Linear Arrangements

Introduction

When it comes to seating arrangements, a linear arrangements calls for objects or people to be placed in a straight line. Candidates must arrange people in a row or several rows according to the given criteria when answering questions involving linear arrangement reasoning. Examinees are required to ascertain the precise locations of items and their relative positions to each other based on the information given.

A variety of competitive tests will use verbal and logical reasoning questions to assess your ability to reason logically and solve problems. Interviewers frequently pose questions about linear arrangement reasoning. It is essential to be familiar with seating arrangement questions and answers when getting ready for your test or interview. Let us now explore the many sorts of seating arrangements and learn from examples and answers.

Single Row

In a single row Linear arrangements, people are seated next to each other in a straight line. The most common convention is that all the people face north. If no direction is given, you have to assume that the people face north. Assume eight people in a straight line e.g. A, B, C, D, E, F, G, H. Based on this, some of the common terminology (also applicable to the other question types in this chapter and other chapters like rectangular and square arrangements) is:

1. B is at extreme left and H is at extreme right end or B and H are at the corner or B and H are separated by the maximum possible distance. This can also be expressed as – B (or H) has exactly one neighbour.
2. A is to the immediate right to B and D is to the right of B. Note the difference between these. Similarly, E is to the immediate left of G and F is to the left of G.
3. EDF or FDE means D is between F and E or F and E are neighbours of D.
4. A _ _ E or E _ _  A means that there are two people between them or A is third to the right/left of E and A is three places away from E.

Always look for the clue that provides information about extreme position. For instance, if a statement says – A is at an extreme end and B is to his immediate right, then A is at the extreme left end. On the other hand, if a statement says A is at the extreme end and is to the right of B, then A is at the extreme right end.

Example:

Seven teachers of seven different subjects – Economics, Audit, Accountancy, Psychology, Mathematics, History and Strategic Management – sit in a row, but not necessarily in the same order. It is also known:

1. The Mathematics teacher is to the left of Strategic Management teacher.
2. The teacher of Economics is not an immediate neighbour of the Accountancy teacher.
3. The History teacher, who is at one of the extreme ends, is to the immediate right of the Accountancy teacher.
4. The Economic teacher is third to the right of the Psychology teacher.

Sol. Psychology   Mathematics   Audit   Economics   Strategic Management   Accountancy History

The teacher of which subject sits exactly between the Economics and Accountancy teacher?

• Strategic Management

(Since no direction is given, assume that the people face north)

Strategy for LRDI (CAT 2024)

Multiple Rows

In multiple row linear arrangements, some people (commonly 10 or 12 or 8) are placed in two rows facing each other, such that each person faces exactly one person. Hence, you need to treat them like two single row problems simultaneously. A major difference between the two is that you are given clues that establish the relation between people in the two rows e.g. A is opposite B or A is diagonally opposite B (which implies that both are at opposite extremes of different rows) or A is second to the right of the person opposite B, and so on. Like single rows, you can have more than one parameter e.g. city, sport etc.

Example:

There are two rows (row 1 and row 2) of books in a library facing each other and each row has five books. Each row has one book each on Physics, Chemistry, Economics, History and Mathematics. The books are equally spaced. Economics in row 2 is at a corner. In both rows, Economics is next to Chemistry, which is next to Physics. Physics is the only book that faces itself in the two rows. The two History books are at the maximum possible distance from each other. Only one more set of books on the same subject is diagonally opposite each other.

Question 1: Which book is opposite the Chemistry book of row 1?

(1) Economics

(2) History

(3) Mathematics

(4) Physics

(5) Cannot be determined

Solution: Denote the books by their initials i.e. P, C, M, H, E.

Let row 1 be the upper row and row 2 be the lower row.

HMPCE or ECРМН

Thus, Mathematics is opposite the Chemistry book of row 1.

Hence, option 3.

Floor Arrangements

These are the same as arranging the people in a straight line but the linear arrangements here is vertical rather than horizontal. Hence, all the solving techniques are the same. The clues may be framed as – “A lives three floors above B” or “Two people live between A and C” or *No one lives above D” and so on.

In linear arrangements the major variation in floor based questions is where you are given a “vacant” floor i.e. there may be floors in the building but only seven people live in it. So one floor is empty. This also has to be solved using the same techniques but you need to be careful of certain sentences. In a normal floor arrangement, “two people live between A and B” means the same thing as “There are two floors between A and B”. However, if a floor is vacant, “two people live between A and B” can mean:

A           or          A

D                        D

Vacant                C

C                         B

B

Thus, is does not necessarily mean that “there are two floors between A and B”. As seen above, there can even be three floors between A and B, if the vacant floor is between them.

Example :

Answer the questions below based on the given information:

Seven people – A to F – live in a seven-storeyed building having floors numbered one to Seven. The lowest floor is floor 1 while the topmost floor is floor 7. Only one person lives on a floor. They study in different colleges – Khalsa, KC, Vivekanand, Rizvi, Ruparel, Thakur ana SIES, in any order. Also:

1. Exactly one person lives between the people who study in Vivekanand and Thakur.
2. H lives on an even numbered floor and the person who studies in Vivekanand lives on an odd numbered floor.
3. The person who studies in Khalsa lives on the second floor.
4. E neither studies in SIES nor in Ruparel.
5. The person who studies in Vivekanand does not live on the first floor and the person who studies in SIES does not live on the topmost floor.
6. C lives immediately above the person who studies in Vivekanand.
7. D lives immediately below H.
8. G lives on the fifth floor.
9. The person who studies in Ruparel stays immediately above A.
10. The person who studies in SIES lives immediately above the person who studies in KC.
11. A lives on an odd numbered floor.

Question 1: Who studies in SIES?

• C
• G
• B
• D
• A

Solution: in floor based questions, the floor number is the primary parameter based on which you should make a table. Also, add the directly given data.

Thus, B studies in SIES.

Hence, option 3.

Question 2: The person living immediately below C studies in which college?

• Ruparel
• Thakur
• KC
• Khalsa
• Vivekanand

Solution: A lives immediately below C and studies in Vivekanand.

Hence, option 5

Question 3: Which of the following combinations is correct?

• D-1 Thakur
• B-7 Vivekanand
• E-2 Khalsa
• G-5 SIES
• B-6 Rizvi

Solution: Among the given combinations, only D-1 – Thakur is correct.

Hence, option 1

Strategy Tips

For single row, assume north facing if nothing is given. If some people face south, the interpretation of right and left changes.

Understand the implications of the various types of statements related to right/left.

For multiple row questions, first try to identify the people in each row or column Diagonally opposite people are at corners, unless explicitly mentioned.

For floor based questions, be careful when handling vacant floors. Always make the floor numbers as the main reference value for your table.

For questions on scheduling, make the unit of time (day, month, hour, etc) as the main reference value for the table. You may need to make mathematical calculations as well.

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