Iteration

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Iteration

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Title: Iteration

1
Iteration

Chapter 4
Spring 2007
CS 101
Aaron Bloomfield

2
Java looping

Options
while
do-while
for
Allow programs to control how many times a
statement list is executed

3
Averaging values
4
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
Dont how big of a list to process
Need ability to repeat code as needed

5
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

6
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 gt 0)
// add value to running total
// processed another value
// prepare next iteration - get next value
// display result
if (valuesProcessed gt 0)

int valuesProcessed 0
double valueSum 0
// set up the input
Scanner stdin new Scanner (System.in)
// prompt user for values
System.out.println("Enter positive numbers 1 per
line.\n"
"Indicate end of the list with a negative
number.")
// get first value
double value stdin.nextDouble()
// process values one-by-one
while (value gt 0)
valueSum value
valuesProcessed
value stdin.nextDouble()

9
Program Demo

NumberAverage.java

10
While syntax and semantics
Expression

Action
while

(

)

11
While semantics for averaging problem
// process values one-by-one while ( value gt 0 )
// add value to running total valueSum
value // we processed another
value valueProcessed // prepare to iterate
get the next input value stdin.nextDouble()

12
While Semantics
Expression
false
true
Action
13
Execution Trace
Suppose input contains 4.5 0.5 1.3 -1
Suppose input contains 4.5 0.5 1.3 -1
Suppose input contains 4.5 0.5 1.3 -1
Suppose input contains 4.5 0.5 1.3 -1
Suppose input contains 4.5 0.5 1.3 -1
0
valuesProcessed
1
2
3
4.5
valueSum
0
5.0
6.3
int valuesProcessed 0 double valueSum
0 double value stdin.nextDouble() while
(value gt 0) valueSum value valuesProces
sed value stdin.nextDouble() if
(valuesProcessed gt 0) double average
valueSum / valuesProcessed System.out.println("A
verage " average) else System.out.println
("No list to average")
int valuesProcessed 0 double valueSum
0 double value stdin.nextDouble() while
(value gt 0) valueSum value valuesProces
sed value stdin.nextDouble() if
(valuesProcessed gt 0) double average
valueSum / valuesProcessed System.out.println("A
verage " average)
value
4.5
0.5
1.3
-1
average
2.1
14
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

15
Converting text to lower case
16
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)

17
Sample run
18
Program Demo

LowerCaseDisplay.java

19
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)
20
Program trace
converted (currentConversion "\n")
21
Another optical illusion
22
Loop Design Reading From a File
23
Loop design

Questions to consider in loop design and analysis
What initialization is necessary for the loops
test expression?
What initialization is necessary for the loops
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?

24
Reading a file

Background

Scanner fileIn new Scanner (new File (filename)
)
25
Reading a file

Class File
Allows access to files (etc.) on a hard drive
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

26
Reading a file

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.o
ut.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
27
Todays demotivators
28
End of lecture on 23 February 2007
29
The For statement
30
The For Statement
currentTerm 1
int
for ( int i 0 i lt 5 i )
System.out.println(currentTerm)
currentTerm 2

31
ForInit
ForExpr
true
false
Action
ForUpdate
32
for statement syntax
for
ForInit

ForExpression

ForUpdate

Action

(

)
33
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!
Well see when they are different in a bit

34
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
35
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
36
Execution Trace
i
0
1
2
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

int i 0
i lt 3
i
for (
)
int i 0
i lt 3
i
Variable i has gone out of scope it is local
to the loop
37
for vs. while

An example when a for loop can be directly
translated into a while loop
int count
for ( count 0 count lt 10 count )
System.out.println (count)
Translates to
int count
count 0
while (count lt 10)
System.out.println (count)
count

38
for vs. while

An example when a for loop CANNOT be directly
translated into a while loop
for ( int count 0 count lt 10 count )
System.out.println (count)
Would (mostly) translate as
int count 0
while (count lt 10)
System.out.println (count)
count

only difference
count is NOT defined here
count IS defined here
39
for loop indexing

Java (and C and C) indexes everything from zero
Thus, a for loop like this
for ( int i 0 i lt 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 lt 10 i ) ...

40
Nested loops

int m 2
int n 3
for (int i 0 i lt n i)
System.out.println("i is " i)
for (int j 0 j lt m j)
System.out.println(" j is " j)

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

41
Nested loops

int m 2
int n 4
for (int i 0 i lt n i)
System.out.println("i is " i)
for (int j 0 j lt i 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

42
How well do you understand for loops?

Very well! This stuff is easy!
Fairly well with a little review, Ill be good
Okay. Its not great, but its not horrible,
either
Not well. Im kinda confused
Not at all. Im soooooo lost

43
From Dubai
44
do-while loops
45
The do-while statement

Syntax
do Action
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

46
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

47
Guessing a number

This program will allow the user to guess the
number the computer has thought of
Main code block
do
System.out.print ("Enter your guess ")
guessedNumber stdin.nextInt()
count
while ( guessedNumber ! theNumber )

48
Program Demo

GuessMyNumber.java

49
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
50
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

51
How well do you understand do-while loops?

Very well! This stuff is easy!
Fairly well with a little review, Ill be good
Okay. Its not great, but its not horrible,
either
Not well. Im kinda confused
Not at all. Im soooooo lost

52
Todays demotivators
53
End of lecture on 26 February 2007
54
Loop controls
55
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 lt 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

56
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 lt 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

57
Four Hobos
58
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 NPRs Sunday Morning Edition puzzle person

59
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?

60
Analysis

We are looking for integer solutions to the
formula
a2b2c2d2 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 doesnt 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

61
Implementation

public class FourHobos
public static void main (String args)
for ( int a 1 a lt 14 a )
for ( int b 1 b lt 14 b )
for ( int c 1 c lt 14 c )
for ( int d 1 d lt 14 d )
if ( (a lt b) (b lt c) (c lt d) )
if ( aabbccdd 200 )
System.out.println ("(" a ", " b
", " c ", " d ")")

62
Program Demo

FourHobos.java

63
Results

The output
(2, 4, 6, 12)
(6, 6, 8, 8)
Not surprisingly, the smart hobo picks the short
straw of the first combination

64
Todays demotivators
65
Alternate implementation

We are going to rewrite the old code in the inner
most for loop
if ( (a lt b) (b lt c) (c lt d) )
if ( aabbccdd 200 )
System.out.println ("(" a ", " b
", " c ", " d ")")
First, consider the negation of
( (a lt b) (b lt c) (c lt d) )
Its ( !(a lt b) !(b lt c) !(c lt d) )
Or ( (a gt b) (b gt c) (c gt d) )

66
Alternate implementation

This is the new code for the inner-most for loop
if ( (a gt b) (b gt c) (c gt d) )
continue
if ( aabbccdd ! 200 )
continue
System.out.println ("(" a ", " b ", "
c ", " d ")")

67
How well do you understand four hobos?

Very well! This stuff is easy!
Fairly well with a little review, Ill be good
Okay. Its not great, but its not horrible,
either
Not well. Im kinda confused
Not at all. Im soooooo lost

68
The 2006 Ig Nobel Prizes

Ornithology
Nutrition
Peace
Acoustics
Mathematics
Literature
Medicine
Physics
Chemistry
Biology

For explaining why woodpeckers dont 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 dont 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
69
3 card poker
70
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

71
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 lt i lt 13 then the card is a club
If 14 lt i lt 26 then the card is a diamond
If 27 lt i lt 39 then the card is a heart
If 40 lt i lt 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.

72
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.

73
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

74
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()
Well 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

75
Hand Methods to Implement

The assignment required the students to implement
the other methods of the Hand class
We havent 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()

76
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

77
Program Demo

HandEvaluation.java

78
How well do you understand 3-card poker?

Very well! This stuff is easy!
Fairly well with a little review, Ill be good
Okay. Its not great, but its not horrible,
either
Not well. Im kinda confused
Not at all. Im soooooo lost

79
All your base are belong to us

Flash animation
Reference http//en.wikipedia.org/wiki/All_your_b
ase_are_belong_to_us

80
End of lecture on 28 Feb 2007

Class was optional on 2 March 2007
Then spring break

81
The Halting Problem
82
Whats 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?

83
(No Transcript)
84
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!
(Im ignoring which way he showed it for now)

85
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.GetNextLotteryNumbe
rs(), 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

86
Take your best guess do you think its possible
to solve the halting problem?

Yes
No
I dont understand what the halting problem is

87
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

88
Halting problem examples will they halt?

First sample program
...println (Alan Turing)
...println (was a genius)
System.exit()
Second sample program
for (int n 0 n lt 10 n)
...println (n)
System.exit()

Third sample program
while (true)
...println (hello world)
System.exit()
Fourth sample program
int x 10
while ( x gt 0 )
...println (hello world)
x x 1
System.exit()

89
Take your best guess do you think its possible
to solve the halting problem?

Yes
No
I dont understand what the halting problem is

90
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 numbers6, 28,
496, 8128, 33550336, 8589869056, 137438691328,
2305843008139952128, 26584559915698317446546926159
53842176, 1915619426082361072947933780843036381309
97321548169216
The last one was 54 digits!
All known perfect numbers are even its an open
(i.e. unsolved) problem if odd perfect numbers
exist
Sequence A000396 in OEIS

91
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 lt 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()
Adapted from http//en.wikipedia.org/wiki/Halting_
problem

92
Take your best guess do you think its possible
to solve the halting problem?

Yes
No
I dont understand what the halting problem is

93
Where does that leave us?

If a human cant figure out how to do the halting
problem, we cant 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?

94
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)
Its using the bytes of program P as the input to
program P

95
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?

96
CheckHalt()s non-existence

A (somewhat condensed) version of class Test
public class Test
main (String args)
if ( Oracle.CheckHalt (Test.java,
Test.java) )
System.exit() else while (true)

Two possibilities
Either class Test halts
Then CheckHalt(Test,Test) returns true (loops
forever)
Which means that class Test loops forever
Contradiction!
Or class Test loops forever
Then CheckHalt(Test,Test) returns false
(halts)
Which means that class Test halts
Contradiction!

97
How well do you understand the halting problem?

Very well! This stuff is easy!
Fairly well with a little review, Ill be good
Okay. Its not great, but its not horrible,
either
Not well. Im kinda confused
Not at all. Im soooooo lost

98
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 its much harder to prove that a program can
never exist
First shown by Alan Turing in 1936
Before digital computers existed!

99
New 2005 demotivatiors!
100
Not going over any more slides in this slide set
101
Triangle counting
102
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

103
Sample execution

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

104
Program Demo

TriangleDemo.java

105
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()

106
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 lt
y lt 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 cant assume that once a property is
present, the others are not.

107
Look at that them there code

TriangleDemo.java

108
How well do you understand triangle counting?

Very well! This stuff is easy!
Fairly well with a little review, Ill be good
Okay. Its not great, but its not horrible,
either
Not well. Im kinda confused
Not at all. Im soooooo lost

109
Fibonacci numbers
110
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

111
Fibonacci sequence in nature
13 8 5 3 2 1
112
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?

113
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)

114
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)