Types of Puzzles in Reasoning: How to Solve a Reasoning Puzzle 1

Submitted by Atanu Chaudhuri on Mon, 23/01/2017 - 01:11

Types of Puzzles in Reasoning: How to Solve a Reasoning Puzzle in easy steps 1

Many types of puzzles in reasoning form an important part in testing problem solving skills

Out of various types of puzzles in reasoning, we'll explain how to solve our first logic assignment reasoning puzzle of 3 variables in easy steps.

This is the first of a series of sessions on how to solve this challenging class of reasoning puzzles easily.

Our overall objective being effective real life problem solving, we'll spend some time on role of reasoning in real life problem solving and types of reasoning problems before taking up the main topic of solving a reasoning puzzle in easy steps.

The sections are,

Reasoning skill is a critical component for effective real life problem solving: Relation between Reasoning used in everyday life and logic.
Use of the term Reasoning in competitive exams: Types of Reasoning problems.
Types of reasoning puzzles.
Assignment logic analysis is the main function in solving a Reasoning puzzle.
How to solve the simple reasoning puzzle on family relations.
How to solve the second more complex 3 variable reasoning puzzle in easy steps.

This being the first session on how to solve reasoning puzzles in easy steps, we have introduced the subject topic in first four sections, and introduced the method of solving a reasoning puzzle by solving a simple puzzle in section 5.

The proper reasoning puzzle of SBI PO class is solved in easy steps in the last section.

You may move directly on to any of the sections by clicking on corresponding internal links and return by clicking on browser back button.

For example, you may skip all of first five sections and straightway move on to solving the main target reasoning puzzle in section 6 by clicking here.

Reasoning skill is a critical component for Effective Real Life Problem Solving

The term Reasoning is a broad sense one and has varying definitions. Its meaning is not precise just like the other popularly used broad sense words in any natural language.

In its simplest form we may state the meaning of the term Reasoning as,

To think logically for taking a decision.

As expected Logical thinking lies at the heart of Reasoning and the goal of reasoning is to take a decision, that is, to solve a problem.

Why is the term Reasoning needed at all when we have the well developed discipline of Logical thinking

Logic, logical thinking, logic analysis and other such phrases involving the term Logic always deal with precise and certain statements, conditions, operations and problem states with no scope of ambiguity and uncertainty.

In real life problem solving though practically nothing is certain. Most situations are ambiguous so much so that even reality may be different from what we perceive.

To deal with this ambiguity in real life problem solving we add one very important term “Sensibility” to the meaning of Reasoning. The more effective usable meaning of Reasoning thus becomes,

To think logically and sensibly for taking a decision.

With the infusion of this rich meaning, the term Reasoning becomes usable by one and all. We, the common folks not knowing anything about the strict discipline of logic, effortlessly and comfortably use such statements as,

In a company board meeting: “This seems to be the most reasonable decision.”
In a family gathering: “It surprises me how reasonable your grandson is at such young age.”

Relation between Reasoning and Logic

Addition of the term Sensibility based on Common Sense that varies from place to place and time to time makes the concept of Reasoning subjective but hugely effective and usable in real world problem situations.

Though more subjective in meaning, Reasoning encompasses the concept of Logic and smoothens the sharp edges of Logic when necessary with the help of its other invaluable component of Sensibility. It thus achieves acceptable and effective results in real world problems that won’t have been possible at all with logic alone.

To cite an extreme example, in affective or emotion-rich problem situations use of pure logic is generally impossible with catastrophic results while well-developed sensible reasoning may produce amazing results.

Logic and sensibility reside in Reasoning with Logic as base concept layer and Sensibility as rich concept layer together making up the whole of the concept of Reasoning.

reasoning puzzles 1-1

This way the term Reasoning is used in everyday life.

Use of the term Reasoning in Competitive tests: Types of Reasoning problems

All leading competitive tests attempt to judge the problem solving ability of the test-taker. Maths being the most suitable resource for testing the problem solving skill, it automatically takes the leading role in the tests.

Additionally to assess the pattern identification skill and logic analysis skill directly rather than indirectly through Maths problems, the test creators fall back upon the concept of Reasoning defining specific types of reasoning problems from puzzles, family relations, sitting arrangements to statement analysis and paragraph analysis in a natural language (generally English).

In all these problems, Problem analysis, Strategy formulation, Pattern identification and Logic analysis skills play key roles.

Typically the standalone topic of Reasoning in these tests includes the problem types of,

Analogy
Classification
Coding-decoding
Word arrangement
Venn diagram
Dice problems
Figure and image analysis
Statement analysis
Paragraph analysis
Syllogism..., and,
Reasoning puzzles.

These are the non-verbal reasoning problems that form the major part of reasoning tests. And the last in the list Reasoning puzzles represent what we call as the serious and most difficult type of reasoning problems.

We would discuss only reasoning puzzles from this point on.

Reasoning puzzles in turn come in different types. Following are the usual types of reasoning puzzles.

Types of Reasoning puzzles

Purely assignment logic analysis puzzles: Complexity of this general type of reasoning puzzles depend on number of variables, number of values of a variable, whether variable values are grouped, and so on. The first reasoning puzzle we'll solve will be an assignment logic analysis puzzle of three variables.
Reasoning puzzles on Family relations: This is a favored type of puzzles in many competitive tests.
Reasoning puzzles involving linear seating arrangement: This can be made more complicated in the form of two rows of members sitting face to face, members sitting on a rectangular table and so on.
Reasoning puzzles involving circular seating arrangement on a round table: This type of puzzles may be made more difficult by introducing mixed FACING IN and FACING OUT of the members (though in reality this doesn't happen).
Reasoning puzzles involving floor stay arrangement in a building.

These are the main types that may be adapted and mixed together to form a new puzzle that will surprise you in your test.

Assignment logic analysis is the main function in solving a Reasoning puzzle

In Assignment logic analysis problems there is a set of objects that are to be assigned (or mapped) to a second set of objects determined by a set of logic statements or conditions.

The objects that are to be assigned belong to the to-be-assigned object set (TBA object set) and the objects to which the to-be-assigned objects are to be assigned belong to the to-be-assigned-to object set (or TBAT object set). In case of one-to-one unique assignment between two object sets, any of the two object sets are interchangeable with the other with respect to assignment as assignment is a mutual two directional operation.

An example:

Given a set of men who are husbands and a set of women who are the wives, whose husband is who is to be found out based on a set of relation statements. Two assignment object sets are in this case, the husband set and the wife set, and the condition set is the set of relation statements. Obviously, in this problem assignment between the two sets is mutual and bi-directional.

Typically this type of problems fall under the type of Family relation problems; and usual approach to solve these problems is to use what we call logic tables (or Decision tables, Truth tables and so on) with one of the object set values as rows and the other as columns (husband name as row names and wife names as column names for example).

Complexity increases if number of objects in one assignment object set becomes larger than the number of objects in the second assignment object set. In this case we have to deal with the more complex one-to-many assignment.

Complexity may also increase by leaps and bounds if number of assignment object sets itself increases to 3, 4, 5, and so on. A problem with 6 assignment object sets has the dimension of 6 and we call such problems as six-dimensional assignment logic analysis problem.

Later we will explain the efficient solution of such a six-dimensional assignment logic analysis problem. Typically by conventional classification this type of problems is classified as Puzzles and is found to be generally confusing and difficult to solve by the students.

In this first part of the series on assignment logic analysis we will remove this general difficulty in solving puzzles and family relation problems through the use of collapsed column logic analysis technique. To show the mechanism of this efficient logic analysis process, we will solve two problems, the first a family relation problem taken from the Case example 1 in the session Complex logic analysis, and the second an SBI PO level puzzle.

For ease of understanding we will first present an expanded logic table approach to solving the first simple Family relation analysis type problem and then will solve the problem again more efficiently using the collapsed column logic analysis technique in its simplest form.

In the second more complex problem of proper Reasoning puzzle type, we will show the ease and power of the collapsed column logic analysis technique in its richer form.

How to solve the simple Reasoning puzzle on Family relations: Who is whose husband

Problem statement

Four married couples belong to a theater club. The wives are Puja, Sonia, Kaveri and Monica. The husbands are Naveen, Farukh, Tapan and Manav. Who is married to whom?

Use the following clues.

Naveen is Monica’s brother.
Monica and Tapan were once engaged, but broke up when Monica met her present husband.
Kaveri has a sister but her husband is an only child.
Puja is married to Manav.

Before you go through the solution try to solve the problem yourself. It is not a difficult problem and will give you an idea of purely logic based problem solving.

Conventional logic table based solution

The logic table that we will use here is a four column by four row table with 16 cells as shown.

reasoning puzzles 1-2

There are sixteen relationship cells between four possible couples out of which only four cells will have a Y indicating married relation. Rest 12 will have Xs indicating no-spouse relationship.

First with a ‘Y’ in Puja-Manav cell from the fourth statement, we cross out all other cells in Manav row and Puja column.

Next from first and second statement, we cross out Tapan and Naveen cells of Monica column. Remaining Farukh cell automatically is then paired with Monica. But this crosses out the two remaining cells of Farukh also.

At the third and last step, the third statement along with the first statement implication that Naveen is not a single child decides a ‘Y’ in Kaveri-Tapan cell. The last is then a Y in Sonia-Naveen cell.

Advantages of this problem solving tool of logic table are,

You do not lose track of the results of your analysis at any stage.
Pictorially and unambiguously you account for all the results of your analysis step by step resulting in clarity in analysis and solution.

But in any case, you have to go through the process of analyzing the logic statements optimally and deal with a 4 by 4 matrix of 16 assignment cells.

Application of Collapsed Column Assignment Logic Analysis Technique in a Simple form

As before we will first analyze the effect of the fourth statement in the first step, but instead of four columns, we will use only one column in which names of possible wife will be written alongside the name of a husband. Effectively, the four columns are collapsed into only one column. As soon as we are able to get a unique assignment, we will cross out the corresponding name of the wife from the list of names of wives. In addition to the single assignment column we will use a second column to note the intermediate results of statement analysis. The problem state after the first step is as below,

reasoning puzzles 1-3

Likewise in the second step Statement 1 and Statement 2 are analyzed together as, Monica is appearing in both statements. The problem state after the actions in this step are taken is as below,

reasoning puzzles 1-4

In the third and final step the Statement 3 is analyzed to arrive at the final marital information of the four couples,

reasoning puzzles 1-5

For ease of explanation we have shown the stages separately. In actual exam-hall problem solving we will use only one table the state of which will quickly converge towards the solution with minimum amount of writing and only the key analytical effort.

We will now take the more complex SBI PO level reasoning puzzle problem to highlight the power and ease of the collapsed column technique.

How to solve the second more complex 3 variable reasoning puzzle in easy steps: Which student in what Standard studies which subject

Problem description

A, B, C, D, E, G and I are seven friends who study in three different standards namely 5th, 6th and 7th such that not less than two friends study in the same standard. Each friend also has a different subject namely History, Civics, English, Marathi, Hindi, Maths and Economics but not necessarily in the same order.

Conditional statements

A likes Maths and studies in 5th standard with only one other friend who likes Marathi.
I studies with two other friends.
Both the friends who study with I like languages (here languages include only Hindi, Marathi and English).
D studies in the 6th standard with only one person and does not like Civics.
E studies with only one friend.
The one who likes History does not study in 5th or 6th standard.
E does not like languages.
C does not like English, Hindi or Civics.

Questions

Question 1. Which combination represents E’s favorite subject and standard in which he studies?

Civics and 7th
Economics and 5th
Civics and 6th
History and 7th
Economics and 7th

Question 2. Which of the following is I’s favorite subject?

History
Civics
Marathi
Either English or Marathi
Either English or Hindi

Question 3. Who among the following studies in the 7th standard?

G
C
E
D
Either D or B

Question 4. Which of the following combinations is definitely correct?

I and Hindi
G and English
C and Marathi
B and Hindi
E and Economics

Question 5. Which of the following subjects does G like?

Either Maths or Marathi
Either Hindi or English
Either Hindi or Civics
Either Hindi or Marathi
Either Civics or Economics

Solution to Reasoning puzzle problem 2: Problem analysis

This problem belongs to the class we call Assignment logic analysis where there is one or more than one set of objects that are to be assigned to one or more than one set of objects. The assignments or mappings are determined by a set of conditional logic statements.

In our problem here, the seven friends form the to-be-assigned objects which are to be correctly assigned or mapped to the to-be-assigned-to objects of seven subjects as well as the set of three Class Standards.

The condition or logic statements determine the correct mapping.

The given problem thus falls under the second level of complexity with two to-be-assigned-to object sets and a total number of 3 object sets. It thus is a three-dimensional problem.

Solution objective and Strategies

Our main objective is not only to answer correctly but also to do it in as little time as possible adhering to Efficient Problem Solving philosophy.

Strategy 1.

As there are five questions covering the description of the problem comprehensively, we don’t have any other option than to complete the assignment fully and comprehensively.

If there were only one question, we could have tried to analyze the logic statements and fill up only a part of the logic table for reaching the goal.

Strategy 2 and 3.

The aspects on which we would cut-short the time to fill up the complete logic table will then be,

An efficient representation of the three dimensional table, aiding understanding and filling quickly with least amount of writing (writing and spread of table here increases time), and,
An efficient sequence of analyzing (or processing) the condition or logic statements.

Lastly we have to be able to carry out fast and accurate elementary logic analysis at each step of processing one or more than one conditional statements. This is an essential skill requirement, not a strategy, though for quick solution you need to first form the optimum logic representation table as well as select the statements in a sequence for solution in minimum time. These two also are then skill based.

Crucial to quick logic puzzle solving is usually, optimal sequencing.

Problem solving execution: Strategy 2 implementation: Compact representation

Out of three sets of objects involved in assignments, set of Students forms the to-be-assigned objects, and sets of Standards and Subjects form the two to-be-assigned-to object sets out of which we will take the Standards as the primary to-be-assigned-to object set and these values will form the row labels.

As hinted we won’t represent the problem solving data in a spread out logic table. Instead we will use a compact form which can be written on scratch paper quickly and understood and used for the next stage with ease and accuracy.

As row label values we will select the three standards 5th, 6th, and 7th. The other two sets, Students and Subjects will be placed in three columns. This is because the breakup of 7 students among three standards can only be 3, 2 and 2 by the phrase "not less than two friends study in the same standard", and so each standard may have a maximum of three students, that is, each row may have a maximum of 3 columns (it may also be two columns). You see, the analysis has already started even at this early phase of problem representation.

Each row will be a combined complex one meaning, for each standard and each student column for the standard, there will be two cells, one for the student name and the other for the subject name the student studies. Thus instead of a single cell at the cross-section of a column and row there will be multiple cells. By this technique of representation, we can handle problems of higher dimension also with ease.

The cells can hold specific assigned values or intermediate results, and ultimately only the specific assigned values when all assignments are over.

Now we will analyze and deal with the eight conditional statements, possibly not serially.

Problem solving execution: Strategy 3 implementation

Step 1. We will process those logic statements first that directly make definitive assignment of a to-be-assigned object to a to-be-assigned-to object of primary set. Such a statement with maximum degree (number of objects referred to) will be selected first.

Only the first statement makes a direct and full assignment (to the two to-be-assigned-to objects standard and subject) and also carries a large amount of information. So we will process the first statement in the first step.

After processing the first statement, “A likes Maths and studies in 5th standard with only one other friend who likes Marathi” our logic table will look like,

reasoning puzzles 1-6

We have not forgotten to write the values of the two sets of students and subjects above the logic table and crossing out the values that are used up at this stage. This action helps in tracking the values that are yet to be assigned.

You will notice that we could identify the 5th standard to be a two student standard which we noted within brackets with the standard name (no information should be lost).

Step 2. In this second step we have to process the 4th statement as this is the only one left with certain student assignment to a specific standard. After processing the statement "D studies in the 6th standard with only one person and does not like Civics", the logic table will look like,

reasoning puzzles 1-7

In this second step we could not assign a specific subject to student D, but noted the "not Civics" condition against its value and also identified the 6th standard as the second two student standard. The 7th standard automatically then turns out to be the three student standard. We cross out the value D from the list of student names.

Step 3. There are now two pairs of statements with common student names. In Step 3, we select the statement 2 and 3 involving student I as these statements deal with all three students of 7th standard in some way or other thus achieving maximum simplification.

After applying these two statements 2 and 3 the logic table will have the state,

reasoning puzzles 1-8

Student I is assigned to 7th class directly as it is the only 3 student class and I studies with two other friends. Also the two friends of I like languages English, Hindi or Marathi out of which Marathi has already been assigned. So we assign English and Hindi against the two remaining subject cells other than that of I in 7th standard.

Last action in this 3rd step gets us the two subjects [History, Economics] as possible subject value subset for student D, as with four subjects already assigned and D studying "not Civics" implies D studies either of these two subjects, [History, Economics]. Later we will see the usefulness of this small but significant analytical action.

The student I and subjects English and Hindi are crossed out from object value lists.

Step 4. In the 4th step we select statements 5 and 7 as these two have student E common. After applying the effect of these two conditional statements the logic table reaches the following state.

reasoning puzzles 1-9

As E studies with only one friend, its standard must be 5th or 6th. In 5th standard one cell is still to be assigned a student name but the subject Marathi, a language subject, has already been assigned to this place. By the 7th statement we come to know that E does not like languages and thus E can be assigned uniquely only in 6th standard. This is what we call the simplest level of logic analysis or elementary logic analysis.

Step 5. In 5th step we have only two statements left out of which statement 6 catches our attention as it states that the History loving student does not study in either 5th or 6th standard. With this we can conclude that student I in 7th standard must be the student liking History as subjects of his two other friends have already been assigned. The logic table now is,

reasoning puzzles 1-10

After assigning subject History to student I, the possible subject value subset [History, Economics] for D reduces to only Economics by canceling subject value History. Now only our earlier action of recording the value subset [History, Economics] against student D as possible subjects results in a unique assignment that is continuously sought for.

This is the Value Subset Analysis technique similar to Digit Subset Analysis technique we learned while solving hard Sudoku games. This way we have used the powerful domain mapping technique. From Sudoku domain we have abstracted a powerful technique and applied it after suitable minor modification to logic analysis domain.

Lastly in this step 5 we further note that only the subject Civics is left to be assigned. So E gets Civics. The corresponding object values are crossed out and we are left with only one conditional statement and three student values yet to be assigned.

Step 6. In the 6th and final step, by applying statement 8, as C does not like English, Hindi or Civics, we can assign C only in 5th standard and not in 7th standard. Note that in this conditional statement the fragment "or Civics" is superfluous.

We have exhausted all the conditional statement but two students B and G are yet to be assigned. What should we do?

The only option now is to assign both these as possible student value subset [B, G] to the two vacant slots in 7th standard. The final logic table state is then,

reasoning puzzles 1-11

This is the maximum extent of assignment that is possible with the given information. Now it should take only about a minute's time to answer the five questions.

Answers

Question 1. Which combination represents E’s favorite subject and standard in which he studies?

Answer 1. Option 3: Civics and 6th.

Question 2. Which of the following is I’s favorite subject?

Answer 2. Option 1: History.

Question 3. Who among the following studies in the 7th standard?

Answer 3. Option 1: G. Option 5 mentions B who studies in 7th standard, but this option taken as a whole is incorrect as the answer of this question because it also mentions the possibility of D studying in 7th standard that is not true.

Question 4. Which of the following combinations is definitely correct?

Answer 4. Option 3: C and Marathi. Both the option 2 and 4 are statements of possibilities as B or G could be assigned to a subject with certainty. Answer needed a choice that is definitely correct with no uncertainty.

Question 5. Which of the following subjects does G like?

Answer 5. Option 2: Either Hindi or English.

Logic analysis puzzles can have potential error traps anywhere. Even after correctly assigning the object sets, you need to be careful while answering. In answering the questions 3 and 4, mistake could happen unless logic analysis is done to eliminate the partially possible choices.

Note that we could not assign the logic table with complete certainty as [B, G] student pair were assigned to two places in Standard 7th with favorite subjects as “Either Hindi or English” but we could not tell which subject B or G definitely likes. But as expected the questions also were asked considering this uncertainty.

Once you form the final state of the logic table, answering the questions should be quick.

Important to note

In practice you won’t write all the logic table states separately that we have done for detailed explanation. You will reach the final table at one place, only modifying the assignments at each step. As the above process is a step by step clear method, if you can quickly analyze the statements and follow the method systematically, you will reach the final state of the logic table quickly, correctly and with assurance.

Recommendation

As you might have noted, the collapsed column logic analysis technique is not only systematic, clear, and efficient it is also versatile as it can not only help you solve logic puzzles quickly, it can also be used for solving family relation problems efficiently.

To get a firm grip on the powerful technique though you might like to go through this material once more and more importantly solve a few other similar problems setting a time limit.

End note

Solving reasoning puzzles does not need knowledge on any subject—it is just identifying useful patterns by analysis of the problem and using appropriate methods. It improves problem solving skill, because patterns and methods lie at the heart of any problem solving.

Other resources for learning how to discover useful patterns and solve logic analysis problems

Einstein's puzzle or Einstein's riddle

The puzzle popularly known as Einstein's puzzle or Einstein's riddle is a six object set assignment logic analysis problem. Going through the problem and its efficient solution using collapsed column logic analysis technique in the session Method based solution of Einstein's logic analysis puzzle whose fish should be a good learning experience.

Playing Sudoku

As a powerful method of enhancing useful pattern identification and logic analysis skill, play Sudoku in a controlled manner. But beware, this great learning game, popularly called Rubik's Cube of 21st Century, is addictive.

To learn how to play Sudoku, you may refer to our Sudoku pages starting from the very beginning and proceeding to hard level games.