# Iteration - PowerPoint PPT Presentation

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Iteration. Chapter 4 Spring 2007 CS 101 Aaron Bloomfield. Java looping. Options while do-while for Allow programs to control how many times a statement list is executed. Averaging values. Averaging. Problem

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## Iteration

Chapter 4

Spring 2007

CS 101

Aaron Bloomfield

### Java looping

• Options

• while

• do-while

• for

• Allow programs to control how many times a statement list is executed

## Averaging values

### Averaging

• Problem

• Extract a list of positive numbers from standard input and produce their average

• Numbers are one per line

• A negative number acts as a sentinel to indicate that there are no more numbers to process

• Observations

• Cannot supply sufficient code using just assignments and conditional constructs to solve the problem

• Don’t how big of a list to process

• Need ability to repeat code as needed

### Averaging

• Algorithm

• Prepare for processing

• Get first input

• While there is an input to process do {

• Process current input

• Get the next input

• }

• Perform final processing

### Averaging

• Problem

• Extract a list of positive numbers from standard input and produce their average

• Numbers are one per line

• A negative number acts as a sentinel to indicate that there are no more numbers to process

• Sample run

Enter positive numbers one per line.

Indicate end of list with a negative number.

4.5

0.5

1.3

-1

Average 2.1

public class NumberAverage {

// main(): application entry point

public static void main(String[] args) {

// set up the input

// prompt user for values

// get first value

// process values one-by-one

while (value >= 0) {

// add value to running total

// processed another value

// prepare next iteration - get next value

}

// display result

if (valuesProcessed > 0)

// compute and display average

else

// indicate no average to display

}

}

int valuesProcessed = 0;

double valueSum = 0;

// set up the input

Scanner stdin = new Scanner (System.in);

// prompt user for values

System.out.println("Enterpositivenumbers1perline.\n"

+ "Indicate end of the list with a negative number.");

// get first value

double value = stdin.nextDouble();

// process values one-by-one

while (value >= 0) {

valueSum += value;

++valuesProcessed;

value = stdin.nextDouble();

}

// display result

if (valuesProcessed > 0) {

double average = valueSum / valuesProcessed;

System.out.println("Average: " + average);

} else {

System.out.println("No list to average");

}

### Program Demo

• NumberAverage.java

Logical expression that

Action is either a single

determines whether Action

statement or a statement

is to be executed

list within braces

### While syntax and semantics

Expression

Action

while

(

)

Test expression is evaluated at the

start of each iteration of the loop.

If test expression is true, these statements

are executed. Afterward, the test expression

is reevaluated and the process repeats

### While semantics for averaging problem

// process values one-by-one

while ( value >= 0 ) {

// add value to running total

valueSum += value;

// we processed another value

++valueProcessed;

// prepare to iterate – get the next input

value = stdin.nextDouble();

}

Expression is

evaluated at the

start of each

iteration of the

loop

If Expression is

true, Action is

executed

If Expression is

false, program

execution

continues with

next statement

Expression

false

true

Action

### Execution Trace

Suppose input contains: 4.50.51.3-1

Suppose input contains: 4.50.51.3 -1

Suppose input contains: 4.5 0.5 1.3 -1

Suppose input contains: 4.5 0.5 1.3 -1

Suppose input contains: 4.50.5 1.3 -1

valuesProcessed

1

0

3

2

4.5

valueSum

0

6.3

5.0

int valuesProcessed = 0;

double valueSum = 0;

double value = stdin.nextDouble();

while (value >= 0) {

valueSum += value;

++valuesProcessed;

value = stdin.nextDouble();

}

if (valuesProcessed > 0) {

double average = valueSum / valuesProcessed;

System.out.println("Average: " + average);

}

else {

System.out.println("No list to average");

}

int valuesProcessed = 0;

double valueSum = 0;

double value = stdin.nextDouble();

while (value >= 0) {

valueSum += value;

++valuesProcessed;

value = stdin.nextDouble();

if (valuesProcessed > 0) {

double average = valueSum / valuesProcessed;

System.out.println("Average: " + average);

value

1.3

4.5

-1

0.5

average

2.1

### What do these pictures mean?

• Light beer

• Dandy lions

• Assaulted peanut

• Eggplant

• Dr. Pepper

• Pool table

• Tap dancers

• Card shark

• King of pop

• I Pod

• Gator aide

• Knight mare

• Hole milk

## Converting text to lower case

### Converting text to strictly lowercase

public static void main(String[] args) {

Scanner stdin = new Scanner (System.in);

System.out.println("Enter input to be converted:");

String converted = "";

while (stdin.hasNext()) {

String currentLine = stdin.nextLine();

String currentConversion =

currentLine.toLowerCase();

converted += (currentConversion + "\n");

}

System.out.println("\nConversion is:\n" +

converted);

}

An empty line

was entered

A Ctrl+z was

entered. I

t is the

Windows escape

sequence for

indicating

end-of-file

### Program Demo

• LowerCaseDisplay.java

### Program trace

public static void main(String[] args) {

Scanner stdin = new Scanner (System.in);

System.out.println("Enter input to be converted:");

String converted = "";

while (stdin.hasNext()) {

String currentLine = stdin.nextLine();

String currentConversion =

currentLine.toLowerCase();

converted += (currentConversion + "\n");

}

System.out.println("\nConversion is:\n" +

converted);

}

public static void main(String[] args) {

Scanner stdin = new Scanner (System.in);

System.out.println("Enter input to be converted:");

String converted = "";

while (stdin.hasNext()) {

String currentLine = stdin.nextLine();

String currentConversion =

currentLine.toLowerCase();

converted += (currentConversion + "\n");

}

System.out.println("\nConversion is:\n" +

converted);

}

The append assignment operator updates the representation

of converted to include the current input line

Representation of lower case

Newline character is needed

conversion of current input line

because method nextLine()

"strips" them from the input

### Program trace

converted += (currentConversion + "\n");

## Loop Design & Reading From a File

### Loop design

• Questions to consider in loop design and analysis

• What initialization is necessary for the loop’s test expression?

• What initialization is necessary for the loop’s processing?

• What causes the loop to terminate?

• What actions should the loop perform?

• What actions are necessary to prepare for the next iteration of the loop?

• What conditions are true and what conditions are false when the loop is terminated?

• When the loop completes what actions are need to prepare for subsequent program processing?

Same Scanner class!

filename is a String

It’s in the java.io package

• Background

Scanner fileIn = new Scanner (new File (filename) );

• Class File

• Constructor File (String s)

• Opens the file with name s so that values can be extracted

• Name can be either an absolute pathname or a pathname relative to the current working folder

Scanner stdin = new Scanner (System.in);

System.out.print("Filename: ");

String filename = stdin.nextLine();

Scanner fileIn = new Scanner (new File (filename));

String currentLine = fileIn.nextLine();

while (currentLine != null) {

System.out.println(currentLine);

currentLine = fileIn.nextLine();

}

Scanner stdin = new Scanner (System.in);

System.out.print("Filename: ");

String filename = stdin.nextLine();

Scanner fileIn = new Scanner (new File (filename));

String currentLine = fileIn.nextLine();

while (currentLine != null) {

System.out.println(currentLine);

currentLine = fileIn.nextLine();

}

Set up standard input stream

Determine file name

Set up file stream

Process lines one by one

Get first line

Make sure got a line to process

Display current line

Get next line

Make sure got a line to process

If not, loop is done

Close the file stream

## The For statement

The body of the loop iterates

while the test expression is

Initialization step

true

is performed only

After each iteration of the

once -- just prior

body of the loop, the update

to the first

expression is reevaluated

evaluation of the

test expression

The body of the loop displays the

current term in the number series.

It then determines what is to be the

new current number in the series

### The For Statement

int

currentTerm = 1;

for ( int i = 0; i < 5; ++i ) {

System.out.println(currentTerm);

currentTerm *= 2;

}

Evaluated once

at the beginning

of the for

statements's

The ForExpr is

execution

evaluated at the

start of each

iteration of the

loop

If ForExpr is true,

Action is

executed

After the Action

If ForExpr is

has completed,

false, program

the

execution

PostExpression

continues with

is evaluated

next statement

After evaluating the

PostExpression, the next

iteration of the loop starts

ForInit

ForExpr

true

false

Action

ForUpdate

Logical test expression that determines whether the action and update step are

executed

Initialization step prepares for the

first evaluation of the test

Update step is performed after

expression

the execution of the loop body

The body of the loop iterates whenever

the test expression evaluates to true

ForInit

ForExpression

ForUpdate

Action

for

(

;

;

)

### for vs. while

• A for statement is almost like a while statement

for ( ForInit; ForExpression; ForUpdate ) Action

is ALMOST the same as:

ForInit;

while ( ForExpression ) {

Action;

ForUpdate;

}

• This is not an absolute equivalence!

• We’ll see when they are different in a bit

### Variable declaration

• You can declare a variable in any block:

while ( true ) {

int n = 0;

n++;

System.out.println (n);

}

System.out.println (n);

Variable n gets created (and initialized) each time

Thus, println() always prints out 1

Variable n is not defined once while loop ends

As n is not defined here, this causes an error

### Variable declaration

• You can declare a variable in any block:

if ( true ) {

int n = 0;

n++;

System.out.println (n);

}

System.out.println (n);

Only difference from last slide

### Execution Trace

i

0

2

1

3

System.out.println("i is " + i);

}

System.out.println("all done");

System.out.println("i is " + i);

}

System.out.println("all done");

i is 0

i is 1

i is 2

all done

for (

int i = 0;

int i = 0;

i < 3;

i < 3;

++i

++i

) {

Variable i has gone out of scope – it

is local to the loop

### for vs. while

• An example when a for loop can be directly translated into a while loop:

int count;

for ( count = 0;count < 10; count++ ) {

System.out.println (count);

}

• Translates to:

int count;

count = 0;

while (count < 10) {

System.out.println (count);

count++;

}

### for vs. while

• An example when a for loop CANNOT be directly translated into a while loop:

for ( int count = 0;count < 10; count++ ) {

System.out.println (count);

}

• Would (mostly) translate as:

int count = 0;

while (count < 10) {

System.out.println (count);

count++;

}

only difference

count is NOT defined here

count IS defined here

### for loop indexing

• Java (and C and C++) indexes everything from zero

• Thus, a for loop like this:

for ( int i = 0; i < 10; i++ ) { ... }

• Will perform the action with i being value 0 through 9, but not 10

• To do a for loop from 1 to 10, it would look like this:

for ( int i = 1; i <= 10; i++ ) { ... }

i is 0

j is 0

j is 1

i is 1

j is 0

j is 1

i is 2

j is 0

j is 1

### Nested loops

int m = 2;

int n = 3;

for (int i = 0; i < n; ++i) {

System.out.println("i is " + i);

for (int j = 0; j < m; ++j) {

System.out.println(" j is " + j);

}

}

i is 0

i is 1

j is 0

i is 2

j is 0

j is 1

i is 3

j is 0

j is 1

j is 2

### Nested loops

int m = 2;

int n = 4;

for (int i = 0; i < n; ++i) {

System.out.println("i is " + i);

for (int j = 0; j < i; ++j) {

System.out.println(" j is " + j);

}

}

### How well do you understand for loops?

• Very well! This stuff is easy!

• Fairly well – with a little review, I’ll be good

• Okay. It’s not great, but it’s not horrible, either

• Not well. I’m kinda confused

• Not at all. I’m soooooo lost

## do-while loops

Action

true

Expression

false

### The do-while statement

• Syntax

doAction

while(Expression)

• Semantics

• Execute Action

• If Expression is true then execute Action again

• Repeat this process until Expression evaluates to false

• Action is either a single statement or a group of statements within braces

### Picking off digits

• Consider

System.out.print("Enter a positive number: ");

int number = stdin.nextInt();

do {

int digit = number % 10;

System.out.println(digit);

number = number / 10;

} while (number != 0);

• Sample behavior

Enter a positive number: 1129

9

2

1

1

### Guessing a number

• This program will allow the user to guess the number the computer has “thought” of

• Main code block:

do {

guessedNumber = stdin.nextInt();

count++;

} while ( guessedNumber != theNumber );

### Program Demo

• GuessMyNumber.java

### while vs. do-while

• If the condition is false:

• while will not execute the action

• do-while will execute it once

while ( false ) {

System.out.println (“foo”);

}

do {

System.out.println (“foo”);

} while ( false );

never executed

executed once

### while vs. do-while

• A do-while statement can be translated into a while statement as follows:

do {

Action;

} while ( WhileExpression );

• can be translated into:

boolean flag = true;

while ( WhileExpression || flag ) {

flag = false;

Action;

}

### How well do you understand do-while loops?

• Very well! This stuff is easy!

• Fairly well – with a little review, I’ll be good

• Okay. It’s not great, but it’s not horrible, either

• Not well. I’m kinda confused

• Not at all. I’m soooooo lost

## Loop controls

### The continue keyword

• The continue keyword will immediately start the next iteration of the loop

• The rest of the current loop is not executed

• But the ForUpdate part is, if continue is in a for loop

for ( int a = 0; a <= 10; a++ ) {

if ( a % 2 == 0 ) {

continue;

}

System.out.println (a + " is odd");

}

• Output:1 is odd

3 is odd

5 is odd

7 is odd

9 is odd

### The break keyword

• The break keyword will immediately stop the execution of the loop

• Execution resumes after the end of the loop

for ( int a = 0; a <= 10; a++ ) {

if ( a == 5 ) {

break;

}

System.out.println (a + " is less than five");

}

• Output:0 is less than five

1 is less than five

2 is less than five

3 is less than five

4 is less than five

## Four Hobos

### Four Hobos

• An example of a program that uses nested for loops

• Credited to Will Shortz, crossword puzzle editor of the New York Times

• And NPR’s Sunday Morning Edition puzzle person

### Problem

• Four hobos want to split up 200 hours of work

• The smart hobo suggests that they draw straws with numbers on it

• If a straw has the number 3, then they work for 3 hours on 3 days (a total of 9 hours)

• The smart hobo manages to draw the shortest straw

• How many ways are there to split up such work?

• Which one did the smart hobo choose?

### Analysis

• We are looking for integer solutions to the formula:

a2+b2+c2+d2 = 200

• Where a is the number of hours & days the first hobo worked, b for the second hobo, etc.

• We know the following:

• Each number must be at least 1

• No number can be greater than 200 = 14

• That order doesn’t matter

• The combination (1,2,1,2) is the same as (2,1,2,1)

• Both combinations have two short and two long straws

• We will implement this with nested for loops

### Implementation

public class FourHobos {

public static void main (String[] args) {

for ( int a = 1; a <= 14; a++ ) {

for ( int b = 1; b <= 14; b++ ) {

for ( int c = 1; c <= 14; c++ ) {

for ( int d = 1; d <= 14; d++ ) {

if ( (a <= b) && (b <= c) && (c <= d) ) {

if ( a*a+b*b+c*c+d*d == 200 ) {

System.out.println ("(" + a + ", " + b

+ ", " + c + ", " + d + ")");

}

}

}

}

}

}

}

}

### Program Demo

• FourHobos.java

### Results

• The output:

(2, 4, 6, 12)

(6, 6, 8, 8)

• Not surprisingly, the smart hobo picks the short straw of the first combination

### Alternate implementation

• We are going to rewrite the old code in the inner most for loop:

if ( (a <= b) && (b <= c) && (c <= d) ) {

if ( a*a+b*b+c*c+d*d == 200 ) {

System.out.println ("(" + a + ", " + b

+ ", " + c + ", " + d + ")");

}

}

• First, consider the negation of

( (a <= b) && (b <= c) && (c <= d) )

• It’s ( !(a <= b) || !(b <= c) || !(c <= d) )

• Or ( (a > b) || (b > c) || (c > d) )

### Alternate implementation

• This is the new code for the inner-most for loop:

if ( (a > b) || (b > c) || (c > d) ) {

continue;

}

if ( a*a+b*b+c*c+d*d != 200 ) {

continue;

}

System.out.println ("(" + a + ", " + b + ", "

+ c + ", " + d + ")");

### How well do you understand four hobos?

• Very well! This stuff is easy!

• Fairly well – with a little review, I’ll be good

• Okay. It’s not great, but it’s not horrible, either

• Not well. I’m kinda confused

• Not at all. I’m soooooo lost

### The 2006 Ig Nobel Prizes

• Ornithology

• Nutrition

• Peace

• Acoustics

• Mathematics

• Literature

• Medicine

• Physics

• Chemistry

• Biology

For explaining why woodpeckers don’t get headaches

For showing that Kuwaiti dung beetles are finicky eaters

For development of a high-pitched electronic teen-ager repellent (and, later, ring tones)

For experiments to determine why people don’t like the sound of fingernails scraping on a blackboard

For calculating the number of photos you must take to ensure that (almost) nobody in a group will have their eyes closed

For a report entitled, “Consequences of Erudite Vernacular Utilized Irrespective of Necessity: Problems with Using Long Words Needlessly.“

For a medical case report titled, “"Termination of Intractable Hiccups with Digital Rectal Massage“

For studying why dry spaghetti breaks into multiple pieces

For a study entitled, “Ultrasonic Velocity in Cheddar Cheese as Affected by Temperature,"

For showing that the female malaria mosquito is equally attracted to the smells of limburger cheese and human feet

## 3 card poker

### 3 Card Poker

• This is the looping HW from a previous fall

• The problem: count how many of each type of hand in a 3 card poker game

• Standard deck of 52 cards (no jokers)

• Four suits: spades, clubs, diamonds, hearts

• 13 Faces: Ace, 2 through 10, Jack, Queen, King

• Possible 3-card poker hands

• Pair: two of the cards have the same face value

• Flush: all the cards have the same suit

• Straight: the face values of the cards are in succession

• Three of a kind: all three cards have the same face value

• Straight flush: both a flush and a straight

### The Card class

• A Card class was provided

• Represents a single card in the deck

• Constructor: Card(int i)

• If i is in the inclusive interval 1 ... 52 then a card is configured in the following manner

• If 1 <= i <= 13 then the card is a club

• If 14 <= i <= 26 then the card is a diamond

• If 27 <= i <= 39 then the card is a heart

• If 40 <= i <= 52 then the card is a spade

• If i % 13 is 1 then the card is an Ace;

• If i % 13 is 2, then the card is a 2, and so on.

### Card class methods

• String getFace()

• Returns the face of the card as a String

• String getSuit()

• Returns the suit of the card as a String

• int getValue()

• Returns the value of the card

• boolean equals(Object c)

• Returns whether c is a card that has the same face and suit as the invoking card

• String toString()

• Returns a text representation of the card. You may find this method useful during debugging.

### The Hand class

• A Hand class was (partially) provided

• Represents the three cards the player is holding

• Constuctor: Hand(Card c1, Card c2, Card c3)

• Takes those cards and puts them in sorted order

### Provided Hand methods

• public Card getLow()

• Gets the low card in the hand

• public Card getMiddle()

• Gets the middle card in the hand

• public Card getHigh()

• Gets the high card in the hand

• public String toString()

• We’ll see the use of the toString() method later

• public boolean isValid()

• Returns if the hand is a valid hand (no two cards that are the same)

• public boolean isNothing()

• Returns if the hand is not one of the “winning” hands described before

### Hand Methods to Implement

• The assignment required the students to implement the other methods of the Hand class

• We haven’t seen this yet

• The methods returned true if the Hand contained a “winning” combination of cards

• public boolean isPair()

• public boolean isThree()

• public boolean isStraight()

• public boolean isFlush()

• public boolean isStraightFlush()

### Class HandEvaluation

• Required nested for loops to count the total number of each hand

• Note that the code for this part may not appear on the website

### Program Demo

• HandEvaluation.java

### How well do you understand 3-card poker?

• Very well! This stuff is easy!

• Fairly well – with a little review, I’ll be good

• Okay. It’s not great, but it’s not horrible, either

• Not well. I’m kinda confused

• Not at all. I’m soooooo lost

### All your base are belong to us

• Flash animation

• Reference: http://en.wikipedia.org/wiki/All_your_base_are_belong_to_us

## The Halting Problem

### What’s wrong with this program?

public class LoopsForever {

public static void main (String args[]) {

while ( true ) {

System.out.println ();

}

}

}

• Given a more complicated program, how do we tell if it gets stuck in an infinite loop?

• Such as when an application “hangs”?

### The Halting problem

• Given a Java program P, and input I

• Let P be a filename for a program file on a disk somewhere

• Let I be a filename for a file that contains all the input the program takes in

• Will the program P with input I ever terminate?

• Meaning will program P with input I loop forever or halt?

• Can a computer program determine this?

• Can a human?

• First shown by Alan Turing in 1936

• Before digital computers existed!

• (I’m ignoring which way he showed it for now)

### A few notes

• To “solve” the halting problem means we have a method Oracle.CheckHalt (String P, String I)

• Let Oracle be a class that can give lots of (truthful) answers

• Oracle.PredictFuture(), Oracle.GetNextLotteryNumbers(), etc.

• P is the (filename of the) program we are checking for halting

• I is the (filename of the) input to that program

• And it will return “loops forever” or “halts”

• As a boolean: true means “loops forever”, false means “halts”

• Note it must work for any (Java) program, not just some programs

• Or simple programs

### Take your best guess – do you think it’s possible to solve the halting problem?

• Yes

• No

• I don’t understand what the halting problem is

### Can a human determine if a program halts?

• Given a program of 10 lines or less, can a human determine if it halts?

• Assuming no tricks – the program is completely understandable

• And assuming the computer works properly, of course

• And we ignore the fact that an int will max out at 4 billion

• As there are ways we can get around this…

• For the sample programs on the next page:

• Assume that the code is in a proper main() method in a proper class

• Assume “…print” stands for “System.out.print”

• Likewise for “…println”

First sample program:

...println (“Alan Turing”);

...println (“was a genius”);

System.exit();

Second sample program:

for (int n = 0; n < 10; n++)

...println (n);

System.exit();

Third sample program

while (true)

...println (“hello world”);

System.exit();

Fourth sample program:

int x = 10;

while ( x > 0 ) {

...println (“hello world”);

x = x + 1;

}

System.exit();

### Take your best guess – do you think it’s possible to solve the halting problem?

• Yes

• No

• I don’t understand what the halting problem is

### Perfect numbers

• Numbers whose divisors (not including the number) add up to the number

• 6 = 1 + 2 + 3

• 28 = 1 + 2 + 4 + 7 + 14

• The list of the first 10 perfect numbers:6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216

• The last one was 54 digits!

• All known perfect numbers are even; it’s an open (i.e. unsolved) problem if odd perfect numbers exist

• Sequence A000396 in OEIS

### Odd perfect number search

• Will this program ever halt?

int n = 1; // arbitrary-precision integer

while (true) {

int sumOfFactors = 0;

for ( int factor = 1; factor < n; factor++ )

if ( n % factor == 0 ) // factor is a factor of n

sumOfFactors = sumOfFactors + factor;

if (sumOfFactors == n) then

break;

n = n + 2;

}

System.out.exit();

### Take your best guess – do you think it’s possible to solve the halting problem?

• Yes

• No

• I don’t understand what the halting problem is

### Where does that leave us?

• If a human can’t figure out how to do the halting problem, we can’t make a computer do it for us

• It turns out that it is impossible to write such a CheckHalt() method

• But how to prove this?

### CheckHalt()’s non-existence

• Consider a program P with input I

• Suppose that a method Oracle.CheckHalt(P,I) exists

• Tests if P(I) will either “loop forever” or “halt”

• A program is a series of bits

• And thus can be considered data as well

• Thus, we can call CheckHalt(P,P)

• It’s using the bytes of program P as the input to program P

### CheckHalt()’s non-existence

• Consider a new program:

public class Test {

public static void main (String args[]) {

if ( Oracle.CheckHalt(“Test.java”, “Test.java”) )

// if Test.java loops forever

System.exit();// then halt

else// else if Test.java halts

while (true) { }// then loop forever

}

}

• Do we agree that class Test is a valid program?

A (somewhat condensed) version of class Test:

public class Test {

… main … (String args[]) {

if ( Oracle.CheckHalt (“Test.java”,

“Test.java”) )

System.exit(); elsewhile (true) { }

}

}

Two possibilities:

Either class Test halts…

Then CheckHalt(Test,Test) returns true (“loops forever”)…

Which means that class Test loops forever

Or class Test loops forever…

Then CheckHalt(Test,Test) returns false (“halts”)…

Which means that class Test halts

### How well do you understand the halting problem?

• Very well! This stuff is easy!

• Fairly well – with a little review, I’ll be good

• Okay. It’s not great, but it’s not horrible, either

• Not well. I’m kinda confused

• Not at all. I’m soooooo lost

### Why do we care about the halting problem?

• It was the first algorithm that was shown to not be able to exist by a computer

• You can prove something exists by showing an example (a correct program)

• But it’s much harder to prove that a program can never exist

• First shown by Alan Turing in 1936

• Before digital computers existed!

## Triangle counting

### The programming assignment

• This was the looping HW from two springs ago

• List all the possible triangles from (1,1,1) to (n,n,n)

• Where n is an inputted number

• In particular, list their triangle type

• Types are: equilateral, isosceles, right, and scalene

Enter n: 5

(1,1,1) isosceles equilateral

(1,2,2) isosceles

(1,3,3) isosceles

(1,4,4) isosceles

(1,5,5) isosceles

(2,2,2) isosceles equilateral

(2,2,3) isosceles

(2,3,3) isosceles

(2,3,4) scalene

(2,4,4) isosceles

(2,4,5) scalene

(2,5,5) isosceles

(3,3,3) isosceles equilateral

(3,3,4) isosceles

(3,3,5) isosceles

(3,4,4) isosceles

(3,4,5) right scalene

(3,5,5) isosceles

(4,4,4) isosceles equilateral

(4,4,5) isosceles

(4,5,5) isosceles

(5,5,5) isosceles equilateral

### Program Demo

• TriangleDemo.java

### The Triangle class

• That semester we went over classes by this homework

• So they had to finish the class

• We will be seeing class creation after spring break

• Methods in the class:

• public Triangle()

• public Triangle (int x, int y, int z)

• public boolean isTriangle()

• public boolean isRight()

• public boolean isIsosceles()

• public boolean isScalene()

• public boolean isEquilateral()

• public String toString()

### The TriangleDemo class

• Contained a main() method that tested all the triangles

• Steps required:

• Check if the sides are in sorted order (i.e. x < y < z)

• If not, then no output should be provided for that collection of side lengths

• Create a new Triangle object using the current side lengths

• Check if it is a valid triangle

• If it is not, then no output should be provided for that collection of side lengths

• Otherwise, indicate which properties the triangle possesses

• Some side length values will correspond to more than 1 triangle

• e.g., (3, 3, 3) is both isosceles and equilateral

• Thus, we can’t assume that once a property is present, the others are not.

### Look at that them there code…

• TriangleDemo.java

### How well do you understand triangle counting?

• Very well! This stuff is easy!

• Fairly well – with a little review, I’ll be good

• Okay. It’s not great, but it’s not horrible, either

• Not well. I’m kinda confused

• Not at all. I’m soooooo lost

## Fibonacci numbers

### Fibonacci sequence

• Sequences can be neither geometric or arithmetic

• Fn = Fn-1 + Fn-2, where the first two terms are 1

• Alternative, F(n) = F(n-1) + F(n-2)

• Each term is the sum of the previous two terms

• Sequence: { 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … }

• This is the Fibonacci sequence

• Full formula:

13

8

5

3

2

1

### Reproducing rabbits

• You have one pair of rabbits on an island

• The rabbits repeat the following:

• Get pregnant one month

• Give birth (to another pair) the next month

• This process repeats indefinitely (no deaths)

• Rabbits get pregnant the month they are born

• How many rabbits are there after 10 months?

### Reproducing rabbits

• First month: 1 pair

• The original pair

• Second month: 1 pair

• The original (and now pregnant) pair

• Third month: 2 pairs

• The child pair (which is pregnant) and the parent pair (recovering)

• Fourth month: 3 pairs

• “Grandchildren”: Children from the baby pair (now pregnant)

• Child pair (recovering)

• Parent pair (pregnant)

• Fifth month: 5 pairs

• Both the grandchildren and the parents reproduced

• 3 pairs are pregnant (child and the two new born rabbits)

### Reproducing rabbits

• Sixth month: 8 pairs

• All 3 new rabbit pairs are pregnant, as well as those not pregnant in the last month (2)

• Seventh month: 13 pairs

• All 5 new rabbit pairs are pregnant, as well as those not pregnant in the last month (3)

• Eighth month: 21 pairs

• All 8 new rabbit pairs are pregnant, as well as those not pregnant in the last month (5)

• Ninth month: 34 pairs

• All 13 new rabbit pairs are pregnant, as well as those not pregnant in the last month (8)

• Tenth month: 55 pairs

• All 21 new rabbit pairs are pregnant, as well as those not pregnant in the last month (13)

### Reproducing rabbits

• Note the sequence:

{ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … }

• The Fibonacci sequence again

### Fibonacci sequence

• Another application:

• Fibonacci references from http://en.wikipedia.org/wiki/Fibonacci_sequence

### Fibonacci sequence

• As the terms increase, the ratio between successive terms approaches 1.618

• This is called the “golden ratio”

• Ratio of human leg length to arm length

• Ratio of successive layers in a conch shell

• Reference: http://en.wikipedia.org/wiki/Golden_ratio

## Number counting

### The programming assignment

• This was the looping HW from last fall

• Get an integer i from the user

• The homework had four parts

• Print all the Fibonacci numbers up to i

• Print all the powers of 2 up to i

• Print all the prime numbers up to i

• Time the previous three parts of the code

### Sample execution

Input an integer i: 10

The 10th Fibonacci number is 55

Computation took 1 ms

2 3 5 7 11 13 17 19 23 29

The 10th prime is 29

Computation took 0 ms

The 10th power of 2 is 1024

Computation took 6 ms

2 4 8 16 32 64 128 256 512 1024

BigInteger: The 10th power of 2 is 1024

Computation took 2 ms

### Background: Prime numbers

• Remember that a prime number is a number that is ONLY divisible by itself and 1

• Note that 1 is not a prime number!

• Thus, 2 is the first prime number

• The first 10 prime numbers: 2 3 5 7 11 13 17 19 23 29

• The easiest way to determine prime numbers is with nested loops

### How to time your code

• Is actually pretty easy:

long start = System.currentTimeMillis();

// do the computation

long stop = System.currentTimeMillis();

long timeTakenMS = stop-start;

• This is in milliseconds, so to do the number of actual seconds:

double timeTakenSec = timeTakenMS / 1000.0;

### Program Demo

• NumberGames.java

• Note what happens when you enter 100

• With the Fibonacci numbers

• With the powers of 2

### BigIntegers

• An int can only go up to 2^31 or about 2*109

• A long can only go up to 2^63, or about 9*1018

• What if we want to go higher?

• 2100 = 1267650600228229401496703205376

• To do this, we can use the BigInteger class

• It can represent integers of any size

• This is called “arbitrary precision”

• Not surprisingly, it’s much slower than using ints and longs

• The Fibonacci number part didn’t use BigIntegers

• That’s why we got -980107325 for the 100th term

• It “flowed over” the limit for ints – called “overflow”

### BigInteger usage

• BigIntegers are in the java.math library

• import java.math.*;

• To get nn:

BigInteger bigN = new BigInteger (String.valueOf(n));

BigInteger biggie = new BigInteger (String.valueOf(1));

for ( int i = 0; i < n; i++ )

biggie = biggie.multiply (bigN);

System.out.println (biggie);

### Look at that them there code…

• NumberGames.java