Problem Solving and Python Programming Unit 1 Complete

Algorithmic Problem Solving

Computer (2 marks)

Electronic machine that,

• Accepts data from the environment
•  Processes the data by performing calculations and operations
•  Generates output result to the environment

Program (2 marks)

• A program is a step by step series of instruction given to the computer to produce a desired output.
• An instruction is an basic commands

Problem-solving (2 marks)

• Problem solving is the sequential process of analyzing information related to given situation and generating appropriate response options

• Steps involves in Problem solving,
• Programmer understands the problem.
• Analyze the problem to find out different way to solve it
• Determines multiple solutions to the problem.
• Decides and selects a single exact solution
• Selected solution is represented in a detailed step-by-step manner
• It is coded using suitable programming language
• Execute the program.

Problem solving Methodology (Software Life cycle) (8 marks)

• There are seven steps in  problem solving Methodology

1. Problem Definition
   • Understand the description of the problem to solve
   • Example:
     • To find average of 5 numbers
     • Asking questions like,
       • What input data is available?
       • Is anything missing?
       • What output data I trying to produce?
       • What I am going to compute?
2. Problem Analysis
   • Original problem is  analyzed  and divided into many sub problems
   • sub problems are easier to solve
   • each sub problems is divided into further smaller ones
   • this fragmentation continued to achieve simple solution
3. Designing the Problem
   • Designing is a  process used  to develop a program
   • It requires 3 steps, they are
     1. Algorithm: step-by-step description of the solution.
     2. Flowchart: Diagrammatic representation of the solution.
     3. Pseudo code: It is written for selected solution. It gives logic of the program
4. Coding (2 marks)
   • Algorithms can’t executed directly by the computer, it has to be translated into a programming languages.
   • The process of translating the algorithms into programs using programming languages is known as coding.
5. Program testing and debugging
   • Testing means running the program.
   • Check the output results with different inputs.
   •  If the output is incorrect, modify the program to get correct results.
   • Debugging is the process of finding and correcting the errors in the program.
6. Documentation
   • It includes,
     •  problem definition
     • Design documents
     • History of program development
     • User manual
7. Program Maintenance
   • Finding and eliminating previously undetected program errors.
   • Modify the current program, often to improve its performance.
   • Adding new features to the program
   • Updating the documentation

ALGORITHM (16 marks & 8 marks)  

Algorithm (2 marks)

      “It is an ordered sequence of finite, well defined, unambiguous instruction for completing a task”

• It is a step-by-step procedure for solving a task.
• English like representation of the logic.
• Steps must be,
  • Ordered
  • Unambiguous
  • Finite in number
• For a particular task, different algorithms can be written.
• Different algorithms differ in their requirements of time and space.

Example: To find the greatest among 3 numbers.  (8 marks)

Algorithm

Steps:

1. Start
2. Read 3 numbers A, B, C.
3. Compare A & B. If A is greater, perform step 4 else perform step 5.
4. Compare A & C. If A is greater, output “A is greatest”
5.  Else output “C is greatest”.
6.  Perform step 6.
7. Compare B & C. If B is greater, output “B is greatest”
8. Else output “C is greatest”.
9. Stop

Characteristics of Algorithm (2 marks)

• Simple and easy to understand
• Should be clear and unambiguous
• It look like normal English
• Includes finite sequence of steps in order
• Instruction should not be repeated infinitely
• Should covers logic of the problem
• Desired result should be obtained at the end of the algorithm

Qualities of Algorithm (2 marks)

       The following are the primary factor used to judge the quality of the algorithm

1. Time
   • To execute programs, computer takes some amount of time
   • The lesser is the time required ,the better is the algorithm
2. Memory
   • To execute programs, computer takes some amount of memory storage
   • The lesser is the memory required ,the better is the algorithm
3. Accuracy
   • Multiple algorithm may provide suitable solution to the problem
   • Some of these may provide accurate results than other algorithm
   • Such algorithm are suitable
4. Sequence
   • Instruction in the algorithm must be in order
   • Some instruction of an algorithm may be repeated number of times until a particular condition satisfy.
5. Generability
   • The designed algorithm must solve a single isolated problem.
   • So algorithms must be generalized.

Advantage of Algorithm (2 marks)

• Simple to understand
• Step wise description of the solution
• Easy to debug
• Independent of programming languages
• Using algorithm problem can be broken down into smaller pieces or steps
• Each step of the algorithm can be easily coded into its equivalent high level language

Disadvantage of Algorithm (2 marks)

• Time consuming process
• Difficult to show branching and looping statement
• Algorithm is not computer program, rather a concept of how a program should be written

BUILDING BLOCKS OF AN ALGORITHM (16 marks & 8 marks)  

Building Blocks of an Algorithm

• Algorithm can be constructed from basic building blocks
• The building blocks of algorithms are (2 marks)
  1. Instruction/statements
  2. State
  3. Control flow
     • Sequence
     • Selection (conditional)
     • Iteration(looping)
  4. Function

1. Instruction/statements
   • An effective procedure for solving a problem in a finite number of steps
   • Instruction must be in order
   • Time taken to execute all instruction of the algorithm should be finite and within a reasonable limit.
2. State
   • It specify the state or condition for program termination
   • It also specify the position in the algorithm
3. Control Flow (2 marks)
   • It is an order in which individual statements, instruction are evaluated
   • Control flow allow the program to make choice, change direction or repeat action
   • The basic control flow needed for writing good and efficient algorithm are
     • Sequence
     • Selection (conditional)
     • Iteration(looping)
   • Sequence (2 marks)
     • In sequence control flow,
       • Instructions are executed in linear order
       • One after the other
     • Example: Algorithm to find sum of two numbers (2 marks)
       1. Step 1: Start
       2. Step 2: Read two numbers A and B
       3. Step 3: Calculate sum= A + B
       4. Step 4: Print the sum value
       5. Step 5: Stop
   • Selection (conditional) (2 marks)
     • In selection control flow,
       • Instructions are executed based on condition
       • If condition is true ,one path is followed
       • If condition is false, another path is followed
     • Example: Algorithm to find biggest of two numbers (2 marks)
       1. Step 1: Start
       2. Step 2: Read two numbers A and B
       3. Step 3: if (A>B) then  Print ‘A’ is bigger else Print ‘B’ is bigger
       4. Step 4: Stop
   • Iteration (looping) (2 marks)
     • In iterative control flow, a same set of instruction executed repeatedly until it satisfy particular condition
     • Example: Algorithm to compute and print average of 10 numbers (2 marks)
       • Step 1: Start
       • Step 2: total=0, average=0
       • Step 3: for 1 to 10
       • Step 4: Read number
       • Step 5: total=total + number
       • Step 6: END for
       • Step 7: average=total/10
       • Step 8: Print average
       • Step 9: Stop
4. Function (2 marks)
   • For complex problems, the task is divided into number of smaller sub-task during algorithm design
   • A function is a block of organized, reusable code that is used to perform single, related actions.
   • Provide better modularity for application
   • Provide high degree of code reuse
   • Example: Algorithm to find addition of two numbers
     • Step 1: Start
     • Step 2: Read two numbers ‘a’ & ‘b’
     • Step 3: Call addition (a, b)
     • Step 4: Print the sum value
     • Step 5: Stop
     • addition (a, b)
     • Step 1: Sum= a + b
     • Step 2: return sum

PSEUDOCODE (16 marks & 8 marks)  

Pseudocode (2 marks)

              “Pseudocode is an outline of a program written in a form that can be easily converted into real programming statement”

• Pseudo means ‘false or imitation’
• Code means ‘instruction’
• Pseudocode is
•  short
• Readable
• Formally styled English languages used for explaining an algorithm
• Short hand way of describing a computer program
• It is not based on any programming languages
• It gives structure of the program before the actual coding
• It is also called program design language

Keywords used in Pseudocode (2 marks)

• Pseudocode uses  some keywords to denote programming processes, they are
  • Input: INPUT, GET, READ, OBTAIN
  • Output: PRINT, DISPLAY, SHOW, OUTPUT
  • Compute: COMPUTE, CALCULATE, DETERMINE
  • Initialization: SET, INITIALIZE
  • Add one: INCREMENT

Example: Pseudocode to find the addition of three numbers.  (2 marks)

• READ A, B, C
• COMPUTE the sum by adding A, B, C
• DISPLAY the sum

Rules for writing Pseudocode (2 marks)

(Or)

Guidelines for  preparing Pseudocode (2 marks)

• Statements should be written in English and programming language independent
• Write one statement per line
• Capitalize initial keywords
• Each set of instruction must be written from top to bottom
• End multiline structure
• Should describe logical plan to develop a program

Control structure used in Pseudocode (16 marks & 8 marks)

There are three control structures used in pseudocode, they are

Sequence (2 marks)

• In sequence control flow,
• Instructions are executed in linear order
• One after the other

 Example: Pseudocode to find the sum of two numbers (2 marks)

1. START
2. READ the values of A and B
3. CALCULATE sum by adding A with B
4. PRINT the sum value
5. STOP

Selection (conditional) (2 marks)

• In selection control flow,
  • Instructions are executed based on condition
  • If condition is true ,one path is followed
  • If condition is false, another path is followed

Example: Pseudocode to find biggest of two numbers (2 marks)

• START
• READ the values of A and B
• IF (A>B) THEN
  • PRINT ‘A’ is bigger
  • ELSE
  • PRINT ‘B’ is bigger
• STOP

Iteration (looping) (2 marks)

• In iterative control flow, a same set of instruction executed repeatedly until it satisfy particular condition

Example: Pseudocode to compute and print an average of 10 numbers (2 marks)

1. START
2. INITIALIZE total=0, average=0
3. FOR (1 to 10)
4. READ number
5. COMPUTE total =total + number
6. END FOR
7. COMPUTE average=total/10
8. PRINT average
9. STOP

Advantage of Pseudocode (2 marks)

• It can be written easily
• It is compact and easy to modify
• It can be readable
• Easy to understand the general working of the program.
• Gives the sketch of structure of program.
• For writing pseudo code, programmer need not know the programming language.

Disadvantage of Pseudocode (2 marks)

• It is not visual
• Not get the picture of design
• No standard rules for writing pseudocode
• Not used to understand the flow of program control

FLOW CHART (16 marks & 8 marks)  

Flow Chart (2 marks)

• Diagrammatic representation of the logic for solving a task.
• Has boxes with lines connected.
• Boxes represent operations.
• Lines show the flow of control.
• Purpose:
  • To make the logic of the program clearer in a visual form.

Need for Flowchart (2 marks)

• Process can be explained clearly through symbols and text in flow chart
• Provide effective program documentation and maintenance
• Easy to review and debug the program
• Logic of the program is communicated in much better way than algorithm
• Easy to understand

Flowchart symbols (2 marks)

• Each symbol is for a specific purpose.
• Most commonly used flowchart symbols are,
  • Process
  • Decision
  • Data
  • Terminator
  • Connector
  • Flow lines

Symbol Name	Description
Terminator	Start or end of the program
Process	Operation or action step
Data	Input or output operation
Decision	Decision making or branching
Connector	Connector or joining of two-part of the program
Flow lines	Indicate the direction of flow

Rules (guidelines) for drawing flowchart (2 marks)

• Should have a start and end.
• Direction of flow must be from top to bottom and left to right.
• Relevant symbols must be used while drawing a flowchart.
• Should be clear, neat and easy to follow
• Only one flow line should come out from process symbol
• Only one flow line should enter a decision symbol
• Only one flow line is used in terminal symbol
• For complex problem, connector symbol is used to reduce number of flow lines in flowchart.

Control structure used in Flowchart (16 marks & 8 marks)

There are three control structures used in a flowchart, they are

Sequence (2 marks)

1. In sequence control flow,
   • Instructions are executed in linear order
   • One after the other

 Example: flowchart to find the sum of two numbers (2 marks)

Selection (conditional) (2 marks)

• In selection control flow,
  • Instructions are executed based on condition
  • If condition is true ,one path is followed
  • If condition is false, another path is followed

Example: flowchart to find biggest of two numbers (2 marks)

Iteration (looping) (2 marks)

• In iterative control flow, a same set of instruction executed repeatedly until it satisfy particular condition

Example: flowchart to compute and print average of 10 numbers (2 marks)

Advantage of Flowchart (2 marks)

• Makes logic clear
• Visual representation
• Useful for coding
• Appropriate documentation.
• Act as a guide or blue print for program development
• Remove repeated and misplaced steps
• Increase in efficiency

Disadvantage of Flowchart (2 marks)

• Complex and long flowchart may run into multiple pages
• which is difficult to understand and follow
• difficult to modify
• Excessive use of the connector will confuse the programmers
• Cost of operation is high
• Updating is not regular

Difference between Algorithm, Flowchart and Pseudo code (2 marks)

Algorithm	Flowchart	Pseudo code
Algorithm is a sequence of instruction to solve a particular problem.	Flowchart is the graphical representation of algorithm.	Pseudo code is readable, English like representation of algorithm.
Can be represented using a flowchart or pseudo code.	It uses different kind of symbols for representation	It uses structured constructs of programming language for representation.
The user need  knowledge to write algorithm for a task	User does not require any programming language knowledge to draw or understand a  flowchart	User does not require any programming language knowledge to write or understand a pseudo code.

PROGRAMMING LANGUAGES (8 marks & 2 marks)  

Programming language (2 marks)

• Computer programming languages are used to communicate instruction to a computer to perform a task.
• Programming language is like English language that can be understood by the computer through the translator of that language.
• It is based on certain syntactic and semantic rules
• Programming language forces the user to write very simple and exact instruction

Types of Programming language (2 marks)

• Programming language can be divided into three major categories
  1. Machine language
  2. Assembly language
  3. High level language(HLL)

Machine language  (2 marks)

• It is native language of computer
• It is in binary form
• It uses only 0’s and 1’s to represent data and instruction
• It is also called Low Level Language(LLL)
• It is very difficult to write or read instruction in binaries

  Example:    0100   00011001

Advantage

• Instruction can executed directly by the computer
• Fast in execution
• Bitwise operation is possible

Disadvantage

• It is machine dependent
• Not compactable
• Difficult to write
• Takes more time to write machine code

Assembly language  (2 marks)

• Instruction are written with mnemonics to simplify the program
• Use letters instead of 0’s and 1’s to run a machine
• The general format of an assembly instruction is

[Label]  <opcode>  <operands> [;Comments]Code language: HTML, XML (xml)

  Example:   

                    Label                     opcode         operands                   Comments

                     BEGIN                   ADD               A, B                    ADD B to A

• Mnemonics are converted to binaries with help of a translator known as assembler.

Advantage

• Easy to understand and use
• Less error
• Faster
• More control on hardware

 Disadvantage

• Machine dependent
• Harder to learn
• Less efficient
• No standardization
• No support for modern software technology

High level language  (2 marks)

• Instruction are written using English language with symbols and digits
• It is also called procedural language
• The commonly used high-level languages are FORTRAN, BASIC, COBOL, C, C++, Java etc.
• The high-level languages are converted to machine language using translators
• Types of  translator
  • Two types of translators are used, they are
    • Compiler
    • Interpreter

 Compiler: (2 marks)

• Translate entire program of high-level languages into machine language
• Convert instruction from human understandable form to machine understandable form.
• Most of the languages use compiler
• It is faster

 Interpreter: (2 marks)

• Translate each line of program instruction to machine language
• Slower
• Very few languages use interpreter

Difference between compiler and interpreter: (2 marks)

Advantage

• Readability
• Machine independent
• Easy debugging
• Easier to maintain
• Low development cost

Disadvantage

• Poor control on hardware
• Less efficient

ALGORITHMIC PROBLEM SOLVING (16 marks & 8 marks)  

Algorithmic Problem Solving

• An algorithm is a plan for solving a problem
• The development of an algorithm is the key step in algorithmic problem solving
• algorithmic problem solving is about the formation and solution of problems
• algorithmic problem solving process consist of six major steps, they are

1. Obtain a description of the problem

• Algorithmic problem solving process starts by obtaining a description of the problem
• Understanding the problem
• Clarify the doubts after leading the problem description
• Identifying and collecting information
• Correct algorithm should work for all possible input

2. Analyze the problem

• Determine both the starting and the ending points for solving the problem
• Analyze both time and space efficiency
• Asking the following question often to determine the starting point
  • What data are available?
  • Where the data from?
  • What are the rules exist for working with the data?
• When determining the ending point algorithm needs to describe the characteristic of a solution.
• Asking the questions like
  • What new facts will we have?
  • What items would have changed?
  • What things will no longer exist?

3.  Develop a high-level algorithm

• Algorithm is plan for solving a problem, but plans come in several levels of detail
• Developing high-level algorithm which includes major part of a solution
• Leaves the detail until later

4. Refine the algorithm

• Provide enough detail to solve the problem
• For complex problem go through the process several time, develop intermediate level algorithm
• Each time more details are added to the previous algorithm, for further refinement
• Stepwise refinement is the process of developing a detailed algorithm by gradually adding details to a high level algorithm

5. Review the algorithm

• Review the algorithm step by step to determine whether or not ,it will solve the original problem
• Asking these questions and seeking their answers is a good way to develop skills that can be applied to the next problem.
  • Does this algorithm solve a very specific problem or general problem?
    • Example: for computing the area of a circle If radius is 5.2 meters (formula π*5.22) solves a very specific problem (formula π*R*R) solves a more general problem.
  • Can this algorithm be simplified?
    • Example:One formula for computing the perimeter of a rectangle is: length + width + length + width
    • A simpler formula would be: 2.0 * (length + width)
  • Is this solution similar to the solution to another problem? How are they alike? How are they different?
    • Example: consider the following two formulae:
      • Triangle area = 0.5 * base * height
      • Rectangle area=length * width
    • Similarities: Each computes an area. Each multiplies two measurements.
    • Differences: Different measurements are used. The triangle formula contains 0.5.

6. Code the algorithm

• Once an algorithm developed ,it is coded using some programming languages
• Each line of algorithm is translate to computer programs using programming languages
• Executing the program
• Checking errors and testing with different inputs

Simple strategies for Developing an Algorithm (16 marks & 8 marks)

Simple strategies for developing an Algorithm

• There are various kinds of algorithm developing technique formulated and used for different types of problems.
• They are,
  1. Sequence structure
  2. Selection structure
  3. Iteration structure
  4. Recursion structure
• Sequence (straight line) structure
  • In sequence control flow,
    • Instructions are executed in linear order
    • One after the other
    • The logic flow is from top to bottom

Example:

Algorithm to find the sum of two numbers

• Step 1: Start
• Step 2: Read two numbers A and B
• Step 3: Calculate sum= A + B
• Step 4: Print the sum value
• Step 5: Stop

Selection (decision or branching)structure

• In selection control flow,
• Instructions are executed based on condition
• If condition is true ,one path is followed
• If condition is false, another path is followed

Example:

Algorithm to find biggest of two numbers

• Step 1: Start
• Step 2: Read two numbers A and B
• Step 3: IF (A>B) then
  • Print ‘A’ is bigger
  • ELSE
  • Print ‘B’ is bigger
• Step 4: Stop

There are Three Selection Structures in Python

• IF
• IF-ELSE
• ELIF

1.  IF

• Check the condition
• If condition is true execute the statement
• If condition is false, the statement is not executed

2. IF-ELSE

• Check the condition
• If condition is true execute the statement
• If condition is false, execute the else statement

3. ELIF

• Check multiple expressions
• Execute a block of code as soon as one of the conditions is true.

Iteration (looping) structure

• In iterative control flow, a same set of instruction executed repeatedly until it satisfy particular condition

Example:

Algorithm to compute and print an average of 10 numbers

• Step 1: Start
• Step 2: total=0, average=0
• Step 3: FOR 1 to 10
• Step 4: Read number
• Step 5: total=total + number
• Step 6: END for
• Step 7: average=total/10
• Step 8: Print average
• Step 9: Stop

There are two repetition structures in python, they are

• WHILE
• FOR

a) WHILE

• It is also called entry controlled loop
• Check the condition
• If condition is true execute the body of the loop
• Process continues until the condition becomes false

b) FOR

• Consists of a header and a set of statements called the body
• Header contains information that controls the number of times that the body executes

Recursion 

• In recursion technique a function or procedure is called by itself again and again until a given condition is satisfied.

 Example: Recursive algorithm for finding the factorial of a number

• Step 1: Start
• Step 2: Read number n
• Step 3: Call factorial (n)
• Step 4: Print factorial f
• Step 5: Stop
• factorial (n)
• Step 1: If n==1 then return 1
• Step 2: Else
• f=n*factorial (n-1)
• Step 3: Return f

ILLUSTRATIVE PROBLEMS (16 marks & 8 marks)  

Find Minimum number in a list (8 marks)

Problem statement

• To find minimum value in an list, take the first element and compare its value against the values of the other elements
• Repeat the process till the end of the list
• Finally the minimum value in the list is obtained

 Algorithm

• Step 1: Start
• Step 2: Assign the first value of the list as minimum value
• Step 3: Compare this value to the other values starting from second value
• Step 4: When a value is smaller than the present minimum value is found, it becomes the new minimum
• Step 5: Repeat the steps till the end of the list
• Step 6: Print the minimum value
• Step 7: Stop

Example

Consider a list

Step 1:

• Take the first value in the list as Min value

Step 2:

• Compare the Min value with the next value in a list 40 < min value, so interchange min value

Step 3:

• Compare the Min value with the next value in a list 5 < min value, so interchange min value

 Step 4:

• Compare the Min value with the next value in a list 9 > min value, so need to interchange min value

Step 5:

• Compare the Min value with the last value in a list 45 > min value, so need to interchange min value

Finally the minimum value in this list is 5

Insert a card in a list of sorted cards (8 marks)

Problem statement

• To insert a card in the sorted card, increase the list size with 1.
• Insert new card in the appropriate position by comparing each element’s value with the new one.
• When the position is found, move the remaining elements by one position up and the card can be inserted

Algorithm

• Step 1: Start
• Step 2: If it is the first element, it is already sorted
• Step 3: Compare with all elements in the sorted sub-list
• Step 4: Shift all elements in the sorted sub-list that is greater than the value
• Step 5: Insert the value
• Step 6: Repeat until list is sorted
• Step 7: Display all the card elements of a new list
• Step 8: Stop

Example

  Consider an array from index 0 through index 5 is already sorted

Need to insert the element currently in index 6 into this sorted array

Step 1:

 Store the element at index 6 into a variable called key

Step 2:

• Compare key with the element at position 5.
• Key value (5) < Element at 5^th position (13), so shift this element to position 6

Step 3:

• Compare key with the element at position 4.
• Key value (5) < Element at 4^th position (10), so shift this element to position 5

Step 4:

• Compare key with the element at position 3.
• Key value (5) < Element at 3^rd position (8), so shift this element to position 4

Step 5:

• Compare key with the element at position 2.
• Key value (5) < Element at 2^nd position (7), so shift this element to position 3

Step 6:

• Compare key with the element at position 1.
• Key value (5) > Element at 1^st position (3), so no need to shift it over

• instead, drop key into the position immediately to the right of this element (that is, into position 2)

Guessing an integer number in a range (8 marks)

Problem statement

• Objective is to randomly generate integer number from 0 to n
• The player have to guess the number
• If player guess number correctly ,output an appropriate message
• If guess number is less than random number generated, output the message “Your guess is lower than the number. Guess again”
• Otherwise output the message “Your guess is higher than the number. Guess again”
• This process is repeated until the player enters the correct number

Algorithm

• Step 1: Start
• Step 2: Generate a random number and call it num
• Step 3: Repeat the following steps until the player has guessed the correct number
  1. Enter the number to guess
  2. if(guess is equal to num)
  3. Print “You guessed the correct number”
  4. Otherwise
  5. if (guess is less than num)
  6. Print “You guess lower than the number.
     • Guess again!”
  7. Otherwise
  8. Print “You guess higher than the number.
     • Guess again!”
• Step 4: Stop

Example

Consider a player want to guess a number between 1 and 100

The number selected for guess is 82

Step 1:  choose the middle number from the range 1 to 100, middle no is 50

Step 2:  choose the middle number from the range 51 to 100, middle no is 75

Step 3:  choose the middle number from the range 76 to 100, middle no is 88

Step 4:  choose the middle number from the range 76 to 87, middle no is 81

Step 5:  choose the middle number from the range 82 to 87, middle no is 84

Step 6:  choose the middle number from the range 76 to 87

Middle no is 82.5, which is rounded to 82

                     

Tower of Hanoi Problem (8 marks)

Problem statement

• Tower of Hanoi is one of the classical problems of computer science
• The problem states that,
• There are three stands(Stand 1,2 and 3) on which a set of disks, each with a different diameter are placed
• Initially, the disks are stacked on Stand 1,in order of size with the larger disk at the bottom

The initial structure of the Tower of Hanoi with three disks is shown below

Rules to move the disk

• The ‘Tower of Hanoi problem’ is to find the sequence of disk moves so that all the disks moved from stand-1 to stand-3 using the following rules
  • Move only one disk at a time
  • A larger disk cannot be placed on top of a smaller disk.
  • Only the “top” disk can be removed

• The recurrence relation for solving the Tower of Hanoi problem can be written as

Algorithm

• Step 1:  START
• Step 2:  Procedure Hanoi (disk, source, dest, aux)
• Step 3:  If disk == 0 THEN
  • Step 3.1:  move disk from source to dest
• Step 4:  ELSE
  • Step 4.1:  Hanoi (disk-1, source, aux, dest)
  • Step 4.2:  move disk from source to dest
  • Step 4.3:  Hanoi (disk-1, aux, dest, source)
• Step 5:  END If
• Step 6:  END Procedure
• Step 7:  STOP

 Example

• The general strategy for solvingthe Tower of Hanoi problem with three disks as  follows

Move 1: move disk 3 to post C 

Move 2: move disk 2 to post B

Move 3: move disk 3 to post B 

Move 4: move disk 1 to post C 

Move 5: move disk 3 to post A

Move 6: move disk 2 to post C 

Move 7: move disk 3 to post C

Algorithm and Pseudo code

Roots of quadratic equation

Algorithm

• Step 1: Start
• Step 2: Read the value of a, b, c
• Step 3: d = b*b – 4*a*c
• Step 4: if d < 0 then
  • Display the Roots are Imaginary.
• Step 5:  else if d = 0 then
  • Display Roots are Equal.
  • r = -b / 2*a
  • display r
• Step 6:  else
  • r1 = -b + √d / 2*a
  • r2 = -b – √d / 2*a
  • display Roots are real and r1, r2
• Step 7: Stop

Pseudo code

• START
• READ the value of a, b, c
• CALCULATE d = b*b – 4*a*c
• IF d < 0 THEN
  • DISPLAY the Roots are Imaginary.
• ELSE if d = 0 then
  • DISPLAY Roots are Equal.
  • DISPLAY, r = -b / 2*a
• ELSE
  • r1 = -b + √d / 2*a
  • r2 = -b – √d / 2*a
  • DISPLAY Roots are real and r1, r2
• STOP