Important Concepts in Logical Reasoning

Important Concepts in Logical Reasoning

Ever since the start of time, man has always been surrounded by logic (time and logic seseemsto have coexisted) — it exists all around us, and is all encompassing. There is i mplicit logic in all human and natural activity—right from the primary level of logic seen in our day-to-day lives to its very advanced form which operates the machines & tools we use for our day-to-day work. Every subject we study, every product we build, every activity we undertake is guided by its own inherent logic. 

For instance, when faced with the situation of starting a car, switching on an electric appliance like an electric bulb, etc. we use logic inherently. In fact, it would be difficult to imagine life today without even a thought on most of the logical structures that we use Inherently.

LOGIC AND LANGUAGE

The ancient Greeks were the first to study logic as a subject in depth. The lack of any

systematic notation for the process of logic during its initial development led the Greeks

to rely extensively on the use of language to explain logic. Each one of us even today

instinctively uses language to explain what we are doing. Thus we use logical language

in each of the following cases—

1. Where is my key?

Ans: It’s on the blue table.

Or

It’s in the second shelf from the bottom inside your cupboard.

2. Could you direct me to Mr Mehta’s house?

Ans: Proceed straight from here and take the right turn from the second crossing.

Move about 100 metres after the right turn and you will reach a ‘Y’ junction.

Take the left leg of the Y and this road leads to a dead end. Mr Mehta lives on

the second floor of the second last house on the left of this road.

3. If I put water in a working refrigerator it will become cold.

4. To turn on the car, one needs to switch on the ignition.

Thus, each of us comes across millions of such everyday situations where we use logic

inherently as part of our day-to-day language.

The study of logic by the Greeks was largely confined to the study and documentation of

logical language. However, the problems of understanding logic through language are

very high, since this approach becomes extremely complex and unwieldy the moment the

logical string becomes longer. This complexity was the reason that when Aristotle

summed up Greek logic in his Treatise on Logic in the 4

th century BC, many of the

greatest minds were at a loss to understand it.

Symbolic logic

It was only in the late 19

th century when Gottlob Frege brought about a revolution in the

whole field of reasoning by inventing ‘symbolic logic’— the use of symbols to

represent ideas; that the next phase of development in logical thought started. With this improvement of notation, logical and mathematical ideas could be precisely written down for perhaps the first time. The inconsistencies and vagueness of language were overcome through the use of symbols to denote logical thoughts. The development ofsymbolic logic further led to the development of ‘logical thinking’.Consider the statement “ If a car has poor air conditioning or low fuel efficiency then its not a nice car, then the fact that a car is nice means that it will have neither poor airconditioning nor low fuel efficiency. 

This long language string used to express the above logic can be condensed using the

following symbols:

P (poor A/C) & L (low fuel efficiency) & N(C) (Nice car).

Thus, if P or L then not N (C) then N (C) means not P or L.

The words OR, NOT & AND all can have their own logical symbols. 

Thus OR is ‘V’, NOT is ~ AND is +.

Thus , the above sentence can be condensed in its logical form as:

(PvL) Æ (~N(C)) Æ N(C) Æ (~P) & (~L).

In fact, to reduce ambiguity, tediousness and complexity of language based

interpretation, solving logical problems involves as its first step the interpretation of the

language of the question and conversion of the language of the question into symbolic

form. Moreover, this also aids in cutting down the time required to solve complex

reasoning questions.

Thus, when the language of logic is converted to symbolic form, then the truth of such a

sentence depends only on its logical form and not on its content.

Any similar sentence, (using totally different language) which uses the same logical

form will be true under the same circumstances. The process of mastering reasoning

then has to take as its first step the understanding of the process of conversion of

language into symbolic logic. In fact, the process of solving logical problems is greatly

eased through the use of symbols to document the language.

Some Standard Symbols that You can Use for Logical Language

NOT ~

OR v

AND +

If and Only if ∫

Not, if π

If, Then …

Besides, you would do well to keep in mind the following factors while creating a

symbolic framework for your question:

Æ Proper Nouns should always be denoted by capitals.

Æ In Questions where the sex of the proper noun is crucial to the solving of the

question, you can denote the female by underlining the capital letter used.

Æ As you start solving Reasoning Questions, create your own set of symbols for

standard relational sentence structures. (See the highlighted box):

(1) Is next to

Thus, A is next to B can be: AB v BA

(2) (a) Is to the immediate left of

Thus, A is to the immediate left of B can be: AB

(b) Is to the left of

Thus, A is to the left of B can be: A___B

(3) (a) Are at the ends

Thus A and B are at the ends means:

(A _________ B) v (B _________A)

(b) Is at the extreme left

Thus A is at the extreme left means:

(A______________)

(4) Is more than/taller than/greater than

Thus A is greater than/ taller than/ more than B & C can be symbolically

represented by:

A > (B + C) [Note here that we do not know the relationship between B & C]

(5) Is less than/shorter than/lower than

Thus A is less than/shorter than/lower than B & C can be symbolically

represented by:

A < (B + C) [Again, note here that we do not know the relationship between B

& C]

(6) Is between

Thus, A is between B & C means that A can be anywhere between B & C.

(B___A ___C) v (C___A ___B)

(7) Is the son of

Thus, A is the son of Æ

(8) Is the daughter of

Thus, A is the daughter of B Æ

(9) Is the Parent of

Thus, A is the parent of B Æ V

(10) Is the spouse of

Thus, A is the spouse of B Æ A—B v A—B

(11) Is the wife of

Thus, A is the wife of B Æ A—B

(12) Is the sibling of

Thus, A is the sibling of B Æ (A..B) v (A .. B) v (A..B) v (A..B)

Some Important Logical Manoeuvers:

(1) If, Then

The condition If A, then B leads to the following valid conclusions

Valid Reasoning 1: A therefore B.

Valid Reasoning 2: Not B, therefore not A.

At the same time If A then B also throws up the following invalid conclusions.

Invalid Reasoning 1: B, therefore A.

Invalid Reasoning 2: Not A, therefore not B.

The above structures of logical thought can be illustrated through the following

examples:

If A teaches, B will go to the movies.

Valid Reasoning 1: A teaches, therefore B will go to the movies.

Valid Reasoning 2: B has not gone to the movies, therefore A did not teach.

Invalid Reasoning 1: B went to the movies, therefore A must have taught. This

is an invalid line of reasoning.

Invalid Reasoning 2: A did not teach, therefore B did not go to the movies.

This too is invalid.

(2) If and Only If

The condition If and Only If A, then B leads to the following valid conclusions.

Valid Reasoning 1: A, therefore B.

Valid Reasoning 2: Not B, therefore not A.

Valid Reasoning 3: B therefore A.

Valid Reasoning 4: Not B therefore not A.

The above structures of logical thought can be illustrated through the following examples:

If and only if A teaches, B will go to the movies.

Valid Reasoning 1: A teaches, therefore B will go to the movies.

Valid Reasoning 2: B has not gone to the movies, therefore A did not teach.

Valid Reasoning 3: B went to the movies, therefore A must have taught.

Valid Reasoning 4: A did not teach, therefore B did not go to the movies.

(3) Either Or

Either A or B

Valid Reasoning 1: Not A then B

Valid Reasoning 2: Not B then A

Invalid Reasoning 1: A then Not B

Invalid Reasoning 2: B then Not A

The above structures of logical thought can be illustrated through the following

examples:

Either A teaches or B goes to the movies.

Valid Reasoning 1: A does not teach, therefore B will go to the movies.

Valid Reasoning 2: B has not gone to the movies, therefore A must have

taught.

Invalid Reasoning 1: A taught, then B did not go to the movies.

Invalid Reasoning 2: B went to the movies, then A did not teach.

(4) If, Then Not

If A then Not B:

Valid Reasoning 1: A then not B

Invalid Reasoning 1: Not B then A

Invalid Reasoning 2: B then Not A

The above structures of logical thought can be illustrated through the following

examples:

If A teaches, then B will not go to the movies.

Valid Reasoning 1: A teaches, therefore B will not go to the movies.

Invalid Reasoning 1: B has not gone to the movies, therefore A taught.

Invalid Reasoning 2: B went to the movies, therefore A did not teach.

As a student it is important for you to follow a few standard steps while solving logical

questions:

1. Take a complete ‘preview of the situation’ clearly understanding the context.

Remember to accept the situation as it is given.

2. Read each and every part of the question carefully. You should concentrate hard

and focus fully while reading the question. This is a very important prerequisite

for solving questions on logic since very often in long sentences there will be

individual single words which will transform the meaning of the sentences. If

you fail to take into account these words, the end result will be errors in logic &

deduction.

Let us now proceed to understand how all this applies to real life problem solving

through examining questions which have been asked in different competitive exams and

CAT.

Example 1 A party is held at the house of the Mehtas. There were five other couples

present (besides Mr and Mrs Mehta), and many, but not all, pairs of people shook

hands. Nobody shook hands with anyone twice, and nobody shook hands with his/her

spouse. Both the host and hostess shook some hands.

At the end of the party, Mr Mehta polls each person present to see how many hands each

person (other than himself) shook. Each person gives a different answer. Determine how

many hands Mrs Mehta must have shaken.

(Can we prove that it was not Mrs Mehta who shook 10 hands?)

Solution Let there be 5 couples:

A — A

B — B

C — C

D — D

E — E

& M — M

Deduction 1 From the condition that nobody shook hands with his/her spouse, it is clear

that none of the twelve people in the party shook more than 10 hands.

(Since, nobody shakes hands with himself or his/her spouse, it leaves a maximum of 10

people to shake hands with).

Deduction 2 Mr Mehta has asked the question to eleven different people and each of

them has given a different answer. Also, the highest answer anyone could have possibly

given is 10. Hence, the only way to distribute different numbers of hand shakes amongst

the 11 people is:

0,1,2,3,4,5,6,7,8,9,10. [ Note: somebody shook 0 hands and somebody shook 10].

Deduction 3 Since, the host & hostess have both shaken some hands, the person who

shook ‘0’ hands cannot be either M or M. It has to be one of the other 10 people in the

party.

[At this point you need to realise that in the context of this problemA,A B,B,C, C, D, D,

E & E are alike, i e. there is no logical difference amongst these 10 and you have

exactly the same information about each of these 10 people. However, Mrs Mehta is

different because she stands out as the hostess as well as the wife of the person who has

asked the question].

Since all ten guests are the same, assume ‘A’ shook no hands. This leads us to the

following deduction.

Deduction 4 Take any one person apart from A & A; say B. B will not shake hands with

himself & his wife. Besides B will also not shake hands with A (who has shaken no

hands). Thus, B can shake a maximum of 9 hands and will thus not be the person to

shake 10 hands.

What applies to B, applies to B, C, C, D, D, E, E and M.

Hence, A is the only person who could have shaken 10 hands. Hence, amongst the

couple A & A, if we suppose that A had shaken 0 hands, then A must have shaken 10

hands.

Note: The main result here is that, out of the people to whom M has asked the question,

and amongst whom we have to distribute the numbers 0 to 10 there has to be a couple

who has had 0 & 10 handshakes. It could be any of the five couples, but it cannot be M

who has either 0 or 10 handshakes.

We now proceed, using the same line of reason as follows.

Deduction 5 Suppose B has 1 handshake — he must have shaken hands with A (who

has shaken everybody’s hands she can).

Then, B wouldn’t have shaken hands with anyone out of A, B, C, C, D, D, E, E & M. At

this point the following picture emerges:

A — A — 10

B — 1 B —

C — C —

D — D —

E E

M

Numbers left to be allocated — 2, 3, 4, 5, 6, 7, 8, 9.

Considering C, as a general case, he cannot shake hands with C, A (Who shook no

hands) & B (Who shook hands only with A. This is mandatory since A has shaken hands

with 10 people).

Thus, C can shake hands with a maximum of 8 people and this deduction will be true for

C, D, D ,E , E & M too. Hence, the only person who could get 9 handshakes is B.

Thus, we conclude that just like 0 and 10 handshakes were in one pair, similarly 1 and 9

handshakes too have to be part of one pair of husband and wife.

Similar deductions, will lead to the realisation that 2 & 8, 3 & 7 and 4 & 6 handshakes

will also occur for couples amongst the 11 people questioned.

Hence, M must have shaken 5 hands.

The above question was solved on the basis of a series of deductions, which were

based on a series of Logical Form (LF), then logical structures.

Let us consider another example:

Example 2 Consider the following grid:

L P

M O Q

N R

Each letter in the above grid represents a different digit from 0 to 9, such that

L ¥ M ¥ N = M ¥ O ¥ Q = P ¥ Q ¥ R. Find the value of ‘O’.

Solution In order to solve such a question, one needs to proceed systematically making

one deduction at a time.

Deduction 1 There are seven alphabets and ten digits. We need to somehow eliminate 3

of these 10 to define the 7 digits required to be allocated.

It is obvious, that ‘0’ cannot be used since if we make any alphabet ‘0’ we will end up

with a product that ‘0’ in one or a maximum of two of the three cases.

Deduction 2 You need to find a product which can be made in three different ways.

Deduction 3 Two of these three ways have to be independent of each other with no

matching digits and the third way has to be drawn out of one digit each from the first

two ways and one independent digit that has not been used.

Also, at this point there are 9 digits and 2 more to be eliminated.

Now we move to the process of Trial & Error.

Notes on Interpretation

Trial & Error is one of the most useful processes for solving questions based on

reasoning. Principally there are three ways for carrying out trial & error search.

(1) Complete trial & error.

(2) Directed trial & error.

(3) Blind trial & error.

The blind trial and error is what most students practice and hence are unable to solve

logical questions. Since they do not use their deductive logic to do a more focused

search, they end up going round in circles while trying to solve such questions.

Instead directed trial & error and comprehensive trial & error are superior problem

solving processes and therefore score above the complete trial and error method.

Application of Directed Trial & Error The question above is a classical situation

warranting a directed trial & error. Hence, this is best illustrated through the example.

At this stage we are at a situation where we know: 1, 2, 3, 4, 5, 6, 7, 8 and 9 have to be

allocated to L, M, N, O, P, Q and R.

At this point, take a call as to whether you want to use 9 or not? Nine and 1 are different

from the other numbers primarily because they are the highest and lowest numbers

respectively and also because 9 gives us “maximum room for manoeuvering” the

question as compared to the other numbers, while 1 is a useful tool for a third multiplier

if we do not want to change the value of the multiplication.

Note: The student should analogise the thinking process applied here to the thinking

process used to unravel a ball of wool which has got entangled.

To disentangle an entangled ball of wool, we need to search the end of the ball. Once

you identify either end, the remaining process of disentangling the ball requires very

elementary logic coupled with patience and perseverance. The logic for picking up the

end point of the ball of wool is that it is different from all other points in the ball.

Similarly, in reasoning questions, we need to identify things/objects/people which are

different from other things/objects/ people and start our solution from there.

After that, the whole process becomes one of use of elementary logic for elementary

deductions coupled with patience and perseverance.

In the question under consideration, if we take a call on using ‘9’ and decide to do so,

we can then deduce that the required product has to be a multiple of 9.

[Remember that at this point of time we have ignored the line of thinking that neglects 9.

We will have to consider it, if we do not get an answer by using 9].

Since the product has to be a multiple of 9, assume that the product is 36. But, this

product eliminates the use of 8,7 & 5 and leaves only 6 digits. Going below 36 as a

product for further trial and error will further reduce the number of possibilities. Hence,

let us try to go to the higher extreme & try to experiment with the number 72.

We see that 72 = 3

2 ¥ 2

3 and can be formed by 9 ¥ 8 ¥ 1 or 6 ¥ 3 ¥ 4 or 9 ¥ 4 ¥ 2.

This satisfies our deduction that 72 gives three ways of solving the question. Also, the

second requirement that there should be two ways which are independent of each other

and a third way which uses one term each from the two independent ways and one

unique term is also satisfied.

Since 6 ¥ 3 ¥ 4 is independent of 9 ¥ 8 ¥ 1 in all its digits. Also, 9 ¥ 4 ¥ 2 uses the no. 9

from 9 ¥ 8 ¥ 1 & 4 from 6 ¥ 3 ¥ 4.

Hence, the following possibility emerges:

L = 6 P = 8

M = 4 O = 2 Q = 9

N = 3 R = 1

You need to understand here that, the digit ‘2’ is the only one which is fixed to its place

in the grid. All the other digits can be changed. Thus, we can have alternative

arrangements like:

L = 3 P = 1

M = 4 O = 2 Q = 9

N = 6 R = 8

OR

L = 6 P = 6

M = 9 O = 2 Q = 4

N = 1 R = 3

The only thing that is fixed is the number 2 for O.

Example 3 Let us now consider another question, which is a classic case of complete

or comprehensive trial & error method.

Two people A & B are playing a game. Both A & B are logical people. There are two

boxes on the table. One of them contains 9 balls, the other contains 4 balls. In this game,

the players are supposed to take alternate turns of picking up balls according to the

following rules:

(a) Pick up as many balls as you want to from any box.

(b) Pick up an equal number of balls from both boxes.(if you pick from both boxes).

The person who picks up the last ball wins the game. In his/her turn it is mandatory to

pick up at least one ball. A has to play first. What should he do to ensure a win?

Solution

Deduction 1 The rules of the game define that there are two legal moves:

Picking up an equal number from each box or picking up any number from either box (at

least 1).

Deduction 2 A has 17 possible moves to make and since the question asks for one

particular move that will ensure a win, one of these 17 must be the winning move. It is

at this stage that you should realise that the question calls for a comprehensive trial &

error which should result in the elimination of 16 possibilities.

The starting position is 9, 4.

A’s options at the start of the game can be basically divided into three options:

I Pick up balls from the first box. II Pick up balls from the second box. III Pick up balls from both the boxes.

The position after A plays his move can be documented as follows (in terms of the

number of balls B has in front of him.):

[There are 9 moves in option I.] (Balls from first box are picked by A.) If

(a) A picks up 1 ball, B will be left with 8, 4.

(b) A picks up 2 balls, B will be left with 7, 4

(c) A picks up 3 balls, B will be left with 6, 4

(d) A picks up 4 balls, B will be left with 5, 4

(e) A picks up 5 balls, B will be left with 4, 4

(f) A picks up 6 balls, B will be left with 3, 4

(g) A picks up 7 balls, B will be left with 2, 4

(h) A picks up 8 balls, B will be left with 1, 4

(i) A picks up 9 balls, B will be left with 0, 4

Similarly, if he picks up balls from the second box(option II) he will have an end result

of: If

(a) A picks up 1 ball, B will be left with 9, 3

(b) A picks up 2 balls, B will be left with 9, 2

(c) A picks up 3 balls, B will be left with 9, 1

(d) A picks up 4 balls, B will be left with 9, 0

And for option III, If:

(a) A picks up one ball each from both boxes, B is left with 8, 3

(b) A picks up two balls each from both boxes, B is left with 7, 2

(c) A picks up three balls each from both boxes, B is left with 6, 1

(d) A picks up four balls each from both boxes, B is left with 5, 0

Out of these 17 options, the options of leaving (0,4), (9,0) and (5,0) are infeasible since

Deduction 3 A cannot leave B with a situation in which B can make into (1, 2) or (2, 1).

Evaluating the 13 options left for A, we get that (1, 4), (2, 4), (3, 4), (5, 4), (9, 1) , (9,

2), (6, 1) & (6, 2) are situations from which B can reach (2, 1) or (1, 2) in one move.

This means B will win if A leaves him with any of these eight situations.

Thus, A will eliminate these eight options from his list of 13 and come down to five

options which need further checking. These are: (6, 4), (7, 4), (8, 4), (9, 3) and (8, 3).

Notes on Interpretation

Normally CAT questions only use 10% of this logic. However, let us now consider a

question (Example 1.4) which appeared in CAT 2004 and was found to be extremely

tough to crack. In fact, the question has been put down as unsolvable by most famous

national level coaching centres on their website.

Example 4 The year was 2006. All six teams in Pool A of World Cup Hockey play

each other exactly once. Each win earns a team three points, a draw earns one point and

a loss earns zero points. The two teams with the highest points qualify for the

semifinals. In case of a tie, the team with the highest goal difference (goals for–goals

against) qualifies. In the opening match, Spain lost to Germany. After the second round

(after each team played two matches),the (pool A) table looked as shown on the next

page.

In the third round, Spain played Pakistan, Argentina played Germany, and New Zealand

played South Africa. All the third round matches were drawn. The following are some

results from the fourth and fifth round matches.

(a) Spain won both the fourth and fifth round matches.

(b) Both Argentina and Germany won their fifth round matches by 3 goals to 0.

(c) Pakistan won both the fourth and fifth round matches by 1 goal to 0.

Solution For solving the above question, one requires tremendous alacrity, logical

consistently and above all a cool head.

Also, the solution of the above question is dependent on your ability to interpret the Deduction If Germany had won the first game 2–1 against Spain, Spain would have

won its second round match by 4–0, while if Germany won by 1–0, then Spain would

have won its second round match 5–1 (since Spain has Goals For = 5 and Goals Against

= 2).

Further, since only two teams — New Zealand and South Africa have conceded 4 or

more goals, Spain must have played one of them. Looking into South Africa’s G.F/G.A

columns, if South Africa had conceded 4 goals in the second round, then it should have

won the first round (1,0). But, South Africa has lost both rounds.

Hence, Spain played its second round against New Zealand. Further, if this is true, no

other team can play New Zealand in round two.

At this stage, the following possibilities emerge.

Team Germany

Round 1 vs. Spain Won 1–0 or 2–1

Round 2 vs. Pak/S.A. Won 2–1 or 1–0

Round 3 vs. Argentina Draw

Team Spain

Round 1 vs. Germany Lost 0–1 or 2–1

Round 2 vs. New Zealand Won 5–1 or 4–0

Round 3 vs. Pakistan Draw

Team New Zealand

Round 1 vs. Arg/Pakistan Lost 0–1 or 1–2

Round 2 vs. Spain Lost 1–5 or 0–4

Round 3 vs. South Africa Draw

Deduction Team Pakistan won one round and lost one and GF/GA 2/1. Hence, won 2–0

and lost 0–1.

Now, since New Zealand played its first round against Pakistan or Argentina it could

not have lost 1–2. This is because in the case of Pakistan, if Pakistan had won 2–1

against NZ in round 1, its round 2 would have been a draw.

Further, Argentina has conceded no goals. Hence, it could not have won 2–1 against

N.Z.

This means that N.Z. must have lost 0–1 in its first match to Argentina (that cannot

happen against Pakistan, because Pakistan cannot win 1–0 in the first round since it will

result in a 1–1 draw in round 2).

Consequently N.Z. lost 1–5 in its second match to Spain and hence Spain must have lost 0–1 to Germany.

The following scenario emerges from these deductions:

Team Germany

Round 1 vs. Spain Won 1–0

Round 2 vs. S.A. Won 2–1

Note: Here Germany’s round 2 has to be vs. S.A., since Pakistan cannot lose 2–1

Team Spain

Round 1 vs. Germany Lost 0–1

Round 2 vs. N.Z. Won 5–1

Team N.Z.

Round 1 vs. Argentina Lost 0–1

Round 2 vs. Spain Lost 1–5

Team Pakistan

Round 1 vs. S.A. Won 2–0

Round 2 vs. Argentina Lost 0–1

The first three rounds are as under:

Round 1 matches:

Germany beat Spain 1–0

Argentina beat N.Z. 1–0

Pakistan beat S.A. 2–0

Round 2 matches:

Spain beat N.Z. 5–1

Argentina beat Pak 1–0

Germany beat S.A. 2–1

Putting all deductions into one table, the following picture emerges:

Germany Argentina Spain Pak New Zealand S. Africa

Germany — D#3 W(1–0)#1 W(2–1)#

Argentina D#3 — W(1–0)#2 W(1–0)#1

Spain L(0–1)#1 — D#3 W(9–1)#2

Pakistan L(0–1)#2 D#3 — W(2–0)#1

New Zealand L(0–1)#1 L(1–5)#2 — D#3

S. Africa L(1–2)#2 L(0–2)#1 D#3 

Based on these deductions, the following questions can be answered. 

1. Which one of the following statements is true about the matches played in the

first two rounds?

(a) Pakistan beat South Africa by 2 goals to 1.

(b) Argentina beat Pakistan by 1 goal to 0.

(c) Germany beat Pakistan by 2 goals to 1.

(d) Germany best Spain by 2 goals to 1.

2. Which one of the following statements is true about the matches played in the

first two rounds?

(a) Germany beat New Zealand by 1 goal to 0.

(b) Spain beat New Zealand by 4 goals to 0.

(c) Spain beat South Africa by 2 goals to 0.

(d) Germany beat South Africa by 2 goals to 1.

3. Which team finished at the top of the pool after five rounds of matches?

(a) Argentina (b) Germany

(c) Spain (d) Cannot be determined

Spain must be top of the pool since it has the best goal difference even in it’s

worst case scenario.

4. If Pakistan qualified as one of the two teams from Pool A, which was the other

team that qualified?

(a) Argentina (b) Germany

(c) Spain (d) Cannot be determined

Notes on Interpretation

This question has an ambiguity since according to the deduction Notes on Interpretation

This question has an ambiguity since according to the deductions, Spain and Germany

both should be above Pakistan in terms of goal difference and hence Pakistan cannot

qualify. However, if Pakistan qualifies, so do both Spain and Germany.

The above question was basically testing the ability of the student to analyse data. In the

very same paper (CAT 2004), another question on data analysis went as follows:

Example 5 Prof. Singh has been tracking the number of visitors to his homepage. His

service provider has provided him with the following data on the country of origin of

the visitors and the university they belong to:

Number of Visitors/Day

University 1 2 3

University 1 1 0 0

University 2 2 0 0

University 3 0 1 0

University 4 0 0 2

University 5 1 0 0

University 6 1 0 1

University 7 2 0 0

University 8 0 2 0

Number of Visitors/Day

Country 1 2 3

Canada 2 0 0

Netherlands 1 1 0

India 1 2 0

UK 2 0 2

USA 1 0 1

Deduction 1 Looking at Day 3, University 4 must belong to the UK and University 6

must belong to the USA.

Deduction 2 From Day 2 it is clear that University 8 has to be an Indian University

while University 3 has to be from Netherlands.

Deduction 3 From the analysis of Day 1 data , University 2 and University 7 should be

distributed amongst UK and Canada in either order, i.e. 2 belongs to UK and 7 to

Canada or 2 belongs to Canada and 7 to UK. [Symbolically, (2UK + 7 Canada) vs (2

Canada + 7 UK)]

Deduction 4 The visitor from USA on Day 1 must come from University 6. Hence,

University 1 and University 5 should be distributed between India and Netherlands.

With this set of deductions, we get the following table. Using this table the answers to

the following questions become quite elementary.

Number of Visitors/Day

University 1

University 1 I v N

University 2 UK v C

University 3 N

University 4 UK

University 5 N v I

University 6 USK

University 7 C v UK

University 8 I

1. To which country does University 5 belong?

(a) India or Netherlands but not USA

(b) India or USA but not Netherlands

(c) Netherlands or USA but not India

(d) India or USA but not UK

2. University 1 can belong to:

(a) UK (b) Canada

(c) Netherlands (d) USA

3. Which among the listed countries can possibly host three of the eight listed universities

(a) None (b) Only UK

(c) Only India (d) Both India and UK

4. Visitors from how many universities from UK visited Prof. Singh’s homepage in three days? 

(a) 1 (b) 2

(c) 3 (d) 4

Arrangement questions are one of the most common question types in logical reasoning.

As the name suggests, questions on arrangements typically involve arranging people or

objects in straight lines or around circles/squares or other geometrical shapes.

The key skills involved in solving questions on arrangements include but are not limited

to:

(i) the ability to visualise the geometrical shape of the arrangement situation.

(ii) the ability to order the clues in the correct order of usage (as explained in the

theory of logical reasoning).

(iii) the ability to perceive what indirect clues are talking about – and how to use

them.

(iv) the ability to convert clues written in language form into visual cues so that you

do not need to read the text again and again. Also, converting the language clues

to visual cues is critical for the purpose of being able to ‘see’ all the clues at

one go.

Illustrated below are the solutions to a few typical questions on arrangements. We

would advise you to first have a look at the question and try to solve the same on your

own before looking at the step by step process of solving the same – illustrated through

the revolutionary “Reaction Tracker” mechanism which is an integral part of this

section of the book.

Note: The Reaction tracker is a blow by blow account of exactly what reaction should

go on in your mind as you solve an individual question in reasoning.

Look at the following questions and try to solve them:

Example 1 Question at an easy level of difficulty

Directions: Study the information given below to answer these questions.

(i) Arnold’s fitness schedule consists of cycling, rowing, gymnasium, jogging and

boxing from Monday to Saturday, each workout is on one day, one day being a

rest day.

(ii) Gymnasium workout is done neither on the first nor on the last day but is done

earlier than rowing.

(iii) Jogging is done on the immediate next day of the rowing day.

(iv) Cycling is done on the immediate previous day of the rest day.

(v) Jogging and boxing were done with a two-day gap between them. 

(vi) Boxing was done on the following day the rest day.

1. Which of the following is a rest day?

(a) Wednesday (b) Tuesday

(c) Friday (d) Thursday

2. Cycling and jogging days have a gap of how many days between them?

(a) Nil (b) Two

(c) Three (d) Four

3. On which day is boxing done?

(a) Thursday (b) Friday

(c) Monday (d) Wednesday

4. Which of the following is a wrong statement?

(a) Gymnasium workout is done on the immediate previous day of rowing.

(b) Jogging is done three days after the day on which boxing was done.

(c) There is a gap of three days between the days on which cycling and rowing

are done.

(d) There is a two days’ gap between the rest day and the day on which

gymnasium workout is done.

5. Which of the following is the correct statement?

(a) Jogging competition is done after rowing.

(b) Cycling is done on Thursday.

(c) No workout is done on Wednesday.

(d) Rowing is done earlier than cycling.

REACTION TRACKER

The starting figure we start with when we read the first statement is:

Monday Tuesday Wednesday Thursday Friday Saturday

Workouts are cycling, rowing, gymnasium, jogging and boxing.

From the second and third clues (gymnasium workout is done neither on the first nor on

the last day but was done earlier than rowing and jogging was done on the day

immediately following the rowing day), we know that rowing and jogging should be

together and also that gymnasium has to be somewhere before this.

Visually this can be represented as:

From the fourth and sixth clues we have:

C-Rest day-B

Note: Putting it in a box signifies that there is no break between the items in the box.

Once we have these two visual representations we can go back to our original figure

and think as follows:

Monday Tuesday Wednesday Thursday Friday Saturday

Since gymnasium has to precede rowing and jogging, and gymnasium is not on the first

day we can have 3 possibilities for placing gymnasium—viz: Tuesday, Wednesday or

Thursday.

Possibility 1:

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Gymnasium

This case is rejected because, once we place gymnasium we would need to place

rowing and boxing in either Wednesday-Thursday or Thursday-Friday or Friday-

Saturday. In each of these cases, we would also need to place a 3 day period having Directions: Study the following information to answer these questions.

(i) Alex, Betsy, Chloe, Dennis, Edward, Fiona, Giles and Herbert are gamblers

sitting around a round table facing the centre.

(ii) Dennis is the neighbour of Alex but not of Herbert.

(iii) Betsy is the neighbour of Fiona and 4

th

to the left of Dennis.

(iv) Edward is the neighbour of Herbert and 3

rd

to the right of Fiona.

(v) Chloe is neither the neighbour of Alex nor of Giles.

1. Which of the following is wrong?

(a) Betsy is to the immediate left of Herbert.

(b) Herbert is to the immediate left of Edward.

(c) Dennis is 4

th

to the right of Fiona.

(d) All are correct.

2. Which of the following is correct?

(a) Dennis is to the immediate left of Giles.

(b) Alex is between Chloe and Dennis.

(c) Fiona is 3

rd

to the right of Dennis.

(d) Edward is between Herbert and Betsy.

3 Which of the following groups has the 2

nd person sitting between the 1

st and the

3

rd?

(a) Alex-Fiona-Chloe

(b) Giles-Alex-Dennis

(c) Betsy-Edward-Herbert

(d) Herbert-Fiona-Betsy

4 Which of the following pairs has the 1

st person sitting to the immediate right of

the second?

(a) Betsy-Herbert (b) Fiona-Betsy

(c) Edwards-Giles (d) Alex-Dennis

5. Which of the following pairs are fourth to one another?

(a) Chloe-Edwards (b) Fiona-Herbert

(c) Dennis-Chloe (d) Dennis-Betsy

6. If Chloe and Giles interchange their positions, which of the following will

indicate Alex’s position?

(a) To the immediate right of Giles

(b) 4

th

to the right of Chloe

(c) 2

nd

to the left of Giles

(d) To the immediate left of Chloe

6-11 :From clues (i) and (iii), we get following two possibilities:

Now, read clue (iv): Edward is third to the right of Fiona. This information rejects Case

I. Using clue (iv), we get the following two possibilities, i.e. Case II (a) and Case lI (b).

Now, from clue (ii), we come to know that Dennis is not the neighbour of Herbert.

Hence, reject Case lI(a). From clue (ii) we get the following two possibilities:

Now, from clue (v), Chloe can’t be the neighbour of Giles. Therefore, reject Case II (c).

Again, since Chloe can’t be the neighbour of Alex, the final seating arrangement will be

as follows:

The answers then become pretty straight forward

1. (c) 2. (c) 3. (b) 4. (d) 5. (d) 6. (d)

Example 3

Directions: These questions are based on the information that follows.

In a row of soldiers facing North, (i) Lambert is 8

th

to the right of Khurusheva; (ii)

Mickey is 16

th

from the left end; (iii) Lambert is 16

th

to the right of Jackson, who is 27

th

from the right end of the row; (iv) Khurusheva is nearer than Mickey to the right end of

the row; (v) there are 5 boys between Mickey and Khurusheva.

1. How many soldiers are there between Jackson and Mickey?

(a) One (b) Two

(c) Three (d) Data inadequate

2. How far away is Khurusheva from the right end of the row?

(a) 30

th

(b) 10

th

(c) 19

th

(d) 18

th

3. How many soldiers are there in the row?

(a) 50 (b) 40

(c) 36 (d) Data inadequate

4. How far away is Jackson from the right end of the row?

(a) Data inadequate (b) 24

th

(c) 25

th

(d) 27

th

REACTION TRACKER

Let us arrange the whole information.

From clue (i), we get

Hence the answers are:

1. (a)

2. (c) From the above information it is obvious that occupies 19

th position from the

right end of the row.

3. (b) Total numbers of soldiers = 13 + 3 + 5 + 1 + 7 + 1 +10 = 40

4. (d)

EXERCISE ON ARRANGEMENTS

Directions for Questions 1–5: (Question Category: Matching Puzzle) Study the

following information and answer the questions that follow:

i. Six picture cards P, Q, R, S, T and U are framed in six different colours – blue,

red green, grey yellow and brown and are arranged from left to right 

Reasoning questions on Rankings involve an ordering of people objects based on their

heights/weights/performance in an exam, etc. As the name suggests, in questions on

rankings you are supposed to place people/objects in a decreasing or increasing order

based on an attribute being measured.

Key skills required in solving Logical Reasoning questions based on rankings:

(i) The ability to visualise the structure in which the rankings have to be created

(ii) The ability to order the clues in the correct order of usage (as explained in the

theory of logical reasoning)

(iii) The ability to perceive what indirect clues are talking about, and find the

appropriate point in the solving process about how to use them

(iv) The ability to convert clues written in language form into visual cues so that you

do not need to read the text again and again, and also are able to ‘see’ all the

clues at one go.

The following illustrated examples (with the reaction tracker used to explain the

solutions) would help you get acquainted with questions based on rankings. For each of

the following questions first try to solve them on your own before looking at the reaction

tracker process of solving the same.

Example 1

Directions for Questions 1 to 4 (Constraint Based Arrangement):

i. Six students A, B, C, D, E and F participated in a self-evaluation test of Quants

and Data Interpretation (D.I.).

ii. The total marks of A in Quants was just above C and in D.I. just above F.

iii. B was just above C in D.I. but he scored less than D in Quants.

iv. F got more marks than D and E in D.I. but did not perform as well in Quants as

in D.I. as compared to D and E.

v. No one is in between C and D in Quant and C and A in D.I.

1. Who got the highest marks in D.I.?

(a) A

(b) B

(c) C

(d) Data inadequate

2. Which of the following students has scored the least in D.I.?

(a) Only D

(b) Only E

(c) Only D or E

(d) None of these

3. Who was just below D in Quants?

(a) B

(b) E

(c) C

(d) Data inadequate

4. Which of the given statements is not necessary to answer the questions?

(a) (ii)

(b) (iii)

(c) (iv)

(d) All are necessary

REACTION TRACKER

From the second statement we have:

QUANTS D.I.

A

C

A

F

From Statement (iii) we have: B just above C in D.I. & B somewhere below D in

Quants. At this point our figure remains the same as we cannot put this information into

the figure.

QUANTS D.I.

A

C

A

F

From Statement (iv) we have:

QUANTS D.I.

A

C

D & E above F

A

F

D & E below F

From Statement (v) we have:

QUANTS D.I.

A

C

D

B

C

A

F

D & E below F

This leads us to the following table:

In Quants In Data Interpretation

A

C

D

Also, B is less than D,

F is below D and E.

Note: E could be placed

anywhere as we don’t have

any information about E.

B

C

A

F

D and E are below F.

Hence B must be first.

The answers are:

1. B (b)

2. D or E (c)

3. Data inadequate (d)

4. All are necessary (d)

REACTION TRACKER

The first direct clue in the question is Clue (vi). Clue (vii) is the next clue to use as

using the last 2 clues we get that H,G and D play table tennis (thus fixing our list of TT

players). Hence, when we read that B does not play football we can deduce that B plays

cricket—as he cannot be playing table tennis since we already know the three people

playing TT. The other direct clue that we can use for placing an individual in our

starting figure based on height is clue (iv). According to this, E does not play cricket

(hence must be football) and he is second to the tallest. At this point we also need to

keep aside the information that E is taller than B outside the purview of the figure for

later use. At this point our figure would be as given below with the additional

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