# Numerical LogicLogical Reasoning

## 1. Introduction

Numerical logic deals with numbers in various situations like money, weight, distance, position and so on. Most problems in numerical logic will specify several conditions on the numbers involved in a given situation. You have to then construct the entire situation using the given conditions and apply basic concepts of mathematics to solve the related questions.

To be able to effectively tackle numerical logic, you will have to use an "if"-"then" approach to cover all the possibilities introduced by the given conditions. Since the conditions given can be quite confusing at times, you need to analyse the problem in a systematic manner and ensure that you do not miss out on any critical nuance in the given data. You should choose the most helpful clue to start solving the problem. Solving becomes easy if the data can be arranged in a structured format.

In this chapter, we will look at the following types of numerical logic questions:

A. Sequential output (algorithm based) problems

B. Distribution type problems

C. Arrangement of numbers

D. Gambling game

E. Odd weight problems

In the next section, we will look at each of these types in greater detail. We will also look at solved examples of each type to illustrate the solving technique.

## 2. Sequential Output (Algorithm Based)

### 2.1. Structure

In question of this type, you are usually given information about a computing machine which accepts an input, processes it according to certain rules and then gives an output. You are given the steps followed for processing the sample input and the output. You are expected to identify the rule or set of rules (algorithm) which governs the machine and then answer questions related to how the machine would process other inputs. The input would usually be a string of words or a set of numbers.

Let us look at an example of such an algorithm. The following are the steps that it follows after taking a number as an input:

STEP 1: Multiply by 5

STEP 2: Subtract from 100

STEP 3: Square the number

STEP 4: Output the number

If we input the number 18, the following operations will be performed:

STEP 1: $18 \times 5$

STEP 2: $100 -90 = 10$

STEP 3: $10 \times 10$

STEP 4: $100$

### 2.2. Solving Technique

STEP 1: Read the given conditions carefully.

STEP 2: Analyse the changes made in each step by the machine on the previous input to reach the output.

STEP 3: Try to infer the algorithm which is followed for making changes at each step.

STEP 4: Now apply that algorithm to the inputs given in the related problems.

### Example:

#### Directions:

A string of words is fed into a computing machine. It processes the input and gives an output in the following manner:

Input: Honesty is the best policy.

Step 1: is Honesty the best policy

Step 2: is the Honesty best policy

Step 3: is the best Honesty policy

Step 4: is the best policy Honesty

Output: is the best policy Honesty

Answer the following questions based on the above information.

#### Question 1:

If the input is “A stitch in time saves nine”, how many steps will be required to get the output?

(A) 3     (B) 5    (C) 6     (D) 4     (E) This string cannot be processed by the given algorithm

#### Solution:

The first step is to identify what algorithm or set of rules is followed by the machine for processing the input and generating the output.

By observing the steps given in this case it is very easy to identify that the first word “Honesty” is moved one place to the right in each step till it reaches the end of the string, while the order of the other words remain the same.

So, if the input given is “A stitch in time saves nine”, the steps followed are:

Step 1: stitch A in time saves nine

Step 2: stitch in A time saves nine

Step 3: stitch in time A saves nine

Step 4: stitch in time saves A nine

Step 5: stitch in time saves nine A

Step 5 gives the output in this case.

Hence, option B.

#### Question 2:

If the input is “A rolling stone gathers no moss”, what will be the fifth word in the string after 4 steps?

(A) Gathers     (B) No     (C) Moss     (D) A     (E) Stone

#### Solution:

For the input “A rolling stone gathers no moss”, the steps followed are:

Step 1: rolling A stone gathers no moss

Step 2: rolling stone A gathers no moss

Step 3: rolling stone gathers A no moss

Step 4: rolling stone gathers no A moss

After 4 steps the fifth word is “A”.

Hence, option D.

Note: The algorithm is that after the nth step, the first word is in the (n+1)th place. So after 4 steps, the first word will be in the fifth place and the answer can be directly marked as A without listing all the steps.

#### Question 3:

The input is “Twenty overs are enough”. Suppose the output of the first cycle is given as an input to the same computing machine and this is done 3 times in succession, what will be the output?

(A) overs are enough Twenty

(B) are enough Twenty Overs

(C) enough Twenty Overs are

(D) Twenty overs enough are

(E) are twenty overs enough

#### Solution:

In this case we need not list each step since only the final output of each cycle is important.

Input: Twenty overs are enough

Output of Cycle 1: overs are enough Twenty

Output of Cycle 2: are enough Twenty overs

Output of Cycle 3: enough Twenty overs are

Hence, option C.

It is important in algorithm-based questions to identify which question requires actual listing of each step and which question can be solved using a short cut, like the third question in this case. This will help you save crucial time.

## 3. Distribution

### 3.1. Structure

In question of this type, you are usually given information about a set of things like money, balls, fruits, cards, tasks and so on, which have to be distributed among a set of recipients like people, boxes, time slots and so on, under a set of constraints.

Let us look at a simple example to understand this question type.

There were 12 balls – some white and some black that were distributed equally between 3 boys such that all the boys got balls of both the colours. Then, the balls were redistributed equally between 4 boys, not taking care that every boy gets balls of both colours.

In this distribution, two of the boys got only black balls. What can be the minimum and maximum number of white balls?

Now, when the 12 balls were distributed between 3 boys making sure that all boys got balls of both colours, each boy had 4 balls.

Thus, minimum number of white balls = 3

And, maximum number of white balls = 9

In the second case, two boys have 3 black balls each. Between the other two boys,

Minimum number of white balls = 2

And, maximum number of white balls = 6

Looking at both the cases, the minimum number of white balls can be 3 while the maximum can be 6.

### 3.2. Solving Technique

Step 1: Read the given conditions carefully.

Step 2: Organize the given data into sets of things that have to be distributed and a set of recipients.

Step 3: Identify the constraints with respect to each set.

Step 4: Begin a stepwise process of distribution according to the conditions given in each individual problem.

### Example:

#### Directions:

5 five hundred rupee notes, 10 hundred rupee notes and 5 fifty rupee notes are distributed among 4 people A, B, C and D such that each person has an equal number of notes.

Answer the following questions based on the above information.

#### Question 1:

If A has Rs.2100 what is the maximum amount that B can have?

(A) 850     (B) 900     (C) 1300     (D) 450     (E) 250

#### Solution:

Analysing the given distribution pattern, we know that each person has five notes.

The only way for A to have Rs.2100 is to have 4 five hundred rupee notes and 1 hundred rupee note $(4 \times 500 + 1 \times 100 = 2100)$.

From the remaining notes, B can have the maximum amount if he gets notes of the highest denominations possible. There is only 1 five hundred rupee note remaining. So the maximum amount that B can have is $1 \times 500 + 4 \times 100 = 900$

Hence, option B.

#### Question 2:

If B has only one type of notes and C has Rs. 850, how much does B have?

(A) 400     (B) 2500     (C) 500     (D) 250     (E) 450

#### Solution:

Analyzing the given distribution pattern, we know that each person has five notes.

Now, C has Rs. 850. This is possible only if he has $1 \times 500 + 3 \times 100 + 1 \times 50 = 850$

$\therefore$ B cannot have either 5 five hundred rupee notes or 5 fifty rupee notes as only 4 of each type are now available.

$\therefore$ B has 5 hundred rupee notes. So the total amount with him is $5 \times 100 = 500$

Hence, option C.

## 4. Arrangement of Numbers

### 4.1. Structure

In questions of this type, you are usually given a structure like a matrix or a group of matrices. You are given conditions for arrangement of numbers inside this structure. You are supposed to arrange numbers in it to answer the questions. This involves a basic understanding of numbers.

For example, arrange the numbers from 1 to 9 in a $3 \times 3$ grid such that the numbers in any row, column or a diagonal add up to 15.

The following grid displays one of the solutions for this problem:

 2 7 6 9 5 1 4 3 8

### 4.2. Solving Technique

Step 1: Read the given conditions carefully.

Step 2: Identify the constraints of the structure and those of the numbers that have to be arranged in the structure.

Step 3: You may need to follow different possible paths of arranging numbers. You can copy the structure and follow this process simultaneously for different conditions.

### Example:

#### Directions:

Enter digits from 1 to 9 into the blank spaces. Every row, column and every $3 \times 3$ square must contain numbers from 1 to 9. Consider that the columns are numbered from 1-9 and the rows from a-i.

Answer the following questions based on the above information.

#### Question 1:

What is the sum of numbers in 3a, 4d, 3g, 1d, 3c, 3f, 5d, 3h, 7d, 3b, 3d, 2d, 3e, 9d, 6d, 8d, 3i?

(A) 88     (B) 76     (C) 87     (D) 82     (E) 90

#### Solution:

If you arrange the given positions in proper sequence you get: 3a, 3b, 3c, 3d, 3e, 3f, 3g, 3h, 3i & 1d, 2d, 4d, 5d, 6d, 7d, 8d, 9d. This means you have to find the sum of numbers in the 3rd column and 4th row counting the common cell 3d only once. According to the rules, any row or column will have numbers from 1-9.

The value of 3d is already given to be 5.

Answer $= 45 + 45 - 3 = 87$

Hence, option C.

#### Question 2:

What is the value in position a8?

(A) 2     (B) 5     (C) 9     (D) 7     (E) 4

#### Solution:

If you look at the top right $3 \times 3$ square in which a8 lies, you see that the missing numbers in that $3 \times 3$ square are 2, 7 and 9. Looking at row (a), you see that a3 already has 9. So, a8 can possibly have 2 or 7.

Note that the other two blank spaces in the top right square are 7c and 9c. You can see that both columns (7) and (9) already have the number 7. So, the only possible position for 7 in the top right $3 \times 3$ square is a8.

Hence, option D.

## 5. Gambling Game

### 5.1. Structure

In questions of this type, you are given information about a set of people playing a game with certain rules. Usually, the initial amount of money with each player is given. The rules of the game include how the money needs to be given to or taken from each player. You are supposed to determine how the game would proceed according to the given conditions and answer questions based on the result of the game.

Let us look at an example of such a game.

Ram and Rahim play a card game. Each of them can put down 1, 2, 3 or 4 cards at a time on the stack. The person who places the 10th card on the stack wins.

If Ram starts the game and both of them play intelligently, who wins the game?

#### Solution:

To understand this problem, let us begin from the last stand i.e. the winner’s hand.

Since a person can put down 1, 2, 3 or 4 cards at a time, it means that if the stack reaches 6, the next person wins.

Thus, if Ram begins, Rahim must make sure that he does not take the stack to six cards. Also, he must ensure that Ram takes the stack to the 6th card or a card higher than that. If he does this, he will win in all cases.

To do this, Rahim simply needs to bring the stack total upto 5 after Ram plays the first turn.

If Ram places one card, Rahim should place four,

If Ram places two cards, Rahim should place three,and so on.

Once Rahim has played his turn, Ram will lose in all cases.

### 5.2. Solving Technique

Step 1: Read the given conditions carefully.

Step 2: Organize the given data into number of players and initial conditions.

Step 3: Identify the constraints of each round of the game.

Step 4: Develop a stepwise process of how the game should proceed according to the conditions given in each individual problem.

### Example:

#### Directions:

A, B and C play a game. Each of them has Rs. 500 in the beginning. At the end of each round of the game, the player who is in third place in that round gives Rs. 20 each to the other two and the player who is second gives Rs. 30 to the player who is first.

Answer the following questions based on the above information.

#### Question 1:

Content

If the three players play 5 rounds of the game, what is the maximum possible amount of money that A could have if B and C have won at least one game each?

(A) 650     (B) 630     (C) 700     (D) 680     (E) 610

#### Solution:

To win the maximum amount of money it is necessary to win the maximum amount of games. Since B and C have won at least one game each, A could at most have won 3 games.

Also according to the rules,

The player who wins gains Rs. 50 (20 from 3rd and 30 from 2nd)

The player who is second loses Rs. 10 (20 from 3rd and -30 to 1st)

The player who is third loses Rs. 40 (- 20 to 1st and -20 to 2nd)

So, to win maximum money under the given constraints, A must win 3 games and come 2nd in the other 2.

Thus the net gain for A $= 50\times 3 + (-10) \times 2 =$ Rs. 130

A starts with Rs. 500

So, A will have at most Rs. 630 after 5 rounds.

Hence, option B.

## 6. Odd Weight

### 6.1. Structure

In questions of this type, sets of items like balls, coins, or bags are given. All elements, except one, in every set have the same weight. You are expected to find the minimum number of times you require to weigh using a spring or a pan balance to find the odd weight item.

Let us look at a simple example to understand this question type better.

There are 9 identical balls, one of which is slightly heavier than the rest. Given a pan balance, what is the minimum number of times one has to weigh to ensure that the heavy ball is found?

To solve this problem, let us divide the 9 balls into 3 groups of 3 balls each.

Now, use the weighing scale to weigh group 1 against group 2.

If group 1 is heavier, then the heavy ball lies in group 1.

If group 2 is heavier, then the heavy ball lies in group 2.

If the scale is balanced, the heavy ball lies in group 3.

Now, take 2 balls from the group that has the heavy ball and weigh them against each other.

The heavy ball will tilt the balance on its side. If both balls weigh the same, then the 3rd ball of that group is the heavy ball.

Thus, a minimum of 2 turns are required to determine the heavy ball from the group.

### 6.2. Solving Technique

To solve these problems you can use the following methods:

1. In case of a pan balance, iterations of dividing the items into groups of 3 are to be performed. Hence, if the number of items is $n$, such that $3^a < n \leq 3^{(a+1)}$, then the minimum number of times you need to weigh to find the odd item is $(a + 1)$.
2. In case of a spring balance, iterations of dividing the items into groups of 2 are to be performed. Hence, if the number of items is $n$, such that $2^a < n \leq 2^{(a+1)}$, then the minimum number of times you need to weigh to find the odd item is $(a + 1).$

### Example:

Of the 80 boxes that Ram has, each except one weighs 1 kg. If he uses a spring balance, what is the minimum number of times he must weigh to find the odd box?

(A) 8     (B) 9     (C) 5     (D) 6     (E) 7

#### Solution:

Since a spring balance is used, the boxes will have to be successively divided into groups of 2, till the odd weight box is found. These iterations are performed as follows:

40     40

20     20

10     10

5       5

2       2      1

At the last step, he will have to weigh thrice to find the faulty box from an odd number of boxes.

$\therefore$ The total number of times he needs to weigh is 7.

Hence, option E.

Alternately,

Since a spring balance is used, 80 can be represented as

$2^6 < 80 < 2^7$

So the total number of times he will have to weigh is 7.

## 7. Strategic Tips

### 7.1. Sequential Output Tips

• For complex algorithms, it is more efficient to make a table with every row listing down the step operations and the columns representing various inputs.
• Be wary of missing operational steps for long algorithms, as such errors would affect the solution of all questions.

### 7.2. Distribution Problem Tips

• First, ascertain the object being distributed and the audience to which it is being distributed.
• If unique values of distributed objects are not obtained, continue solving using a range of values.

### 7.3. Arrangement of Numbers Tips

• The trial and error method works well in these question type.
• Coming up with a set of rules that determine the arrangement is a good ploy but is very time consuming.
• Looking at the options of the questions can provide vital clues for solving these questions.

### 7.4. Gambling Game Tips

• All gambling game questions require an initial analysis which is common to the solution of all questions.
• Gambling game questions cannot be answered independently of each other and any such attempts to do so should be avoided.

### 7.5. Odd Weight Problem Tips

• Try and formulate a method for solving these question types.
• The solution, even though it can be obtained from trial and error method, is less time consuming if worked out through logic.

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