# Complex ArrangementLogical Reasoning

## 1. Introduction

Complex arrangement problems are the most common problem types in all entrance exams containing Logical Resoning section. These are similar to linear arrangement problems as they also deal with certain objects and their properties. But unlike linear arrangement, where the objects had only one property – their positioning, objects in complex arrangement have multiple properties. The information that is provided is of two types:

1. Information relating an object with its property
2. Information that matches two properties of an object.

Using these clues, you are required to match all the objects with their corresponding properties. We will take a simple example to make things clear.

Alex, Bridgette and Carol are three women who live in San Francisco, Los Angeles and San Hose, not necessarily in that order. Carol is a Doctor while the other two have one of the two professions – a Lawyer and a Realtor. The Realtor lives in San Hose. Bridgette lives in San Francisco.

In this case, the three objects are the three women – Alex, Bridgette and Carol. Properties are their city of residence and their profession.

Now, analyse and filter the information that is relevant.

Information:

1. Carol is a Doctor

This is the kind of information that relates an object with a property. It is given that Carol is a Doctor. This implies that Bridgette and Alex cannot be doctors.

Table below can be scrolled horizontally

Objects City of residence Profession
Alex
Bridgette
Carol   Doctor

2. The Realtor lives in San Hose.

This piece of information relates a property type with another property type. It says that the person who is the Realtor lives in San Hose. This further implies that Carol cannot be that person since it is already known that she is a doctor.

3. Bridgette lives in San Francisco.

This information again relates an object with a property type. It says that Bridgette lives in San Francisco. This means that she cannot be the Realtor as the Realtor lives in San Hose. Thus the only one remaining, i.e. Alex has to be the Realtor who lives in San Hose. It also means that Bridgette is a Lawyer and that Carol lives in Los Angeles.

Table below can be scrolled horizontally

Objects City of residence Profession
Alex San House Realtor
Bridgette San Francisco Lawyer
Carol Los Angeles Doctor

Complex Arrangement problem sets have two distinct types of questions that can be asked.

### 1.1. Direct Questions

These questions can be answered on the basis of the unique combination of objects and their properties that is obtained from the analysis of the problem statement. In the above example, a direct question can be framed like this – ‘In which city does Alex live?’

### 1.2. Conditional Questions

These questions are asked when a definite solution for the problem statement cannot be arrived at. Consider the above example without the third piece of information.

If it is not known that Bridgette lives in San Francisco, then the unique combination of objects and their properties cannot be found. A conditional question can be framed like this – ‘If Bridgette lives in San Hose, what is the profession of Alex?’

It follows from this additional information that Bridgette is the Realtor and so, Alex becomes the Lawyer.

## 2. Solving Techniques

There are the two types of solving techniques that can be used to solve complex arrangement problems.

A. Consolidated Table Method

B. Matrix Method

### 2.1. The Consolidated Table Method

In the Consolidated Table Method, we prepare a table where the second column lists down the objects, and the remaining columns have properties associated with the objects. The first column is to note down which property types are eliminated for the object that we are considering in that particular row.

To understand this better, let us go back to the same example that we used in the previous section. The three objects in the question were Alex, Bridgette and Carol. The properties were city in which they live and their profession.

The options for the first property were San Francisco, Los Angeles and San Hose, while that for the second were Lawyer, Doctor and Realtor.

When we start, we will leave the first column blank to write down the property types that are eliminated for a particular object. In the second column, we will list down the names of the three objects i.e. Alex, Bridgette and Carol in this case. We will title the third column as ‘City’ and will write down the corresponding option for city for each object as and when we analyse the data. Similarly, we will title the fourth column as ‘Profession’ and will write down the profession for each object.

The complete consolidated Table will look like this:

Table below can be scrolled horizontally

 Eliminated Name City Profession Doc, SF, Law Alex San Hose Realtor Doc, Rea Bridgette San Francisco Lawyer SH, SF Carol Los Angeles Doctor

The step by step method of solving a question set using the consolidated table method is as follows:

• First, prepare a table as per the rules explained above.
• Read the conditions that are given and find out the information that relates a particular object with a property type. This is direct information and should be filled in the table in the appropriate row. Also, note this down as the eliminated property type against rows of all other objects.
• After listing down all direct information, look for cells in the table that can be filled using the eliminated properties for any object. For example, if two out of three property types have been eliminated for a particular property of an object, then it follows that the object has the third property type.
• After this, look for information that relates two or more property types with each other and look for places in the table where they can be accommodated.
• Finally, a partially complete or a complete table will be obtained which has the objects aligned with their property types. In case of partially complete tables, use short forms of property types separated by ‘/’ to accommodate multiple cases in the solution.
• Use this table to answer the questions that follow. If the question has additional information, modify the table accordingly.

The solved examples that follow will illustrate the consolidated table method of solving complex arrangement problems.

### Example

A rock band has five members. Their names are John, Mark, Andrew, Paul and Simon. The instruments that the band uses are the drums, the flute, the keyboard, the guitar and the bass. Each member plays only one of the instruments. Each member of the band is from one of South Africa, USA, Ireland, France and England, though not necessarily in that order. The hair colours of the members are blonde, green, red, black and yellow, again not necessarily in that order.

The following information is known.

1. John’s hair is red. He is not from Ireland or France.
2. Andrew plays the drums. His hair is not yellow, and he is not from the USA.
3. Paul and Simon have yellow and green hair, not necessarily in that order.
4. Mark plays the guitar. He is from England.
5. Paul and Simon are from France and the USA, not necessarily in that order.
6. The Irishman has blonde hair, and the Frenchman has green hair.
7. The South African plays the bass and the American plays the keyboards. One of them has yellow hair.

Analysis of the information – Preparation of a consolidated table

We first choose short forms as follows:

Drums, flute, keyboard, guitar and bass: d, f, k, g and b respectively.

Blonde, green, red, black and yellow: bl, gr, re, bk and ye respectively.

South Africa, USA, Ireland, France and England: SA, USA, IRE, FR and ENG respectively.

We then draw the table. Our table initially looks like this:

Table below can be scrolled horizontally

 Eliminated Options Name Instrument Hair Colour Country John Mark Andrew Paul Simon

We start analysing conditions.

1. John’s hair is red. He is not from Ireland or France: We will fill the hair colour column for John, and eliminate two country options for him. Note: Remember to eliminate red hair as an option for everyone else.
2. Andrew plays the drums. His hair is not yellow, and he is not from the USA: Fill in this information for Andrew in the corresponding boxes. Eliminate ‘drums’ as an option for the rest.
3. Paul and Simon have yellow and green hair, not necessarily in that order: Thus, we can eliminate ye and gr for everyone else, and re, bl and bk for Paul and Simon.
4. Mark plays the guitar. He is from England: We fill in these two boxes for Mark, and eliminate them for the others.
5. Paul and Simon are from France and the USA, not necessarily in that order: We eliminate the USA and France for everybody else, and eliminate all other countries for Paul and Simon.

The information we get so far is consolidated in the table below:

Table below can be scrolled horizontally

 Eliminated Options Name Instrument Hair colour Country IRE, FR, d, y, gr,g, ENG, USA John Red re, d, ye, gr Mark Guitar England re,ye, USA, gr, ENG, FR Andrew Drums re, d, bk, bl, g, ENG, IRE, SA Paul re, d, bk, bl. g, ENG, IRE, SA Simon

We can see that we have eliminated all the countries except SA for John, and hence he has to be from South Africa only. We write SA against his name, and eliminate SA for everybody else. We then see that all options except IRE are eliminated for Andrew. So he must be from Ireland.

1. The Irishman has blonde hair, and the Frenchman has green hair: We know that the Irishman is Andrew; he thus has blonde hair. Eliminating blonde hair for everybody else, we see that all hair colour options except black have been eliminated for Mark, and hence Mark has to have black coloured hair.We do not know who among Paul or Simon is the Frenchman, and so cannot use the second fact as of now.
2. The South African plays the bass and the American plays the keyboards. One of them has yellow hair: So John plays the bass.

Our table now looks like below:

Table below can be scrolled horizontally

 Eliminated Options Name Instrument Hair colour Country IRE, FR, d, y, gr,g, ENG, USA John Bass Red South Africa re, d, ye, gr, bl Mark Guitar Black England re,ye, USA, gr, ENG, FR Andrew Drums Blonde Ireland re, d, bk, bl, g, ENG, IRE, SA Paul re, d, bk, bl. g, ENG, IRE, SA Simon

Among Paul and Simon, we do not know who is the Frenchman and who is the American. But we know that the Frenchman has green hair and the American plays the keyboards. Thus the American must have yellow (the only remaining colour) coloured hair and the Frenchman must play the flute.

So, we have only two possible cases: one case in which Paul is the American and the other in which he is the Frenchman. The tables corresponding to both cases are as below

Case (1):

Table below can be scrolled horizontally

 Name Instrument Hair colour Country John Bass Red SA Mark Guitar Black England Andrew Drums Blonde Ireland Paul Keyboard Yellow USA Simon Flute Green France

Case (2):

Table below can be scrolled horizontally

 Name Instrument Hair colour Country John Bass Red SA Mark Guitar Black England Andrew Drums Blonde Ireland Paul Flute Green France Simon Keyboard Yellow USA

Having analysed all the information available, we move to the questions.

#### Question 1:

Which country is Andrew from?

(A) Ireland     (B) USA     (C) England     (D) France     (E) South Africa

#### Solution:

It can be seen from the tables for both the cases that Andrew is from Ireland.

Hence, option A is the correct choice.

#### Question 2:

Which member of the band has black hair?

(A) Mark     (B) Andrew     (C) Paul     (D) Simon     (E) None of these

#### Solution:

It can be seen from the tables for both the cases that Mark has black hair.

Hence, option A is the correct choice.

#### Question 3:

From which countries do the Flute and Bass players (in that order) come?

(A) USA and South Africa

(B) Ireland and the USA

(C) The USA and Ireland

(D) France and South Africa

(E) England and France

#### Solution:

It can be seen from the tables for both the cases that the Flute player is from France and that the Bass player is from South Africa.

Hence, option D is the correct choice.

#### Question 4:

If Simon plays the Flute, which member of the band has yellow hair?

(1) Mark     (2) Andrew     (3) Paul     (4) Simon     (5) None of these

#### Solution:

Now, we have come to a point where we have to choose between cases (for this one question only). Simon plays the Flute only in case (1). According to the table for case (1), Paul has yellow hair.

Hence, option C is the correct option.

#### Question 5:

If Paul is French, what instrument does Simon play?

(1) Bass     (2) Guitar     (3) Drums     (4) Keyboard     (5) Flute

#### Solution:

Again, we have to identify the correct case based on the additional fact that is given. Since Paul is French, we have to consider case (2). According to the table of case (2), Simon plays the keyboard.

Hence, option D is the correct option.

### 2.2. The Matrix Method

In the Matrix Method, we prepare a table with the first column as our object heading, and the remaining column headings as the various property types. Each row of the matrix corresponds to one object name. Ticks or crosses are put in the boxes other than in the first column depending on whether the object (the entry in the first column) has that property type or not.

Let us look at the same example taken from the first section. While making the matrix table for this case, the object names i.e. Alex, Bridgette and Carol will be listed down in the first column. The property types – lawyer, doctor and realtor – of the first property i.e. city; and San Francisco, San Hose and Los Angeles – of the second property i.e. profession will follow as the next six columns.

The information will be marked in the matrix table using ticks and crosses. A completed matrix table for this example will look like this:

The step by step method of solving a question set using the matrix method is as follows:

• First, prepare a matrix table listing down the objects and the property types as the headings of all the columns. List the object names as the rows of the table.
• Read the information to find out data that relates an object name directly to a property type. In such cases, put a tick against the property type column for that particular object name. Put crosses in all other rows for that particular column and also for all other sub columns of that property for that object.
• To accommodate information that relates two or more property types with each other, look for rows in the matrix table that carry no ticks or crosses for the properties that are being considered.
• A partial or a complete matrix table will be obtained after all the given information has been represented. Use this table to answer the questions that follow. Accommodate any additional information from the questions in the table to complete it if it is not.

The next few solved examples illustrate the matrix method of solving complex arrangement problems.

### Example

#### Example:

Five persons with names P, M, U, T and X live separately in any one of the following: a palace, a hut, a fort, a house or a hotel. Each one likes two different colours from among the following blue, black, red, yellow and green. U likes red and blue. T likes black.

The person living in a palace does not like black or blue. P likes blue and red. M likes yellow. X lives in a hotel.

M lives in a:

(A) hut     (B) palace     (C) fort     (D) house

#### Solution:

The information can be organized in a tabular form as shown below.

Table below can be scrolled horizontally

 Name U T P M X Colour liked Red, Blue Black Blue, Red Yellow Stays in Hotel

$\therefore$The person living in a palace does not like black or blue implies that person cannot be U, T or P.

$\therefore$The person who lives in a palace will be either M or X. But X lives in a hotel.

$\therefore$ M must live in a palace.

Hence, option B is the correct choice.

## 3. Comparison between Solving Techniques

The above mentioned methods of solving complex arrangement problems differ only in approach and not in reasoning. The suitability of a method depends on various parameters, which you need to consider before you start solving the problem.

The Matrix method leaves very little room for error because, unlike the Table method, it doesn’t require rereading the eliminated options several times. However, a disadvantage of this method is that the size of the table may become very large if the problem deals with lot of properties or property types. In such cases, the matrix method will be very tedious and time-consuming.

The consolidated table method uses a table that is much smaller than the matrix that is formed in the matrix method. Thus, the disadvantages of the matrix method can be overcome. A disadvantage of this method, however, is that you often cannot depict all the given information on the table, and might have to remember a couple of facts while solving the problem. Thus, the method has more scope for error than the matrix method.

It is suggested that you start practicing with both methods but narrow down to a single method that suits you best.

### 3.1. Strategy Tips

• Identify the object and properties of the objects before making the solution table.
• Decide on a solving method as per your suitability and the number of parameters of the problem statement.
• While solving, first always fill in the direct information. This makes it easier since there are lesser property types to consider as some get eliminated.
• When running short on time, read the questions before making the solution table. Many a times, some questions can be answered even before completing the solution table.

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