ch05-program-logic-indefinite-loops.ppt

Building Java Programs
Chapter 5
Program Logic and Indefinite Loops
Copyright (c) Pearson 2013.
All rights reserved.

2
A deceptive problem...
• Write a method printNumbers that prints each number from
1 to a given maximum, separated by commas.
For example, the call:
printNumbers(5)
should print:
1, 2, 3, 4, 5

3
Flawed solutions
• public static void printNumbers(int max) {
for (int i = 1; i <= max; i++) {
System.out.print(i + ", ");
}
System.out.println(); // to end the line of output
}
– Output from printNumbers(5): 1, 2, 3, 4, 5,
• public static void printNumbers(int max) {
for (int i = 1; i <= max; i++) {
System.out.print(", " + i);
}
System.out.println(); // to end the line of output
}
– Output from printNumbers(5): , 1, 2, 3, 4, 5

4
Fence post analogy
• We print n numbers but need only n - 1 commas.
• Similar to building a fence with wires separated by posts:
– If we use a flawed algorithm that repeatedly places a post + wire,
the last post will have an extra dangling wire.
for (length of fence) {
place a post.
place some wire.
}

5
Fencepost loop
• Add a statement outside the loop to place the initial "post."
– Also called a fencepost loop or a "loop-and-a-half" solution.
place a post.
for (length of fence - 1) {
place some wire.
place a post.
}

6
Fencepost method solution
public static void printNumbers(int max) {
System.out.print(1);
for (int i = 2; i <= max; i++) {
}
System.out.println(); // to end the line
}
• Alternate solution: Either first or last "post" can be taken out:
public static void printNumbers(int max) {
for (int i = 1; i <= max - 1; i++) {
System.out.print(i + ", ");
}
System.out.println(max); // to end the line
}

7
Fencepost question
• Modify your method printNumbers into a new method
printPrimes that prints all prime numbers up to a max.
– Example: printPrimes(50) prints
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
43, 47
– If the maximum is less than 2, print no output.
• To help you, write a method countFactors which returns the
number of factors of a given integer.
– countFactors(20) returns 6 due to factors 1, 2, 4, 5, 10, 20.

8
Fencepost answer
// Prints all prime numbers up to the given max.
public static void printPrimes(int max) {
if (max >= 2) {
System.out.print("2");
for (int i = 3; i <= max; i++) {
if (countFactors(i) == 2) {
}
}
System.out.println();
}
}
// Returns how many factors the given number has.
public static int countFactors(int number) {
int count = 0;
for (int i = 1; i <= number; i++) {
if (number % i == 0) {
count++; // i is a factor of number
}
}
return count;
}

10
Categories of loops
• definite loop: Executes a known number of times.
– The for loops we have seen are definite loops.
• Print "hello" 10 times.
• Find all the prime numbers up to an integer n.
• Print each odd number between 5 and 127.
• indefinite loop: One where the number of times its body
repeats is not known in advance.
• Prompt the user until they type a non-negative number.
• Print random numbers until a prime number is printed.
• Repeat until the user has types "q" to quit.

11
The while loop
• while loop: Repeatedly executes its
body as long as a logical test is true.
while (test) {
statement(s);
}
• Example:
int num = 1; // initialization
while (num <= 200) { // test
System.out.print(num + " ");
num = num * 2; // update
}
// output: 1 2 4 8 16 32 64 128

12
Example while loop
// finds the first factor of 91, other than 1
int n = 91;
int factor = 2;
while (n % factor != 0) {
factor++;
}
System.out.println("First factor is " + factor);
// output: First factor is 7
– while is better than for because we don't know how many
times we will need to increment to find the factor.

13
• sentinel: A value that signals the end of user input.
– sentinel loop: Repeats until a sentinel value is seen.
• Example: Write a program that prompts the user for numbers
until the user types 0, then outputs their sum.
– (In this case, 0 is the sentinel value.)
Enter a number (0 to quit): 10
The sum is 60
Sentinel values

14
Flawed sentinel solution
• What's wrong with this solution?
Scanner console = new Scanner(System.in);
int sum = 0;
int number = 1; // "dummy value", anything but 0
while (number != 0) {
System.out.print("Enter a number (0 to quit): ");
number = console.nextInt();
sum = sum + number;
}
System.out.println("The total is " + sum);

15
Changing the sentinel value
• Modify your program to use a sentinel value of -1.
– Example log of execution:
Enter a number (-1 to quit): 15
Enter a number (-1 to quit): -1
The total is 80

16
Changing the sentinel value
• To see the problem, change the sentinel's value to -1:
int sum = 0;
int number = 1; // "dummy value", anything but -1
while (number != -1) {
System.out.print("Enter a number (-1 to quit): ");
sum = sum + number;
}
• Now the solution produces the wrong output. Why?
The total was 79

17
The problem with our code
• Our code uses a pattern like this:
sum = 0.
while (input is not the sentinel) {
prompt for input; read input.
add input to the sum.
}
• On the last pass, the sentinel -1 is added to the sum:
prompt for input; read input (-1).
add input (-1) to the sum.
• This is a fencepost problem.
– Must read N numbers, but only sum the first N-1 of them.

18
A fencepost solution
sum = 0.
prompt for input; read input. // place a "post"
while (input is not the sentinel) {
add input to the sum. // place a "wire"
prompt for input; read input. // place a "post"
}
• Sentinel loops often utilize a fencepost "loop-and-a-half" style
solution by pulling some code out of the loop.

19
Correct sentinel code
int sum = 0;
// pull one prompt/read ("post") out of the loop
int number = console.nextInt();
while (number != -1) {
sum = sum + number; // moved to top of loop
}

20
Sentinel as a constant
public static final int SENTINEL = -1;
...
int sum = 0;
// pull one prompt/read ("post") out of the loop
System.out.print("Enter a number (" + SENTINEL +
" to quit): ");
int number = console.nextInt();
while (number != SENTINEL) {
sum = sum + number; // moved to top of loop
System.out.print("Enter a number (" + SENTINEL +
" to quit): ");
}

22
The Random class
• A Random object generates pseudo-random numbers.
– Class Random is found in the java.util package.
import java.util.*;
– Example:
Random rand = new Random();
int randomNumber = rand.nextInt(10); // 0-9
Method name Description
nextInt() returns a random integer
nextInt(max) returns a random integer in the range [0, max)
in other words, 0 to max-1 inclusive
nextDouble() returns a random real number in the range [0.0, 1.0)

23
Generating random numbers
• Common usage: to get a random number from 1 to N
int n = rand.nextInt(20) + 1; // 1-20 inclusive
• To get a number in arbitrary range [min, max] inclusive:
name.nextInt(size of range) + min
• where (size of range) is (max - min + 1)
– Example: A random integer between 4 and 10 inclusive:
int n = rand.nextInt(7) + 4;

24
Random questions
• Given the following declaration, how would you get:
– A random number between 1 and 47 inclusive?
int random1 = rand.nextInt(47) + 1;
– A random number between 23 and 30 inclusive?
int random2 = rand.nextInt(8) + 23;
– A random even number between 4 and 12 inclusive?
int random3 = rand.nextInt(5) * 2 + 4;

25
Random and other types
• nextDouble method returns a double between 0.0 - 1.0
– Example: Get a random GPA value between 1.5 and 4.0:
double randomGpa = rand.nextDouble() * 2.5 + 1.5;
• Any set of possible values can be mapped to integers
– code to randomly play Rock-Paper-Scissors:
int r = rand.nextInt(3);
if (r == 0) {
System.out.println("Rock");
} else if (r == 1) {
System.out.println("Paper");
} else { // r == 2
System.out.println("Scissors");
}

26
Random question
• Write a program that simulates rolling of two 6-sided dice until
their combined result comes up as 7.
2 + 4 = 6
3 + 5 = 8
5 + 6 = 11
1 + 1 = 2
4 + 3 = 7
You won after 5 tries!

27
Random answer
// Rolls two dice until a sum of 7 is reached.
import java.util.*;
public class Dice {
public static void main(String[] args) {
int tries = 0;
int sum = 0;
while (sum != 7) {
// roll the dice once
int roll1 = rand.nextInt(6) + 1;
sum = roll1 + roll2;
System.out.println(roll1 + " + " + roll2 + " = " + sum);
tries++;
}
System.out.println("You won after " + tries + " tries!");
}
}

28
Random question
• Write a program that plays an adding game.
– Ask user to solve random adding problems with 2-5 numbers.
– The user gets 1 point for a correct answer, 0 for incorrect.
– The program stops after 3 incorrect answers.
4 + 10 + 3 + 10 = 27
9 + 2 = 11
8 + 6 + 7 + 9 = 25
Wrong! The answer was 30
5 + 9 = 13
4 + 9 + 9 = 22
3 + 1 + 7 + 2 = 13
4 + 2 + 10 + 9 + 7 = 42
You earned 4 total points.

29
Random answer
// Asks the user to do adding problems and scores them.
import java.util.*;
public class AddingGame {
// play until user gets 3 wrong
int points = 0;
int wrong = 0;
while (wrong < 3) {
int result = play(console, rand); // play one game
if (result > 0) {
points++;
} else {
wrong++;
}
}
System.out.println("You earned " + points + " total points.");
}

30
Random answer 2
...
// Builds one addition problem and presents it to the user.
// Returns 1 point if you get it right, 0 if wrong.
public static int play(Scanner console, Random rand) {
// print the operands being added, and sum them
int operands = rand.nextInt(4) + 2;
int sum = rand.nextInt(10) + 1;
System.out.print(sum);
for (int i = 2; i <= operands; i++) {
int n = rand.nextInt(10) + 1;
sum += n;
System.out.print(" + " + n);
}
System.out.print(" = ");
// read user's guess and report whether it was correct
int guess = console.nextInt();
if (guess == sum) {
return 1;
} else {
System.out.println("Wrong! The answer was " + total);
return 0;
}
}
}

31
The do/while loop
• do/while loop: Performs its test at the end of each repetition.
– Guarantees that the loop's {} body will run at least once.
do {
statement(s);
} while (test);
// Example: prompt until correct password is typed
String phrase;
do {
System.out.print("Type your password: ");
phrase = console.next();
} while (!phrase.equals("abracadabra"));

32
do/while question
• Modify the previous Dice program to use do/while.
2 + 4 = 6
3 + 5 = 8
5 + 6 = 11
1 + 1 = 2
4 + 3 = 7
You won after 5 tries!
• Is do/while a good fit for our past Sentinel program?

33
do/while answer
// Rolls two dice until a sum of 7 is reached.
import java.util.*;
public class Dice {
int tries = 0;
int sum;
do {
int roll1 = rand.nextInt(6) + 1; // one roll
sum = roll1 + roll2;
System.out.println(roll1 + " + " + roll2 + " = " + sum);
tries++;
} while (sum != 7);
System.out.println("You won after " + tries + " tries!");
}
}

35
Methods that are tests
• Some methods return logical values.
– A call to such a method is used as a test in a loop or if.
System.out.print("Type your first name: ");
String name = console.next();
if (name.startsWith("Dr.")) {
System.out.println("Will you marry me?");
} else if (name.endsWith("Esq.")) {
System.out.println("And I am Ted 'Theodore' Logan!");
}

36
String test methods
String name = console.next();
if (name.contains("Prof")) {
System.out.println("When are your office hours?");
} else if (name.equalsIgnoreCase("STUART")) {
System.out.println("Let's talk about meta!");
}
Method Description
equals(str) whether two strings contain the same characters
equalsIgnoreCase(str) whether two strings contain the same characters,
ignoring upper vs. lower case
startsWith(str) whether one contains other's characters at start
endsWith(str) whether one contains other's characters at end
contains(str) whether the given string is found within this one

37
Type boolean
• boolean: A logical type whose values are true and false.
– A logical test is actually a boolean expression.
– It is legal to:
• create a boolean variable
• pass a boolean value as a parameter
• return a boolean value from methods
• call a method that returns a boolean and use it as a test
boolean minor = (age < 21);
boolean isProf = name.contains("Prof");
boolean lovesCSE = true;
// allow only CSE-loving students over 21
if (minor || isProf || !lovesCSE) {
System.out.println("Can't enter the club!");
}

38
Using boolean
• Why is type boolean useful?
– Can capture a complex logical test result and use it later
– Can write a method that does a complex test and returns it
– Makes code more readable
– Can pass around the result of a logical test (as param/return)
boolean goodAge = age >= 12 && age < 29;
boolean goodHeight = height >= 78 && height < 84;
boolean rich = salary >= 100000.0;
if ((goodAge && goodHeight) || rich) {
System.out.println("Okay, let's go out!");
} else {
System.out.println("It's not you, it's me...");
}

39
Returning boolean
public static boolean isPrime(int n) {
int factors = 0;
for (int i = 1; i <= n; i++) {
if (n % i == 0) {
factors++;
}
}
if (factors == 2) {
return true;
} else {
return false;
}
}
• Calls to methods returning boolean can be used as tests:
if (isPrime(57)) {
...
}

40
Boolean question
• Improve our "rhyme" / "alliterate" program to use boolean
methods to test for rhyming and alliteration.
Type two words: Bare blare
They rhyme!
They alliterate!

41
Boolean answer
if (rhyme(word1, word2)) {
System.out.println("They rhyme!");
}
if (alliterate(word1, word2)) {
System.out.println("They alliterate!");
}
...
// Returns true if s1 and s2 end with the same two letters.
public static boolean rhyme(String s1, String s2) {
if (s2.length() >= 2 && s1.endsWith(s2.substring(s2.length() - 2))) {
return true;
} else {
return false;
}
}
// Returns true if s1 and s2 start with the same letter.
public static boolean alliterate(String s1, String s2) {
if (s1.startsWith(s2.substring(0, 1))) {
return true;
} else {
return false;
}
}

42
"Boolean Zen", part 1
• Students new to boolean often test if a result is true:
if (isPrime(57) == true) { // bad
...
}
• But this is unnecessary and redundant. Preferred:
if (isPrime(57)) { // good
...
}
• A similar pattern can be used for a false test:
if (isPrime(57) == false) { // bad
if (!isPrime(57)) { // good

43
"Boolean Zen", part 2
• Methods that return boolean often have an
if/else that returns true or false:
public static boolean bothOdd(int n1, int n2) {
if (n1 % 2 != 0 && n2 % 2 != 0) {
return true;
} else {
return false;
}
}
– But the code above is unnecessarily verbose.

44
Solution w/ boolean var
• We could store the result of the logical test.
boolean test = (n1 % 2 != 0 && n2 % 2 != 0);
if (test) { // test == true
return true;
} else { // test == false
return false;
}
}
– Notice: Whatever test is, we want to return that.
• If test is true , we want to return true.
• If test is false, we want to return false.

45
Solution w/ "Boolean Zen"
• Observation: The if/else is unnecessary.
– The variable test stores a boolean value;
its value is exactly what you want to return. So return that!
boolean test = (n1 % 2 != 0 && n2 % 2 != 0);
return test;
}
• An even shorter version:
– We don't even need the variable test.
We can just perform the test and return its result in one step.
return (n1 % 2 != 0 && n2 % 2 != 0);
}

46
"Boolean Zen" template
• Replace
public static boolean name(parameters) {
if (test) {
return true;
} else {
return false;
}
}
• with
public static boolean name(parameters) {
return test;
}

47
Improved isPrime method
• The following version utilizes Boolean Zen:
public static boolean isPrime(int n) {
int factors = 0;
for (int i = 1; i <= n; i++) {
if (n % i == 0) {
factors++;
}
}
return factors == 2; // if n has 2 factors, true
}
• Modify our Rhyme program to use Boolean Zen.

48
Boolean Zen answer
System.out.print("Type two words: ");
String word1 = console.next().toLowerCase();
String word2 = console.next().toLowerCase();
if (rhyme(word1, word2)) {
System.out.println("They rhyme!");
}
if (alliterate(word1, word2)) {
System.out.println("They alliterate!");
}
}
return s2.length() >= 2 && s1.endsWith(s2.substring(s2.length() - 2));
}
// Returns true if s1 and s2 start with the same letter.
public static boolean alliterate(String s1, String s2) {
return s1.startsWith(s2.substring(0, 1));
}

49
"Short-circuit" evaluation
• Java stops evaluating a test if it knows the answer.
– && stops early if any part of the test is false
– || stops early if any part of the test is true
• The following test will crash if s2's length is less than 2:
return s1.endsWith(s2.substring(s2.length() - 2)) &&
s1.length() >= 2 && s2.length() >= 2;
}
• The following test will not crash; it stops if length < 2:
return s1.length() >= 2 && s2.length() >= 2 &&
s1.endsWith(s2.substring(s2.length() - 2));
}

50
De Morgan's Law
• De Morgan's Law: Rules used to negate boolean tests.
– Useful when you want the opposite of an existing test.
– Example:
Original Expression Negated Expression Alternative
a && b !a || !b !(a && b)
a || b !a && !b !(a || b)
Original Code Negated Code
if (x == 7 && y > 3) {
...
}
if (x != 7 || y <= 3) {
...
}

51
Boolean practice questions
• Write a method named isVowel that returns whether a
String is a vowel (a, e, i, o, or u), case-insensitively.
– isVowel("q") returns false
– isVowel("A") returns true
– isVowel("e") returns true
• Change the above method into an isNonVowel that returns
whether a String is any character except a vowel.
– isNonVowel("q") returns true
– isNonVowel("A") returns false
– isNonVowel("e") returns false

52
Boolean practice answers
// Enlightened version. I have seen the true way (and false way)
public static boolean isVowel(String s) {
return s.equalsIgnoreCase("a") || s.equalsIgnoreCase("e") ||
s.equalsIgnoreCase("i") || s.equalsIgnoreCase("o") ||
s.equalsIgnoreCase("u");
}
// Enlightened "Boolean Zen" version
public static boolean isNonVowel(String s) {
return !s.equalsIgnoreCase("a") && !s.equalsIgnoreCase("e") &&
!s.equalsIgnoreCase("i") && !s.equalsIgnoreCase("o") &&
!s.equalsIgnoreCase("u");
// or, return !isVowel(s);
}

53
When to return?
• Methods with loops and return values can be tricky.
– When and where should the method return its result?
• Write a method seven that accepts a Random parameter and
uses it to draw up to ten lotto numbers from 1-30.
– If any of the numbers is a lucky 7, the method should stop and
return true. If none of the ten are 7 it should return false.
– The method should print each number as it is drawn.
15 29 18 29 11 3 30 17 19 22 (first call)
29 5 29 4 7 (second call)

54
Flawed solution
// Draws 10 lotto numbers; returns true if one is 7.
public static boolean seven(Random rand) {
for (int i = 1; i <= 10; i++) {
int num = rand.nextInt(30) + 1;
if (num == 7) {
return true;
} else {
return false;
}
}
}
– The method always returns immediately after the first roll.
– This is wrong if that roll isn't a 7; we need to keep rolling.

55
Returning at the right time
// Draws 10 lotto numbers; returns true if one is 7.
public static boolean seven(Random rand) {
for (int i = 1; i <= 10; i++) {
int num = rand.nextInt(30) + 1;
if (num == 7) { // found lucky 7; can exit now
return true;
}
}
return false; // if we get here, there was no 7
}
– Returns true immediately if 7 is found.
– If 7 isn't found, the loop continues drawing lotto numbers.
– If all ten aren't 7, the loop ends and we return false.

56
while loop question
• Write a method digitSum that accepts an integer parameter
and returns the sum of its digits.
– Assume that the number is non-negative.
– Example: digitSum(29107) returns 2+9+1+0+7 or 19
– Hint: Use the % operator to extract a digit from a number.

57
while loop answer
public static int digitSum(int n) {
n = Math.abs(n); // handle negatives
int sum = 0;
while (n > 0) {
sum = sum + (n % 10); // add last digit
n = n / 10; // remove last digit
}
return sum;
}

58
Boolean return questions
• hasAnOddDigit : returns true if any digit of an integer is odd.
– hasAnOddDigit(4822116) returns true
– hasAnOddDigit(2448) returns false
• allDigitsOdd : returns true if every digit of an integer is odd.
– allDigitsOdd(135319) returns true
– allDigitsOdd(9174529) returns false
• isAllVowels : returns true if every char in a String is a vowel.
– isAllVowels("eIeIo") returns true
– isAllVowels("oink") returns false
• These problems are available in our Practice-It! system under 5.x.

59
Boolean return answers
public static boolean hasAnOddDigit(int n) {
while (n != 0) {
if (n % 2 != 0) { // check whether last digit is odd
return true;
}
n = n / 10;
}
return false;
}
public static boolean allDigitsOdd(int n) {
while (n != 0) {
if (n % 2 == 0) { // check whether last digit is even
return false;
}
n = n / 10;
}
return true;
}
public static boolean isAllVowels(String s) {
for (int i = 0; i < s.length(); i++) {
String letter = s.substring(i, i + 1);
if (!isVowel(letter)) {
return false;
}
}
return true;
}

61
Logical assertions
• assertion: A statement that is either true or false.
Examples:
– Java was created in 1995.
– The sky is purple.
– 23 is a prime number.
– 10 is greater than 20.
– x divided by 2 equals 7. (depends on the value of x)
• An assertion might be false ("The sky is purple" above), but it
is still an assertion because it is a true/false statement.

62
Reasoning about assertions
• Suppose you have the following code:
if (x > 3) {
// Point A
x--;
} else {
// Point B
x++;
// Point C
}
// Point D
• What do you know about x's value at the three points?
– Is x > 3? Always? Sometimes? Never?

63
Assertions in code
• We can make assertions about our code and ask whether they are
true at various points in the code.
– Valid answers are ALWAYS, NEVER, or SOMETIMES.
System.out.print("Type a nonnegative number: ");
double number = console.nextDouble();
// Point A: is number < 0.0 here?
while (number < 0.0) {
// Point B: is number < 0.0 here?
System.out.print("Negative; try again: ");
number = console.nextDouble();
// Point C: is number < 0.0 here?
}
// Point D: is number < 0.0 here?
(SOMETIMES)
(ALWAYS)
(SOMETIMES)
(NEVER)

64
Reasoning about assertions
• Right after a variable is initialized, its value is known:
int x = 3;
// is x > 0? ALWAYS
• In general you know nothing about parameters' values:
public static void mystery(int a, int b) {
// is a == 10? SOMETIMES
• But inside an if, while, etc., you may know something:
public static void mystery(int a, int b) {
if (a < 0) {
// is a == 10? NEVER
...
}
}

65
Assertions and loops
• At the start of a loop's body, the loop's test must be true:
while (y < 10) {
// is y < 10? ALWAYS
...
}
• After a loop, the loop's test must be false:
while (y < 10) {
...
}
// is y < 10? NEVER
• Inside a loop's body, the loop's test may become false:
while (y < 10) {
y++;
// is y < 10? SOMETIMES
}

66
"Sometimes"
• Things that cause a variable's value to be unknown
(often leads to "sometimes" answers):
– reading from a Scanner
– reading a number from a Random object
– a parameter's initial value to a method
• If you can reach a part of the program both with the answer
being "yes" and the answer being "no", then the correct
answer is "sometimes".
– If you're unsure, "Sometimes" is a good guess.

67
Assertion example 1
public static void mystery(int x, int y) {
int z = 0;
// Point A
while (x >= y) {
// Point B
x = x - y;
z++;
if (x != y) {
// Point C
z = z * 2;
}
// Point D
}
// Point E
System.out.println(z);
}
x < y x == y z == 0
Point A
Point B
Point C
Point D
Point E
SOMETIMES SOMETIMES ALWAYS
NEVER SOMETIMES SOMETIMES
SOMETIMES NEVER NEVER
SOMETIMES SOMETIMES NEVER
ALWAYS NEVER SOMETIMES
Which of the following assertions are
true at which point(s) in the code?
Choose ALWAYS, NEVER, or SOMETIMES.

68
Assertion example 2
public static int mystery(Scanner console) {
int prev = 0;
int count = 0;
int next = console.nextInt();
// Point A
while (next != 0) {
// Point B
if (next == prev) {
// Point C
count++;
}
prev = next;
next = console.nextInt();
// Point D
}
// Point E
return count;
}
next == 0 prev == 0 next == prev
Point A
Point B
Point C
Point D
Point E
SOMETIMES ALWAYS SOMETIMES
NEVER SOMETIMES SOMETIMES
NEVER NEVER ALWAYS
SOMETIMES NEVER SOMETIMES
ALWAYS SOMETIMES SOMETIMES

69
Assertion example 3
// Assumes y >= 0, and returns x^y
public static int pow(int x, int y) {
int prod = 1;
// Point A
while (y > 0) {
// Point B
if (y % 2 == 0) {
// Point C
x = x * x;
y = y / 2;
// Point D
} else {
// Point E
prod = prod * x;
y--;
// Point F
}
}
// Point G
return prod;
}
y > 0 y % 2 == 0
Point A
Point B
Point C
Point D
Point E
Point F
Point G
y > 0 y % 2 == 0
Point A SOMETIMES SOMETIMES
Point B ALWAYS SOMETIMES
Point C ALWAYS ALWAYS
Point D ALWAYS SOMETIMES
Point E ALWAYS NEVER
Point F SOMETIMES ALWAYS
Point G NEVER ALWAYS

ch05-program-logic-indefinite-loops.ppt

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