how to determine which face will be opposite to each other, by looking at the “know how to determine which face will be opposite to each other, by looking at the “opened-upopened-upcubecube”. For this purpose, there is a very simple rule using which you can tell by looking at the”. For this purpose, there is a very simple rule using which you can tell by looking at theopened-up cubeopened-up cube, which faces will be opposite to each other by just looking at it., which faces will be opposite to each other by just looking at it.

The rule is given below;The rule is given below;

Third is opposite ruleThird 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 cubeWhen you want to find out the opposite face of a face (say X), in figure I, II, III; an opened up cubeis given. We have to find which faces opposite each other when the cube is closed.is given. We have to find which faces opposite each other when the cube is closed.

ExplanationExplanation − −

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 EIn figure (I), the third figure to A is C. So A is opposite of C. So, D and F will be opposite. B and Ewill be opposite.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 willIn figure (II), B is third to D, so B will be opposite to D. Similarly, C will be opposite to E and A willbe opposite to F.be opposite to F.

In figure (III), A is opposite to E, B is opposite F. Hence, C is opposite D.In figure (III), A is opposite to E, B is opposite F. Hence, C is opposite D.

Steps to solve problemSteps to solve problem

We can now solve questions of this type. We know how to find the opposite face by looking at anWe can now solve questions of this type. We know how to find the opposite face by looking at anOpened-up cubeOpened-up cube. We also know that in any view of the cube, opposite faces can’t be together.. 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.Hence, combining two rules, we can easily solve problems.

SummarySummary

With this, the discussion on how to solve questions of type II is completed. You have to use the ruleWith this, the discussion on how to solve questions of type II is completed. You have to use the rulethird to determine which faces are opposite to each other. The following diagram will give thethird to determine which faces are opposite to each other. The following diagram will give thecomplete information about this approach.complete information about this approach.



Type IIIType III

Counting of Cubes (when a varnished solid cube is cut);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.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 coloursThere is another type of question related to cubes wherein a larger cube with different coloursvarnished on different sides, is broken into several smaller cubes and you have to find the numbervarnished on different sides, is broken into several smaller cubes and you have to find the numberof cubes having only one side varnished or two sides varnished.of cubes having only one side varnished or two sides varnished.

Format of this problemFormat of this problem − −

ExampleExample − −

A cube is varnished with three colours green, blue and red on its sides, with every colour varnishedA cube is varnished with three colours green, blue and red on its sides, with every colour varnishedon two opposite faces of the cube. Now the cube is broken into 64 cubes of equal size. Based onon two opposite faces of the cube. Now the cube is broken into 64 cubes of equal size. Based onthis information, answer the following questions −this information, answer the following questions −

1. How many cubes have two sides varnished and remaining sides unvarnished?1. How many cubes have two sides varnished and remaining sides unvarnished?

AA − 18 − 18

BB − 20 − 20

CC − 22 − 22

DD − 24 − 24

2. How many cubes have only one side varnished (with either green or blue colour only)2. How many cubes have only one side varnished (with either green or blue colour only)

AA − 4 − 4

BB − 24 − 24

CC − 16 − 16

DD − 12 − 12

3. How many cubes have no sides varnished?3. How many cubes have no sides varnished?

AA − 0 − 0

BB − 8 − 8



CC − 12 − 12

DD − 64 − 64

reasoning - cube and cuboid

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